zotero/storage/ISUEQUVN/.zotero-ft-cache

THE COLLECTED PAPERS OF ALBERT EINSTEIN
Vol ume 2 The Swiss Years: Writ i ngs , 1900-1909
Anna Beck, Trans lator Peter Havas, Consult ant
Princeton Univers ity Press Princeton, New Jersey

Copyri ght c 1989 by t he Hebrew Uni versi ty of Jerusalem Published by Princeton Uni versi ty Press 41 William Street Princeton, New Jersey 08540 In the United Kingdom: Princeton Universi ty Press, Chichester, West Sussex All Ri ghts Reserved Publication of thi s translation has been aided by a grant from the National Science Foundati on Pri nted i n the Uni ted Stat es of America by Pri nceton Academi c Press
ISBN: 0-69 1- 08549-8 10 9 8 7 6 5 4 3

CONTENTS

Publisher' s Foreword

xi

Pr eface

xi i i

List of Text s

1. Concl usi ons Drawn from t he Phenomena of Capi llar i ty (Folgerungen

aus den Capil lar itat serscheinungen) , Annalen der Physik 4 (1901)

1

2. On the Thermodynami c Theory of the Di ff erence in Potentials

between Metals and Ful l y Dissociated Solut ions of Thei r Salts

and on an Electrical Method fo r Investigat ing Molecular Forces

(Ueber die thermodynami sche Theorie der Potent ialdifferenz

zwischen Metallen und vollstandig dissoci irten Losungen ihrer

Salze und tiber eine el ektrische Methode zur Erforschung der

Molecul arkraft e) , Anna len der Phys ik 8 (1902)

12

3. Kinetic Theory of Thermal Equilibriumand of the Second Law of

Thermodynami cs (Kinetische Theorie des Warmegleichgewichtes und

des zwei ten Hauptsatzes der Thermodynam ik ) , Annal,n der Physik

9 (1902)

30

4 . A Theor y of the Foundat ions of Thermodynamics (Eim• Theorie der

Grundlagen der Thermo<lynami k) , Annal en der Physik 11 (1903)

48

5. On t he General Molecular Theory of Heat (Zur allgemeincn

molekularen Theorie dPr Warme) , Anna len der Ph ysik 14 (1904 )

GS

6. Review of Giuseppe Belluzzo, "Principles of Graphic ThPrmo-

dynamics" ("Principi di t ermod inamica grafica") . De i bla t ter

zu den Anna le n der Physik 29 (1905)

78

7. Rcyiew of Albert Fliegner, "On Clausius' s Law of Entropy"

("Uber den Claus ius'schen Entrop i esatz") . Bei bliitter zu den

Annalen der Phys ik 29 (1905)

79

8. Review of Will iam McFadden Orr, "On Clausius ' Theorem fo r

Irr evers ible Cycles, and on the Incr ease of Entropy, "

lleiblatt er zu den Annalen der Phys ik 29 (1905)

79

9. Review of George Hartl ey Bryan, "The Law of Degradation of

Energy as t he Fundamental Pr incipl e of Thermodynamics, "

Deiblatter zu den Annalen de r Ph ys ik 29 (1905)

80

10. Revi ew of Ni kolay Nikolayevich Schiller, "Some Concerns Regardi ng

t he Theory of Ent ropy Increase Due to t he Diffus ion of Gases

Where t he In itial Pressures of t he Latter Are Equal" (" Einige

Bedenken betreffend di e Theorie der Entropievermehrung <lur ch

Di ff us ion der Gase bei einander gleichen Anfangsspannungen der

letzteren"). Dei bla t t er zu den Anna len der Pll ysik 29 (1905)

81

vi

CONTENTS

11. Rev iew of Jakob Johann Weyr~uch . "On t he speci f ic Heats of

Superheated Water Vapor" ("l;bcr di e spezif ischen Warmen des

tiberhitzten Wasserdampfes") , Beiblat te r zu den Annal en der

Physik 29 (1905 )

82

12 . Review of Jacobus llenricus van' t Hoff, 11The Influence of t he

~hange i n Specif ic Heat on t he Work of Conversion11 ("Einfl uss der

Anderung der spez ifischcn Warme auf die Umwand l ungsarbeit ") ,

Dei blatt er zu den Annal en de r Ph ysik 29 (1905)

82

13. Review of Arturo Giammarco , "A Case of Correspond ing States in

Thermodynamics" ("Un caso di corrispondenza in termodinam ica") ,

Deibl atter zu den Annalen der Ph ysik 29 (1905 )

81

14. On a Heuristic Po int __of View Concerning the Product ion and Trans-

format ion of Light (Uber einen die Erzeugung und Verwandl ung

des Li chtes betreffenden heuristi schen Gesichtspunkt) , Annalen

der Ph ys ik 17 (1905)

86

15. A New Det erminat i on of Mo lec ular Dimensions (University of Zurich

disseration) (Eine neue Des t immung der Mo lekul dimensionen)

104

16 . On t he \lovcmcnt of Small Particles Suspended in St at ionary_

Liquids Required by the Molecul ar-Kinetic Theory of Heat (Uber

di e von der mol ekularkinet ischen Theorie der Warne gefordert e

Dewegung von in ruhendcn Fliissigkeitcn suspendiert en Teilchen ) ,

Anna le n der fhy sik 17 (1905)

123

17. Revi ew of Karl Frcdrik Slotte, "On t he Heat of Fusion" (Uber die

Schme"lzwarme") , Rei blat ter zu den Annal en de r Physik 29 (1905)

135

18. Rev iew of Karl Fredr ik Slotte, "Conclusions Drawn from a Thermo-

dynamic Equation" ("Folgerungen ans ei ner thermodynami schen

Gleichung") , Bei bla tter zu den Annalen der Ph ysik 29 (1905)

135

19. Revi ew of Emile Math ias , 11The Constant a of Rectilinear Diamet ers

and the Laws of Cor responding States" ("La constante a des

diametres rectil ignes et l es lois des etats correspondants"),

De i blat ter zu den Annal en der Ph ysik 29 (1905)

136

20 . Rev iew of Max Planck, "On Claus ius' Theorem fo r I rreversible

Cycles , and on the Increase of Entropy, 11 Bei blii. l te r zu den

Annal en de r fh ysik 29 (1905)

137

21. Review of Edgar Buckingham , "On Certain Diff icul ties Wh ich Are

Encountered in the St udy of Thermodynamics ," Bei bliitt er zu den

Annalen der Physik 29 (1905)

137

22. Revi ew of Paul Langevin, "On a Fundamental Formula of the Kinet ic

Theory" ("Sur une formu l e fondament ale de la theorie cinetique11 ) ,

De i blatte r zu den Annalen der Physik 29 (1905)

138

CONTENTS

vii

23. On the Electrodynamics of Movinf B~dies (Zur Elekt rodynamik

bewegter Korper), Annalen der P yszk 17 (1905)

140

24 . Does the Inertia of a Body Depend upon its Energy Content? (1 st

di e Tragheit eines Korpers von seinem Energiei nhal t abhangig?),

Annalen der Physik 18 (1905 )

172

25. Review of Heinrich Birven, Fundamentals of the Mechan ica l Th eory

of Heat (Crundzuge der mechanischcn Varmetheo rie ) , Beib latter zu

den Annalen der Physik 29 (1905)

175

26. Rev i ew of Auguste Ponsot, "Heat in the Displacement of the Equ i -

l i br ium of a Capillary System" ("Cbaleur dans le deplacement de

l ' equilibre d' un systeme cap il lai re" ) , Dei blatter zu den Annalen

der Physik 29 (1905 )

175

27. Review of Karl Bohlin, "On Impact Considered as the Bas is of

Kinetic Theories of Gas Pressure and of Universal Gravitation"

("Sur le choc, cons idere comme fondement des t heories cinetiques

de la press ion des gaz et de la gravitation universelle") ,

Beiblatter zu den Anna len der Physik 29 (1905 )

176

28 . Review of Georfes Mesl in, "On the Constant in Mariot te and Gay-

Lussac's Law" "Sur la constante de la l oi de Mar iot te et Gay-

Lussac"), Be i blatter zu den Annal en der Physik 29 (1905 )

177

29 . Review of Albert Fliegner, "The Efflux of Hot Water fromContainer

Or ifices ("Das Ausstromen heissen Wassers aus Gefassmiindungen") ,

De i blatt er zu den Anna l en der Physik 29 (1905 )

177

30. Review of Jakob Johann Weyrauch, "An Outl i ne of th e Th eory of

Heat. Vzth Numerous Examples and Applicat ions. Part 1

(Crundriss der farmeth eorie. llit zahlre i chen Be ispi elen und

Anwendungen ), Be i blatt er zu den Anna len der Phys ik 29 (1905)

178

31. Rev iew of Alb~rt Fliegner, "On the Thermal Val ue of Chemical

Processes" ("Uber den li/armewert chemischer Vorgange") ,

Beiblatter zu den Annalen der Physik 29 (1905)

179

32 . On the Theory of Brownian Mot ion (Zur Theorie der Brownschen

Bewegung), Annal en der Ph ysik 19 (1906)

180

33 . "Supplement" to "A New Determinat ion of Molecul ar Dimens ions"

1the version of Document 15 published as an article in Annal en er Ph ysik 19 (1906) : 289 -305] , Annalen der Ph ysik 19 (1906 )

191

34. On the Theory of Light Product ion and Light Absorption (Zur

Theorie der Lichterzeugung und Lichtabsorpt ion ) , Anna len der

Physik 20 (1 906 )

192

vi i i

CONTENTS

35. The Principle of Conservat ion of Mot ion of the Center of

Grav i ty and t he Iner t ia of Energy (Das Pr inzip von der .Erhaltung

der Schwerpunkt sbewegung und di e Tragheit der Energi e) ,

Annal en der Physik 20 (1906)

200

36. On a ~lct hod for the Det erminat ion of t he Ratio of the Transverse

and t he Longitud inal Uass of the Elect ron (Ober eine Methode zur

Best immung des Verhaltnisses der t ransversalen und longitud inalen

Masse des El ekt rons) , Annalen der Physik 21 (1906 )

207

37 . Revi ew of Max Planck, l ect ures on th e Th eory of Th erma l Radi at io n

( Vorlesunge n ii ber di e Thcorie der fliirmes trahlung ) , De ihlatt er zu

den Annale n de r Phys ik 30 (1906)

211

38. Planck's Theory of Radiation and t he Theory of Specifi c Heat

(Die Plancksche Theorie der Strahl ung und die Theorie <lcr

spczifi schen WarmP). An11a len der Physik 22 (1 907)

214

39. On the Lim it of Validi t y of t he Law of Thermodynami c Equ i li brium

and on the Poss ibility of a New Det ermination of t he El ementary

Quanta (Uber die Giilt igke itsgr enze de:, Satzes vom thermodynami sclten

Gleichgewicht und iibPr die Moglichkeit einer neuen Bcst immung

der Elementarquanta) , Anna l en der Physik 22 (1907)

225

40 . Theorct ical Remar ks on Brownian \lot ion (Theoret ische Bemerkung

tibcr die Brownsclte Bcwegung) , Zci tscllri ft f iir Elek t rochemi e und

angewandte physikalische Chemic 13 (1907)

229

41. On the Possi bil i t y of a New Test of th e Rf'lat ivity Pr inc iple (Uber

die Mogl ichkcit ei ner ncuen Pr iifung des Rclativitat sprinzips) ,

Anna len der Physik 23 (1907)

232

42 . Correct ion to My Paper: "Planck's Theory of Radiation, etc."

(Document 381 (Berichtigun* zu meiner Arbei t: "Die Plancksche

'lheoric dcr Strahlung etc. ') , Annalen der Plt ysik 22 (1907)

233

43. Author's abstract of lecture "On t he Nat ure of the Movements of

Mi croscopically Smal l Part icl es Suspended in Liqu ids" ("Ueber

di e Natur der Bewegungen mikroskopisch klei ner, in Flilss igkeiten

suspendierter Teil chen") . Naturfo rs ch ende Gcse l lschaft Dern .

J!i tt ei l ungen (1 907)

235

4'1 . Comments on t he Note of )Ir. Paul Ehrenfest: "The Trans latory

Mot ion of Deformable Electrons and t hP Area Law" (flemcrkungen

zu der Notiz von llrn. Paul Ehrenfest : "Die Translat ion

deformierbarer Elektroncn und der FlAchcnsat z") . Anna le n der

l'liysik 23 (1907)

236

45. O!! t hP Inertia of Energy Required by t he Rel ativity Princi ple

(Uber die vom Relativ i tatspr inzi p gcfordPrte Tragheit der

Energ ie) , Annalen der P/i. ysik 23 (1907)

238

CONTENTS

ix

46 . Review of Jakob Johann Weyrauch, An Outline of the Theory of

Heat . Vi th Numerous Exampl es and Appl ications. Part 2.

(Crundriss der i'iirmetheori e. Jfi t zahlre i chen Beispi el en trnd

Anwendungen), Deibliitter zu den Annal en der Physik 31 (1 907 )

25 1

47 . On the Relativity Priniciple and the Conclusions Drawn from It (Uber das Relativitatsprinzip und die aus demselben gezogene Folgerungen), Jahrbu ch de r Radi oakt ivi tiit und El ektron i k 4 (1907) 252

48. A New Electrostatic Method for the Measurement of Small

Quant ities of Electric ity (Eine neue elektrostat ische Methode

zur Messung kleiner El ektrizi tatsmengen), Ph ysikalische

Zei tschrift 9 (1908)

312

49. Corrections to the Paper "On t he Rel ativ i ty Princip le and..the

Conclusions Drawn from It" (Beri cht igung zu der Arbeit : "Uber

das Relativi tatspr i nzi p und die aus demselben gezogenen

Folgerungen") , Jah rbuch der Radi oakt ivi tiit und El ektron ik

5 (1908)

316

50. Elementary Theory of Brownian Mot ion (Elementare Theorie der

Brownschen Bewegung) , Zei tschri ft fur Elektro chemie und

angewandte physikalis che Chemie 14 (1908)

318

51 . On the Fundamental Electromagnet i c Equat ions for Movi ng Bodies

(Uber di e elektromagnetischen Grundgl eichungen fu r bewegte Korper )

(with Jakob Laub) , Annal en der Physik 26 (1908)

329

52 . On t he Ponderomotive Forces Exerted on Bodies at Rest in the

Elect romagnet i c Fiel d (Uber die im el ektromagnetischen Felde

auf ruhende Ko rper ausgeubten ponderomotorischen Krafte)

(with Jakob Laub) , Anna len der Physik 26 (1908)

339

53 . Correction to t he Paper: "On the Fundamental Electromagnet ic

Equat ions for_ Moving Bod ies" [Document 51] (Bericht igung zur

Abhandlung : "Uber die elektromagnetischen Grundgl eichungen

flir bewegte Korper 11 )

(with Jakob Laub ) , Annal en der Phys ik 26 (1908)

349

54 . Remarks on Our Paper: "On t he Fundamental Electromagnetic

Equations for Mov ing Bodi es" (Do~ument 51]

(Bemerkungen zu unserer Arbe it : "Uber die elektromagnetischen

Grundgle ichungeu f ur bewegte Korper ")[contains also an Addendum]

(with Jakob Laub ) , Annalen der Physik 28 (1909)

350

55. Comment on t he Paper of D. \lir imanoff : "On t he Fundamental Equations . . . " (Bemerkung zu der Arbeit von D. Mi rimanoff : "Uber die Grundgleichungen . . . " ) , Annal en der Ph ysik 28 (1909) 353

56 . On t he Present St atus of t he Radiat ion Problem (Zurn gegenwartigPn Stand des St rahl ungsproblems) , Pliysikali sch e Zei ts chrift 10 (1909 ) 357

X

CONTENTS

57. On the Present Status of the Rad iation Problem

(Zurn gegenwart igen Stand des Strahlungsproblems)

(with Wal ter Ritz) , Ph ys ikali sche Zeitschrift 10 (1909)

376

58. Excerpt from "Discuss ion" following lecture version of Henry

Siedent opf, "On Ul tramiscroscopic Images ," Ph ysikalis cli e

Zei tschri f t 10 (1909)

377

59. Excerpt from "Discussion" following lecture version of Art hur

Szarvass i. "The Theory of El ectromagnetic Phenomena in Movi ng

Bodies and t he Energy Pri nc iple, 11 Phys ikali sche

Zei ts chrift 10 (1 909)

378

60. On t he Development of Our Views Concerning the Nature and

Const itution of Radiat ion (Uber die Ent wickelung unserer

Anschauungen ilber das Wesen und di e Konst itution der Strahlung),

Deutsche Physika lis che Ces ell schaft, Ye rha ndlungen 7 (1909)

379

61 . 11Di scussion11 following lecture vers ion of "On t he Development

of Our Views Concern ing t he Nat ure and Constitut ion of Radiation"

[Document 60] , Physika l isclie Zei tschrif l 10 (1909 )

395

62. "D iscussion" follow ing lect ure ver sion of Fr itz Hasenohrl,

"On the Transformat ion of Kineti c Energy i nto Radiation, "

Ph ysikalische Ze i tschri ft 10 (1909)

399

PUBLISHER'S FQREN)RD
We are pleased to be publishing this second translation volurre of THE
r o ~ PAPERS OF ALBERT EINSTEIN. As with Volurre 1, we strongly urge
readers to use the translat i ons only together with the documentary editi on, \otbi ch provi des the editorial cx:mrentary necessary for a rcore cc:rrplete understanding of the docuirents. Every effort has been made to insure the sci entific accuracy of this transl ation. It i s not intended as a literary translation that can stand al one withcut the docmrentary edition.
We are again grateful to Dr. Anna Beck and Professor Peter Havas for their hard w::>rk and dedication to this project, \otbich i s separate fran the docurrentary edition project. All translati ons appearing in the docuirentary edition were prepared by the editors of that vol urre, and those appearing in this volurre were prepared by Ors. Beck and Havas.
We are also pleased to acknc:Mledge the grant fran the National Science Fcundation that has made thi s publication possible. In particular, we thank Dr. Ronald OVennan of the NSF for his continued interest in the project.
Princeton University Press Septerrber 1989

PREFACE
This volume contains the translat i ons of all documents in Volume 2 of The Collected Papers of Albert Ei ns tein, all of which were originally written in German. It is not sel f-contained and should be read in conjunct ion with t he documentary edit ion and its editorial apparatus. All editorial headnotes and footnotes have been omitted, as have the introductory mat erials and the bibliography. However, we used the bibl iography to check the references cited in the documents and, especially, to correct and compl ete the titl es and bibliographic data given by Einst ein in his reviews of books and articles . In thi s volume we have included the editorial footnote numbers, which appear in brackets in the margin and correspond to the footnotes in the documentary edition. We have not corrected any misprints or ot her errors (including those in the formulas ) if the editors have commented on them. Misspellings of names of per sons have been routinely corrected.
Although some of the documents have been translated before, we have provided new translations here rather than attempt to use any "best" existing translat ion.
The purpose of t he translat ion project, in accordance with the agreement bet ween Princeton Uni versity Press and the National Science Foundation, is t o provide " a careful, accurate translation that is as close to the German or iginal as possible while st ill producing readable English. " This is, therefore, not a "literary" translation but should allow readers who are not fl uent in German to make a schol arly evaluat ion of the content of the documents while also obtaining an appreciation of their flavor.
Many technical expressions used in the original documents are out dated (see the editorial comments in Volume 2); whenever possible, we have not rep l aced them with the modern Engl ish versions but have used the expressions employed in tile technical literature of the t i me, if known, or else we 1>rovided a literal translation. In parti cular, we ret ained the term "electri c

xiv

PREFACE

mass" frequent ly used by Ei nstein for elect ric charge. All formulas were i ncluded in a form as similar to those in the original documents as was possibl e wi th our word processor . We kept the standard German notation used at t he t ime, representing vectors by German (Fractur) l etters and vector products by [ ];for example, we kept [<EiJ] for t he vector denoted in
curr ent l i terat ure by EK H or EK Il.
We are indebted t o John Stachel, the Editor of Vol ume 2, and Robert Schul mann, Associate Edi tor, as well as Walter Lipp incott, Di r ector, and Alice Calapr ice, Senior Ed i tor, of Princet on University Press, for their help and encouragement. We also wish t o t hank Marj orie Zabierek for her part in preparing t he f i nal typescript.
Anna Beck, Translator Peter Havas , Consultant

TEXTS

Doc. 1 CONCLUSIONS DRAWN FROM THE PHENOMENA OF CAPILLARITY
by Al bert Einstein [Annalen der Phys ik 4 (1901) : 513-523]

If we denot e by 1 the amount of mechanical work that we have to supply

t o a l iqu id i n order to i ncrease the free surface by one unit , then 1 is not

the total energy increase of the system, as the fo l lowi ng cyclic process wi l l

show . Let there be a certain amount of liquid of (absol ute) temperature T1
and surface area 01. We now increase isothermal l y the surface o1 to 02,

increase (at constant surface area} surface to 01 and cool the liquid

t he to

t emperature
r1 again.

to If

T2, t hen reduce one assumes t hat

the no

heat i s suppl i ed to the body other t han t hat received on account of its speci-

fic heat, t hen the total heat supplied to t he substance during the cycl ic

process wil l be equal to t he total heat withdrawn. According to the pri ncipl e

of conservation of energy, t he t otal mechan ical work supplied must then also

be zero .

Hence the following equation holds :

or

[ 1]

However, this contradicts experi ence .

[2]

We have, then, no other choice but to assume that t he change in t he sur-

face is associated with an exchange of heat as well, and that t he surface has

a specif ic heat of its own . If we denote by U the energy, by S the en -

tropy of the unit surface of the liqu id, by s t he specific heat of t he

surface , and by w0 the heat necessary t o form a uni t surface, expressed i n mechanical units , then the quant ities

dU = s.0.dT+ {1 + w0 }d0 and
dS = ~ + ,- d0

will be t otal di fferentials. Hence we wi ll have

2

PHENOMENA OP CAPILLARITY

-b-(;s.mO-) -_ 8{1j~o) ,

From these equat ions i t fo l lows that

This i s , however, the total energy necessary t o form a unit surface . Further, we fo rm
[3]
The experimental studi es have shown t hat 1 can be represented with (41 very good approximation as a linear funct ion of temperature , i. e. :
The energy necessary t o forma uni t surface of a l iquid is independent of t he temperature.
It also fo l lows t hat

(5 ) hence : no heat content should be ascr ibed to the surface as such; rather, the energy of t he surface i s of potential natur e. It can be seen already that the qua n t i t y
i s more suited for sto ichiometric invest igations t han is the hitherto used r
at boi ling t mperature . The fact t hat the energy requi red for the format ion of a un it surface barel y varies with the t emperature t eaches us also that the conf igurat ion of mol~cules in the surface layer will not vary with t emperature (apart from changes of t he order of magn it ude of t hermal expansion ).
To f i nd a sto ichiometri c relationship for the quant ity

DOC. 1

3

I proceeded from the simpl est assumpt ions about the nature of molecular

at traction forces and examined their consequences regarding their agreement

with experiment . In this I was guided by the analogy with gravitational

[6]

forces.

Let thus the relative potential of two mol ecules be of the form

where c i s a constant characteristic of the molecul e in question , and ip( r )

i s a f unction of t hei r distance t hat does not depend on t he nat ure of t he

molecules. We assume f urther that

[71

n n

l l ½

cacp ip(ra, p)

o=l P=l

is t he corresponding express ion for n molecu l es. In the special case in which al l mol ecules are alike, this expression becomes

n n

l l ½c2

ip(ro,p) .

a=l /J= l

We furt her make the add it ional assumption t hat the potential of the molecular forces has the same magnitude it would have if the matter were homogeneously distributed in space; this is, however, an assumption which we shoul d expect to be onl y approximately correct. Us i ng it , the above expression converts to

[8]

where N is the number of molecul es per unit volume . If the molecule of our

liquid cons ists of several atoms , then i t shall be poss i ble to put, in analogy

with gravitational forces ,

c

=

Ec
0

,

where

the

c 's denote t he values
0

characteristic for the atoms of the elements. If one also puts 1/ N = v,

where v denotes t he molecular volume, one obtains the fi nal formula

4

PHENOMENA OF CAPILLARITY

II p

= pw

-

½

~ (Ee )2 V

dT. dr' cp( rdT, dT i) .

If we now also assume that the density of the l iquid i s constant up to

i t s surface, wh ich i s made plaus ible by the fact t hat the energy of the

surface is i ndependent of temperature, then we are able to calculate the

pot ential energy per unit volume in the interior of t he liqu id, and that per

uni t surface .

I. e. , if we put

f~.., C..., ½ J;...,

dxdydz .,pUx2+y2+z2] a K,

then the potential energy per uni t volume is

(Ee )2

p = p - K~

oo

v ,t.

If we imagine a l iquid of volume r and surface S, we obtain by
int egrat ion

(9 1

where t he const ant K' denotes

x'=l y' =l z ' =O x=w y=w z=w

I I I I I I [10]

dx . dy . dz. dx'. dy' .dz'

x'=O y'=O z'=-w x=-w y=-w z=O

P] cp[J (x-x 1 )2+( y- y1 )2+(z-z 1

Since not hing is known about cp , we naturally do not get any rel at ionship between K and K'.
One shoul d keep i n mind , t o begin wi t h, that we cannot know whether or not t he molecule of t he liquid contains the n-fold mass of the gas molecule, but it foll ows fromour deri vat ion that thi s does not change our expression for t he potent ial energy of the liquid. Based on the assumpt ions we have j ust made, we obtain the following expression for t he potent ial energy of t he surface :

DOC . 1

5

(Ee )2

or

*. . P = K'----:;fJ-=1- T*,

v Ec0 = J1 - T

~

[ 11]

Since the quantity on the right can be calculated fromR. Schiff's

observat ions for many substances at the boiling temperature, we have ample

mater ial for the det ermination of the quantit ies c0 . I t ook all the dat a
from W. Ostwald ' s book on general chemistry. First, I present here the data [ 12)

that I used for the calculat ion of c for C, H, and O by the least

0

squares method .

The column with the heading

Ec
0

(calc )

gives the

Ec
0

as

determined from chemical formulas using the c t hus obtained . I someric

0

compounds were combined into one value, because their values on the left- hand

side did not differ significantly from each other . The unit was chosen

arbitrarily because it i s not poss ibl e to determine the absolute val ue of c 0
since K' is unknown .

I found:

CO = 46,8.

Formula
8bo~6 C2fl402 Ca11602 C411s0ij Csll10 2 C41160ij
0 C5H1~ 4
Calls C9H10 2 C6H1~03 C1Hs CaH1oO
0 CaH1~02
C5H4 CsH\o C10I 140

Ec

Ec (calc) Name of the compound

0

0

510

524

Li monene

140

145

Formic acid

193

197

Acetic ac id

250

249

Propanoic acid

309

301

But yric acid and i sobutyric ar id

365

352

Val erianic (pentanoic) acid

350

350

Acetic anhydride

505

501

Ethyl oxalat e

494

520

Methyl benzoate

553

562

Ethyl benzoate

471

454

Ethyl -acetoacetate (diacetic ether )

422

419

Ani sole

479

470

Phenetole and methyl cresolate

519

517

Dimethyl resorcinol

345

362

Furfural

348

305

Val eraldehyde

587

574

d-carvone

It can be seen that in almost all cases the deviations barely exceed the experiment al errors and do not show any trend.

6

PHENO\lENA OF CAPILLARITY

After t hat I separately calculated t he val ues for Cl , Dr, and J; t hese determinations are of course l ess r eliable . I found:
cCl = 60 , cllr = 152, CJ= 198.

I pr esent t he data i n t he same way as above:

Formula

Ec

Eca (calc)

Name of the compound

0

CC7G1I1I57CCll

385

379

Chloro benzene

438

434

Chloro toluene

CC73HH75C0Cl l

450 270

434 270

Benzyl chloride Epichlorohydr in

C2011Cl3

358

335

Chloral

C711 50C l

462

484

Benzoyl chloride

C1HGCl 2

492

495

Benzylidene chloride

BC2rH250r

217

304

Bromine

251

254

Ethyl bromide

C3117Dr

311

306

Propy l bromide

Cil7llr

311

306

Isopropyl bromide

C3115Br

302

309

Allyl bromide

Cil50r

353

354

Isobutyl bromide

C5111liDr

425

410

Isoamyl bromide

CGlls r

411

474

Bromo benzene

C71170r

421

526

o-Bromo tol uene

CC321111

4Br 6Br

2 2

345 395

409 461

Ethy l ene bromide Propylene bromide

C2Il5J

288

300

Ethyl i od ide

C3Il7J

343

352

Propyl iodide

C3ll1J

357

352

I sopropyl iodide

C3ll5J

338

355

Ally1 iodide

C4ll9J

428

403

I sobutyl iodide

C5lluJ

464.

455

Tsoamyl iod ide

It seems t o me t hat t he l arger devi ations from our theory occur for

those compounds that have rel atively l arge mol ecul ar masses and small

molecular vol umes.

Based on our assumptions , we found that the expression for the potential

energy per unit volume is

(Ee ) 2

p = p - K~

oo

V" '

wher e K denot es a defin ite quant i t y, which we , however, are not able t o calculate because i t is only def ined completel y by the choi ce of t he c 's.
0

DOC. 1

7

We can therefore set K = 1 and thereby obt ain a definition for t he absolute

values of the c 's . If we take th is into account from now on, we obtain t he 0
follow ing express ion for the magnitude of the potential pertaining to one

equivalent (molecule ):

(Ee )2

p = p - K- -o- '

oo

V

where, of course, Pm denotes another constant. We could now equate the

second member of the right-hand side of thi s equat ion to t he difference

DmJ - Avd, where Dm i s the molecul ar heat of evaporat ion (heat of evaporation x molecular mass ) , J the mechanical equival ent of one calorie,

A the atmospheric pressure in absolute units , anrl vd the molecular volume

of the vapor - if the potent ial energy of t he vapor were zero and if at t he (13)

boiling point the content in kinetic energy woul d not change during the

transit ion fromthe liquid to the gaseous state . The f i rst of these

assumptions seems to me absolutely safe. However, since we have nei ther a

basis for the second assumption nor a possibility to esti mate the quantity in

quest i on, we have no other choice but to use the above quant i ty itself for the

calculat ion.

In the first column of the following table I entered the quantiti es

[14)

~ in thermal units, with D~ denot ing the heat of evaporation minus the
external work of evaporation (in thermal units) . In the second column I
entered the quantit i es Ec , as obtained from cap i l l arity experiments; the 0
th i rd column contains the quot i ents of the two values. I someri c compounds are
once agai n combined into a singl e l ine.

8

PHENmlENA OF CAPILLARITY

Name of t he compound

Form ul a

~

Ec (calc) 0

Quot ient

Isobutyl propanoate Isoarny l acetate Propyl acetate Isobut yl isobutyrate Propyl val Prat e Isobutyl butyrate Isoamyl propanoatc Isoamyl isobutyrate Isobuty] valerat e Isoamyl valerate Benzene Toluene Et hyl benzene m- Xyl ene Propyl benzene Mes i t ylene Cymene Et hy l formate Methyl acet ate Ethy l acetate Methy1 propanoatc Propyl fo rmate Methyl i sobutyrat e Isobutyl formate Ethyl propanoate Propy l acetat e Met hyl butyrate Ethyl isobutyrate Met hyl valerate l sobut yl acetat e Ethyl butyrate Propyl propanoate Isoamyl formate

C1H1402 II II
C8H1602 II II II
C9H1s02 II
C1~H1002 c6 6 C7H8 CsH10
II
C9H12 II
C10Hlf C3H6 2
II
C4Hs02 II II
C5H1002 II II II II
C6H1202 II II II II II

1157

456

2.54

1257

510

2. 47

1367

559

2.45

1464

611

2 . 51

795

310

2. 57

902

372

2.48

1005

424

2.37

1122

475

2.36

1213

527

2.30

719

249

2.89

837

301

2 . 78

882

353

2 . 50

971

405

2.40

Even though t he quot ient in the fifth column is by no mean s a const ant , but is , on the contrary, clear l y dependent on t he constitution of the
compounds , we can nevertheless use t he mat erial on hand to obtai n the fact or,
or at least its order of magni tude, with which we must multiply our c 's t o 0
obtai n them in th P abso lute unit we had chosen . The mean val ue of t he mul tiplier looked for is

(1 5)

2. 51 X ~4 . 17 X 10 7 = 1.62 X 104 .

Si nce t he fo regoing discuss ion shows that the kinetic condi t ions of t he molecules change during evaporat i on (at l east if our expression for t he

DOC. 1

9

potential energy is correct ) , I decided to obtain the absolute quantity c 0
in one more way. I proceeded from the follow i ng idea :

If we compress a liquid i sothermally and i ts heat cont ent does not

change in the process. as we now wish to assume, then the heat re leased during ( 16]

compression equals the sum of the work of compression and the work done by the

molecular force s . We can t herefore calculate the latter work if we can find

the amount of heat released dur ing compression . This we can do wi t h t he hel p

of Carnot ' s pri nciple.

Let the state of the liquid be determined by the pressure p in

absolute units and by the absol ute temperature T; if the value of the heat

suppl ied to the substance during an infinitesimal ly smal l change of state is

dQ in absolute units. and the mechanical work done on the substance is dA,

and if we put

~ * dQ = Xdp + S.dT,
dA = - p. dv = - p { dp + dTJ

= p.v.Kdp - p. v.odT,

[17)

then the condition that dQ/ T and dQ + dA must be total differentials

yiel ds the equations

J[~ = ~[~

and

J <X + pK ) = -/t<S - po) ;

( 18 )

here, as can be seen, X denotes the heat, in mechanical units. suppl ied to the substance during i sothermal compression produced by pressure p = 1, S is the specific heat at constant pressure, K is the coefficient of compress ibility, and a is the coeffic ient of thermal expansion. From these equations, we find

[ 19)

One has to remember that for any phenomena involving compression of liquids, the atmospheric pressure, to which our bodies are usually subj ect ed,

10

PHENOME~A OF CAPILLARITY

can be considered unhesitatingly as infinitesimall y smal l i likewise , compres-

sions in our exper iments are very nearly proportional t o the compress ion

for ces appl i ed . Thus , the phenomena proceed as if the compression forces were

infinitesimal ly small. If t hi s is t aken into account, then our equation

reduces to
(20)

X.dp = - T.o .dp.

If we now apply the assumption that the kinetic energy of the system does not change in i sothermal compress i on , we obtain t he equat ion

X.dp + work of compress ion+ work of the molecular forces= 0.

If P is t he pot ent ial of the molecular forces , t hen the last -ment ioned

work i s

[ 2 1]

aBvP • olJpv • dp

If one inserts herein our expression for the magnitude of the potential of t he molecular f orces and t akes i nto account that the work of compression i s of the order dp2 , one obtains , neglecting this quantit y wh ich is i nfinitesimally smal l of second order,

[22 )

v r Ta (:Eca.)2
If, =

where K denot es the compress ib i l i ty coefficient in absol ute unit s. We thus

obt ain once more a means for the determinat ion of the looked- for proportional -

i ty coefficient f or the quantities

c
0

.

I t ook t he a and

K values for the

[23 ] t emperature of ice from Landolt and Bornstei n's tables . This yields t he

f ollowing values fo r the factor sought:

Xyl ene Cymene Turpentine oil Ethyl ether

1. 71 X 104 1. 71 X 104 1.73 X 104 1. 70 X 104

Et hyl alcohol Methyl alcohol Propyl alcohol Amyl alcohol

1. 70 X 104 1. 74 X 104 1. 82 X 104 2.00 X 104

DOC . 1

11

First of all, it should be noted that the two coefficients obt ained by

different methods show qui te satisfact ory agreement even though they have been

derived from totally different phenomena . The last table shows a very satis- ( 24]

factory agreement of the values; only the higher alcohols show deviat ions.

Thi s is to be expected, because from the deviations of alcohols from

(25]

Mendeleev ' s thermal expansion l aw and from R. Schiff' s stoichiometr i c law of (26]

capil larity, it has already been concl uded earl i er that in these compounds

temperature changes are associated wi th changes in t he size of t he molecules

of t he l iqu id. Hence it i s to be expected that such molecular changes should

also arise dur i ng i sothermal compression, so that for such compounds at t he

same temperature the heat content wil l be a function of vol ume.

I n summary, we may state that our basic assumption stood t he test : To

each atom corresponds a mol ecular attract ion f i el d that i s independent of the

temperature and of t he way in which the atom is chemical l y bound to other

atoms.

Final ly, it should also be point ed out that the constants ca generally increase with increased atomic weight, but not always , and not in a propor-

tional way. The question of whether and how our force s are rel ated to gravi-

tational forces must therefore be left completely open for t he time being. It (27]

should also be added that the introduction of the function <p(r ) , which i s

taken to be independent of the nature of the molecul es, should be understood

as an approximate assumpt ion, and so should the replacement of sums by int e- ( 28]

grals ; i n fact, as the example of water shows, our t heory does not seem to

(29]

hold for substances with small atomi c volumes. Onl y extensive special

investigations can be expected to bring answers to these questions.

Zurich, 13 December 1900. (Received on 16 December 1900)

12

DIFFERENCE rn POTENTIALS

Doc. 2 ON THE THERMODYNAMIC THEORYOf' THE DIFFERENCE IN POTENTIALS l3ETWEEN METALS
AND FULLY DISSOCI ATEDSOLUTIONS OF TIIEIR SALTS ANDON ANELECfRICAL METHODFOR INVESTIGATING MOLECULAR FORCES Ily A. Einstein [Annalen der Pliysik 8 ( 1902): 798-814]

§1. A hypothe t ical extension of the second law of the mechanical theory of heat
The second law of the mechanical t heory of heat can be applied to such physical systems wh i ch are capab le of pass i ng , wi t h any dt>sired approximat ion , through r ever sible cyc lic processes. In accordance with the derivation of th is l aw from the impossib ili ty of converti ng latent heat into mechanical energy, it is here necessary to assume that those processes are realizable . However , i n an important application of the mechani cal t heory of heat , namely the mixing of t wo or more gases by means of semipermeable membranes, i t i s doubtful whether t hi s postulate i s sat i sf ied. The thermodynami c theory of dissociation of gases and the theory of dilu t e solut ions ar e based on t he (l ] assumption t hat this process i s r ealizabl e .
As i s we l l known , the assumption t o be introduced is as follows: For any two gases A and /J it shou ld be poss i ble to produce t wo part itions such
t hat one is permeable fo r A but not for B. while t he other i s permeable for /J but not fo r A. If the mixture consists of more than two components , then
t his assumption becomes even more complicated and i mprobable. Since t he r esults of the t heory have been completely confi rmed by experiment despite the fact that we worked with processes whose r eal izabilit y could indeed be doubted, the question arises whether t he second law could not be appl ied to ideal processes of a certain kind without contradicting experience.
In this sense, on the basis of t he exper i ence obtained , we cert ainly can advance the proposition : One remains in agreement wit h experi Pnce if one extends t he second law to physical mixtures whose individual components are restricted to certain subspaces by conservative fo r ces acting in certain planes . We shall hypothetically generalize this propos it ion t o t he follow ing:

DOC. 2

13

One remains in agreement with experience when one appli es the second l aw

to physi cal mixtures whose indiv idual component s are acted upon by arbit rary

conservative forces.

(2 )

I n the following we will always make use of t his hypothes i s , even when

this does not seem absolutel y necessary.

§2 . On the dependence of the electric potent i al di ffe rence of a comple te ly di ssoci ated salt solut i on and an el ectrode cons i sting of the sol ut e me tal
on the concent rat i on of the so lution and the hydros tatic pressure

Let a solut ion of a completel y dissoc iated salt be cont ained in a

cylindrical vessel whose axi s coincides with the z-axis of a Cartesian

coord inate system. Let vdo be the number of gram-mol ecules of t he salt

dissolved in the vol ume element do , vm do the number of metal ions , and

(3)

vs do t he number of acid ions , where vm and vs are integral multiples of

v, so that we have the fol lowing equations:

vm = nm•v ' vs = ns·v.

Further, l et n.v.E.do be the magnit ude nf t he total posit i ve electric charge

of t he ions in do, and hence also, up to the i nfin ites i mally small , t he

magnitude of the negative charge. Here n is the sum of valencies of the

molecule' s met al ions , and £ t he amount of electricity required for t he

electrolyt i c separation of one gram-molecul e of a univalent ion.

(4)

These equations are certainly valid, since the number of excess ions of

one kind can be neglected .

We shall f urther assume that t he metal and acid ions are acted upon by

an external conservat ive force whose potential per ion has t he magnitude Pm [SJ
and Ps' respectively. Furt hermore, we neglect the var iabi lity of t he density

of t he sol vent with t he pressure and density of t he dissol ved salt, and assume

that a conservative force, whose potent ial per gram-equ ival ent of the solvent

has t he magnitude P0 , act s upon the parts of the sol vent; there shal l be v0do gram-molecules of sol vent in do .
Suppose t hat all force functions depend solely on t he z- coordi nate, and

t hat t he syst em is in electr ical, thermal, and mechanical equili brium. Then

14

DIFFERENCE IN POTENTIALS

t he quantit ies concent ration v, el ectric potent i al r, osmotic pressures of

t he two ion types

Pm and

p
8

,

and

hydro s t at i c

pressure

Po will be func -

t ions of z only.

At each location of t he electrolyte, each of the two types of el ectrons

must t hen be in equi librium separately, wh ich is expressed by the equat ion s

wher e

Tz 1
V

nm adPzm - nE d = 0

dPs

d

az V ns

+ nE -a:r-zz:=r = 0

Pm = v·nm•RT ,
Ps = v•ns•llT,

and where R is a constant common to all ionic speci es .

take t he form
(1)

I w nmRT

1zv

+

dPm
nm Tz

+

nE ad1z.

=

O,

n RT dJ!{ 8

+

ns

dPS
Tz

-

nE ad-zx

=

O

.

Hence t he equations

If Pm and Ps are known for al l z. and v and :r for a part icular z, then equations (1) yiel d v and r as funct ions of z. Also, the condit ion that t he solution as a whole i s in equilibr ium would r esul t in an equation for t he determi nation of the hydrostatic pressure p0 , which need not be written [6 ] down. We only note that the reason t hat dp0 i s independent of dv and dr i s t hat we are free t o postulate arbitr ary conservative forces that act on the molecul es of t he sol vent .
We now imagine that electrodes made of the solute metal and occupying a vanishingly small part of the cross secti on of the cylindr ic vessel are placed in the sol ution at z = z1 and z = z2. The solut ion and the el ectrodes together form a physical system. wh ich we t ake through the fo llow ing revers ible isothermal cyclic process:
1st partial process: We pass t he amount of el ect ric ity n£ infi nitely slowly through the sol ution, using the electrode at z = z1 as anode, and that at z = z2 as cathode.

DOC. 2

15

2nd partial process : The amount of di ssolved metal that has thus been moved electrol yt ical l y from z1 to z2 we now move back mechan i cal l y infinitely sl owl y from z2 to z1.
First of all, it is evident that the process i s strictly revers ible, since all steps are imagined to proceed inf i nit ely slowly, i .e. , the process is compounded of (ideal) states of equilibri um. For such a process the second law requires that the total amount of heat supplied to the system during the cyclic process shall van ish. In conjunction with the s<>cond law, the first. law requires that the sum of all other energies supplied to the system during the cyclic process shall vanish.
During the first partial process the amount of electric work supplied is

where n2 and n1 denote the electric potent ial s of the electrodes.
During the second partial process

is supplied, where K is the force acting in the positive z-direction that is required for the nm metal ions that are to be moved, and which are now in t he metallic state, to keep them at rest at an arbitrary location z. It is easily seen that the following equation will hold for K:

Here vm denotes the volume of one metal ion in the metall ic state. Hence the above work takes on the val ue

Iz1

K. dz

= -

Jz2[ nm

!-zm
az

+

nmvm

Tdpzo]dz

Z2

Z1

16

DIFFERENCE IN POTENTIALS

where the second i ndex denotes the coordinate of the el ectrode. We obtain, hence , t he equat ion
(2 )
If the electric potent ial s in t he cross sect ions of the el ectrodes i nside the solut ion are denoted by :r1 and 1'2 , i ntegrat ion of t he first equation (1 ) yields

- n.£{:r2 - :r1) = nm[Pm2 - Pm1] + nmRTlog[~v 1] ,
where v1 and v2 r efer agai n to the cross sect ions of the electrodes . Adding these equat ions , one obtains

(3 )
Since the v' s and Po are completely i ndependent of each other, th is equation represents the dependence of the potential difference AIT between metal and sol ution on concentration and hydrostat ic pressure. It should be noted that the postul ated forces no longer appear in t he result . If t hey were to appear, the hypot hesis posited in §1 would have been carried ad absurdum. The equat ion obtained can be resolved i nto t wo equations, namely :
at const ant pressure,
at constant concentration.
ThP final formula (3) cou ld have al so been obt ained without t hP hypothes i s proposed in §t had the external forces been identi fi ed with terrestrial grav ity. However, in t hat cas<' v an<l p would not be independent of each other and the r esolut ion i nt o equat ions (4) wou ld not be permi tted.

DOC . 2

17

It should also be briefly noted that the Nernst theory of el ectric

forces inside dissociated electrolytes, taken in conjunction with the fi rst of

equations (4), makes it possible to calculate the electromotive force of the

concentration cell . Thus one arrives at a result that has already been tested

repeatedly ruid that ti l l now has been derived from special assumptions .

(7)

§3 . Un the dep endence of the quantity till on the nature of the acid
Ye shall consider the followi ng ideal state of equi librium: Let us again have a cylindric vessel . Parts I and II shal l each contain a compl etel y di ssociated sal t solution; the two salts shall have an ident ical metal ion (same metal and same electric charge) but a different nrid ion . Between the two parts there shall be a connect ing space V whicb contains both salts

~.

aa~

in solution. Upon the acid ions in V shal l act forces whose potentials

p c1>
5

and

f 8 C2)

depend only on

z, and these forces shall bring about that

only infinitesimal ly few ac id ions of the first and of the second type get

into II and 1, respectively . Furthermore, f C1> and P <2> shall be chosen

8

8

such that the metal ion concentration in the two parts I and II be the same .

Also, let Po = Po .

I

2

If there are per unit volume vm<u and vm12> metal ions that correspond

to the first and second type of salt, respectively, then

(1)

where the subscripts refer to space I and II, respectively . However, the condition for the equilibrium of the metal ions in V
yields

18

DIFFERENCE INPOTENTIALS

dlog (v<1> + v<2>)

d1r

- RT

mdz m - {E Tz = 0

where { denotes t he valency of the metal ion. I ntegrating over V and t aking equat ions (1) into account, we obtain

(2 )

Next we i magine t hat el ectrodes made of t he solute metal are instal led in I

and II, and construct t he follow ing ideal cyclic process :

1st partial process : We send an amount of elect ricity {£ infinitely

slowly t hrough t he syst em , taking the electrode in las anode , and t he other

as cathode .

2nd partial process : The metal thus transport ed electrolytical l y from

z = z to z = z , which has t he mass of one gram-equ ival ent, i s now returned

1

2

mechanical ly to the electrode in z = z1.

By applying t he two laws of t he mechanical theory of heat, one again

reaches t he conclusion t hat the sumof mechanical and electr ical energy

supplied to the system during the cycl i c process van i shes. Si nce, as one Cd.fl

readily see, t he second st ep does not require any energy, one obtains the

equation

(3 )

where rr2 and n1 aga in denote the pot ent ial s of the electrodes . By
subtracting equations (3) and (2) , one obtains

a.nd hence the foll owing t heorem : The potential difference between a metal and a complet ely dissociated
solution of a sal t of t his metal in a given solvent i s independent of the nature of the electronegative component, and depends solely on the concentration of the metal ions. It is assumed, however, that the metal ion of t hese sal ts is charged with t he same amount of electricity.

DOC. 2

19

Before we turn to the study of the dependency of (~IT) on the nature of the solvent, we shal l br ief l y develop the theory of conservative molecular forces in l iquids . I shall borrow the notat ion from a prev ious art i cle on this topic, 1 which shal l at the same time temporarily j ust i fy the hypotheses I am going to introduce.
To each mol ecule of a liqu id or a substance dissolved in a liqu id shal l be assigned a certai n constant c, so that the express ion for the relat ive potential of mol ecular forces of two molecules, which shal l be characterized by t he indices . .. 1 and ... 2. will be

( a)

P = P - c c <;?(r) ,

m 12

where <;?(r) i s a funct ion of distance common to all molecular species . These forces shall s imply superpose, so that the expression for the relat ive potential of n molecules shal l have the form

(b)

l I o=n P=n

Const. - ½

cacp <;?( rap)

o=l P=l

Should all molecules be ident ical, we wou ld obtai n the express ion

(c)

I l o=n P=n

Const . - ½c2

<;?(rap>

o=l P=l

Further, if the laws of interaction and distribution of the mol ecul es are so constituted that it is permissibl e to convert the sums into integrals , then thi s express ion becomes

1A. Einstei n, Ann. d. Physik 4 (1901) : 513.

(9]

20

DIFFERENCE IN POTENTIALS

Here N denotes t he number of molecules per unit volume. If N0 denotes the number of molecules in one gram-equivalent, then N0/ N = v is the molecular [8] volume of t he l i qu id , and if we assume that the investigation involves one gram-equivalent and neglect t he effect of the l iquid surface, our express ion
becomes

We shall now choose the unit for c such t hat this expr ess ion reduces to

(d)

By this choice one obt ai ns absolute units fo r the quantities c. It has been

shown in the prev iously cit ed art icle that one remains in agreement with

experience if one sets c = Ec , where the quantities c r efer to t he atoms

0

0

compos i ng the molecule .

We now want to calculat e the relative at traction potential of a gram-

molecule of an ion wi th respect to its solvent, whi l e making the express

assumption that the at traction fields of t he sol vent molecules do not act upon

the electric charges of the ions. Methods to be developed later will provide

the means by which to decide whether th i s assumption i s permissible .

If cj is the molecular constant of t he ion and c1 that of the sol vent, then the potential of one molecule of t he ion with respect to the

sol vent has the form

l J Const . - cjcl.c.p(r) = const. - cj. clNl dr. c.p(r0 ,d7 ) ,
f,

where Nl denotes t he number of solvent molecules per unit volume. Since
N0/ Nl = vl ' this expr ession becomes

DOC. 2

21

However, s ince a gram-equivalent contai ns N0 molecules of the ion , we obtain for the rel ative pot ent ial of one gram-equivalent of the ion:

Introducing t he solvent concentrat ion 1/v f, = vi' one obt ains the form
(e)
If the sol vent i s a mixture of several liquids , which we shall dist inguish from each other by indices, we obtain

where t he vf, denote the number of gram-molecul es o-f t he individual components of the solvent per uni t volume. The formula (e') holds
approximatel y also if the quant i ties ve vary with pos it ion .

§5. On th e depe,idence of th e el ect r ic pot entia l di ffe renc e ex isting bet ween a metal and a compl etel y dissoci at ed sol ut i o,i
of a sal t of t hi s meta l on tli e natu re of t ile sol ·ven t
Let a cylindric vessel again be divided , as in §3 , i nto spaces I and II and t he connecting space V. Space J shall cont ain a first solvent, fl a. second one , and V a mixtur e of bot h, and fo rces t hat prevent diffusion shall act on the solvents in space V. The vessel shall contain a completely dissociated dissolved salt. In V, on it s anions t here shall act forces whose potenti al shall be cal led P and which shall be chosen such that the salt be
8
of the same concentration in I and II . llle now establish t he condition for the equilibrium of the metal ions. We again t ake t he z-axis parallel to the cylinder axis from J t o II.
The force of electric origin that acts on one gram-equ ivalent will be

-

n nm

E

T,faz-

22

DIFFERENCE IN POTENTIALS

The force exerted on the equi valent by osmotic pressure i s _ RT d log v
dz
The effect of mol ecular forces on the equivalent is

where the superscripts refer to the sol vents. The equilibrium condition sought is t hen

fog -

.n!.m..

E

~
az

-

RT

d

dz

11 + -a#z::{2cmcl0 >vt<. 0

+ 2cmc~2 >11 ~2 >} = 0 .
<- <-

If one int egrates over V and t akes int o account that v i s the same in J
1 and II, and that according to our assumpt ion vi 1> and v 2> vani sh, one
obtai ns

where the superscripts refer to spaces I and 11, respect ively . We now imagine that electrodes made up of the dissolved metal are placed
into I and II, and construct a cycl i c process by sending an amount of el ec-
tri city .!... E through the system and then return ing the transported metal
nm mechanically, which does not require any work i f we assume that the hydrostatic pressure i s t he same in 1 and II . Appl i cat ion of the two laws of t he theory of heat yields
Subtract ion of the two results gives

DOC. 2

23

If each of the two solvents is a mixture of several nonconducting liquids, one obtains somewhat more generally

where now v1 denotes the number of gram-molecules of a component of the solvent in a volume element of the mixed solvent.
Hence the potential difference MI depends on the nature of the sol vent . This dependence can be used as a basis for a method of exploring the mol ecular forces.
§6. A method for the determination of the constant c for metal ions and solvents
Let two completely dissociated salt solutions undergo diffusion in a cylindri cal vessel; these salts shall be indicat ed by subscripts. The sol vent shall be the same throughout the vessel and shall be indicated by the superscript. The vessel shall again be di vided i nto spaces I and II and the connecting space Y. Space I shall contain only the first salt, and II only the second salt; diffusion of the two salts shall take place in space Y. Into spaces I and 11 there shal l be introduced electrodes cons isting of the
respective metal solute and having electric potentials n1 and n2•,
respectively; onto the second electrode shall be soldered a piece of the f i rst
electrode metal, whose potential is n2. Furthermore, we denote the el ectric potentials in the interior of the unmixed solutions in I and JI by 7 1 and
7 2. Ye then have
If one produces exactly the same arrangement exc~pt for us ing a different solvent, which shall be denoted by the superscript <2>, one obtains:

24

DIFFERENCE IN POTENTIALS

Subt racti ng these two expressions and t aking into account the results fo und in §5 , one obtai ns

(Il2 - Ill ) ( 2> - (IT2 - Il l) cu =

{[c:•m]; j {< •2 - •1>'" - <•2 - •1> '" } -

[c:•t}-{c}2lv}" - c}"v}") .

The extens ion required if the sol vents are mixtures i s eas ily obt ai ned as i n §5.
The values of the left -hand side of t hi s equation are obtained directly from experim<'nt. The determination of t he f i r st term of t he right -hand s ide will be dealt wHh in the next paragraph; for the time being , let it on ly be sai d t hat this t erm can be calculat ed f rom the concentrations used and the molecu lar conducti viti es of t he respective ions for t he respective sol vent , provided the arrangement has been suitably chosen. Thus the equat ion makes it possi bl e to calculate t he second tffm on the right-hand s ide.
This we util ize to determine t he constant c f or the metal ions and t o test our hypotheses . We always use the same two solvents in a seri es of experiments of the kind descr ibed . Then for the whol e experimental seri es the quantity

[ 10 )

Hence , if onf' puts n1/nm1 = £1, etc . , t o be equal t he valency of t he

f irst etc. metal ion , the l ast t erm calculated of the right-hand side wi ll be

a relative m£'asure for t he quantity

If one thus examines the combinations of all electrode metals pair by pair, one obtains the quantities

DOC. 2

25

in relat i ve measure .

One obtains in this same measure the quantities cm/t separately by

carrying out an analogous invest igation with a metal in such a way that the

sal t s and electrodes in/ and II contain the same metal, but that t, i .e.,

the val ency (electrical charge) of the metal ion, is different on the two

sides. The val ue of the quanti ties cm in th i s measure can then be obt ained

for the individual metal s. A series of such experiments thus leads to the

rat i os of the em's, i.e . , the constant s for the molecular attract ion of metal

ions . This series of em's must be i ndependent of the nature of the sal t s

used, and the r atios of the em' s thus obtained must be independent of the

nature of thP t wo solvents on which we based the i nvestigat ion . A fu rther

requi rement must be that cm must prove to be independf"nt of the electrical

charge (valen cy) displayed by the ion. If this is the case, the above

assumpt ion that the molecu lar forces do not act upon thf" electrical charges is

correct.

I f one wi shes t o determine the absol ute value of t he quantities cm at least approximately, one can do so by taking the approximat e value of k for

both sol vents from the results of the previously cited paper usi ng the formula

c = Ec . It has to be noted here, of course, that just fo r t he two l iquids 0
most obviously suggesting themselves as solvents, namely water and alcohol, it

has not been poss ible to demonstrate the valid i ty of the law of attraction

from t he phenomena of capil larity, evaporation, and compress ibil ity.

[ 11]

Our results could equal ly well serve as a bas is for studying the solvent

constants c1, however, by basing the investigation on two metal ions and varyi ng the solvent, so t hat then the quantity

i s t o be considered as constant . By also us i ng mixtures for sol vents , the invest igation might be extended to al l electrically noncondnctive liquids. From such experiments it is poss ible to calculate relative val ues of the

26

DIFFERENCE IN POTENTIALS

quantities c t hat pertain to the at oms constituting the liquid molecules. 0
Th is, too, opens ample poss ibilities for testing the theory inasmuch as the
c can be ar bitrarily overdetermined . Here, too, the resul t must be 0
independent of the choice of the metal ions .

Al l that now remains is to study the diffus ion process in the space Y

in greater detail . Let the variable quantities depend on z onl y, where the

z-ax.is of the Cartesian coordinate system we have chosen coinc ides with the

direction of t he axis of our vessel. vm1' v81, vm2' and vs2 shall be the z-dependent concentrat ions (gram-equ i val ents per unit volume) of the four

ionic species, f £, -f E, f E, - f E their electric charges , and r the

~

~

~

8 2

electric potent ial. Since no substantial electric charges occur anywhere, we

have f or all z approximatel y

In add ition, for each ionic species we obtain an equation whi ch states that the i ncrease per unit t ime in t he number of ions of a certain kind present in one volume element equals the difference between the number of mol ecules enter ing and the number of molecules leaving that vol ume element during the same time period:

{a/;m, vm1•oaz

z + f m1vm1E oDzr] = a;mt' ,

1121 <Pl

{a/;s, .4: v S1

uz

0

Z

+

f

v E
St S1

ODZr]

=

; st1 ,

DOC. 2

27

where v with the corresponding subscript denotes the constant velocity imparted by a unit mechanical force to one gram-equ i valent of the corresponding ion in the solution .
In conjunction with the boundary conditions, these four equations completel y determine the process taking place, since they permit the determination of the five quantities

ih iJv

iJ11

____Ei

~ 8

az' at at

uniquely for all times . The general treatment of the problem woul d entail

great diff i culties, however, especi ally since equations (P) are not linear in

the unknowns. However, we are only interested in the determination of T2 - J"1.

We therefore multiply the equations (P) success ively by fmi' - f , tm ' - f ,

and obtain, when taking into account (o) ,

81

2

82

(13]
In view of the fact that
iJv av
_!!!i ~
Dz ' Bz Dz
vani sh wherever diffusion does not take place, integration of this equat ion with respect to z yields
cp = 0 . Since time is to be considered as constant, we may write
[ 14 ]

28

DIFFERENCE IN POTENTIALS

I n general, the expression on the r ight is not a total differential ,

which means that 6Il i s determined not only by t he concentrations in

diff us ion-free r egions but al so by the character of the diffusion process .

However , one can make the integration poss ible by applying an art if ice in the

arrangement .

We imagine that spare V i s divided i nto t hree parts , space (1) , space

(2), and space (3), and that these are separated from each ot her by two

part it ions before the start of t he experiment . Let ( 1) be connected wit h I

and (3) with II, and let the two sal ts be s imultaneously dissolved in (2 ), at

concent rat ions t hat shall be exactly the same as i n I and II, r especti vcly.

Thus , befo re t he exper iment, (1 ) and I cont ain onl y t he first salt in solu-

t ion , IT and (3) onl y t he second, and (2) a mi xture of both. The concen-

t rati on is everywhere constant . At t h~ start of t he experiment the parti tions

are removed and immediately t hereaft er the potent ial di fference between t hP

two electrodes is measured . For t his t ime it is poss ible to integrate over

t he diffus i ng l ayers , because vm and II in t he first diffusing layer,

l

St

and II and 11 in the second, are constant. The integrat ion yiel ds

fll2

52

-V

:f2 - :fl = RT[ "m,

s1

[1 l lg

+

V f m1

2 I m1

I m1

+

Vs1( 2s1IIs 1

vm/- m1 + vs/s1

V f: 2 II + V f: 2 V

m2 m2 m2

S2 S2 S2

V -
m2

vm2

V (2 V +

lg 1 + m2 m2 m,

VS2 ( 2S '}.V S2] } .

V f m2 m2

+

v,':l2 CS2

[

V f 2 V + V t: 2 II

m1 mt m1

S1 St St

The method can be s impl i f ied if it is poss ible to choose the same acid i on of t he samP concentr ation in I and II. If in this case I i s connect ed directl y with space II, onC' has to put for the st art of the diffu s ion process:

o(v + V )

az 81

S2 = 0; II + V

S1

S2

V s

const .

Si milarly, accord ing t o assumpt ion:

f S1

=

(
S2

=

ts

and

vs.

VS2

vs

DOC. 2

29

Equat ion (1) t hen becomes

[1 5]

(1 ' )

Of the equat ions (2) , the f irst and the third remai n unchanged, and the second and t he fourt h yield, by addition,

If the deri vat ives with respect to time are eliminated by means of equation (1 1 ) from t he equat ions (2) thus modif ied, one obtains , as prPviously, an expression for dr , that i s a total differential . Integrat i ng, one gets
where the numerical indices now refer to the integration limi ts . Due to the relations
we obtain even more simply

In conclus ion, I feel the need to apologize for outlining here a skimpy

plan for a laborious investigat ion without contributing anything to its

experi mental solut ion ; but I am not in the pos i tion to do so. Al l the same,

th is work will have achieved its goal if it motivates a researcher to t ackl e

the probl em of molecul ar forces from this direct ion.

[ 16 1

Bern, Apri l 1902 . (Received on 30 April 1902)

30

THEORY OF THERMAL EQUILIBRILlf

Doc . 3 KINETIC THEORY OF THERMAL EQUILIBRIUM AND OF THE SECOND LAW
OF THERMODYNAMICS by A. Ei nstein
[Annalen der Ph ysik 9 (1902) : 417-433]

Great as the achievements of t he kinetic theory of heat have been in t he domain of gas theory, t he sci ence of mechanics has not yet been able to produce an adequate fo undation for the general theory of heat , for one has not yet succeeded in deriving the laws of thermal equilibriumand t he second l aw of thermodynamics us i ng only the equat ions of mechanics and the probabi lity calculus , t hough Maxwell ' s and Bol tzmann ' s theor ies came cl ose t o this goal. [l ] The purpose of the foll owing considerations i s to close this gap. At the same time, t hey will yield an extension of the second law that i s of importance for [2] the applicat ion of thermodynamics . They wi l l also yield the mat hemat i cal expression for ent ropy from the standpoint of mechanics.
§1 . Mechan ical model fo r a physi cal s ys t em
Let us imagine an arbitrary physi cal system that can be represented by a mechanical systemwhose st ate i s uniquely determined by a very large number of [3 ] coordinates p1. .. pn and the corresponding vel ocit i es

Let their energy E consi st of two additive terms , the potential energy V and the kinetic energy l . The former shall be a funct ion of the coordinates alone , and the latt er shal l be a quadrati c function of

DOC. 3

31

whose coefficients are arbitrary funct ions of the p's. Two kinds of external forces shall act upon the masses of the system. One kind of force shal l be
derivable from a potential Ya and shal l represent external condit ions (grav-
i ty, effect of rigid walls without thermal effects. et c. ); their potential may cont ain ti me explicitly, but its derivat ive with respect to time should be very small. The other forces shall not be derivable from a potential and shall vary rapidly. They have to be conceived as the forces that produce the
influx of heat . If such forces do not act, but Ya depends explicitly on
time, then we are deal i ng with an adiabatic process . Also, instead of velocities we will introduce linear function s of them,
t he momenta q1, . .. •qn' as the system's state variabl es, wh i ch are defined by n equations of the form

where l should be conceived as a function of the Pi•···•Pn and Pt•· · ·•P~-

§£. On the distr i but i on of possible states between N identical adi abat i c stat i onarr srs tems, when the energy contents are almost identical.

Imagine infinitel y many (N) systems of the same kind whose energy

content is continuously distributed between definite, very sl ight l y differing

val ues E and E+ bE. External forces that cannot be derived from a poten- [4]

tial shall not be present, and Ya shall not contain t he time explicitl y, so

that t he systemwill be a conservative one. We examine the distri bution of

states, which we assume to be stationary.

We make the assumption that except for the energy

E =

L +

Y a

+

1' . ,
i

or

a

[S1

function of th i s quantity, for the i ndividual system, there does not exist any

f unction of the state variables p and q which remains constant in time; we [6]

shall henceforth cons ider only systems that satisfy this cond i tion. Our

assumption is equivalent to the assumption that the distribut ion of states of

our systems is determi ned by the value of E and is spontaneously established

from any arbitrary initial values of the state variables that satisfy our

condit ion regarding the value of energy. I .e., if there would exist for the

32

fllEORY OF THERMAL EQUILIBRIUM

(7 ) system an addit ional condit ion of t he ki nd ip(p1 , . .. ,qn ) = const . that cannot be reduced to the form ip(E) = const . , then it would obv iously be
possibl e to choose initial condi tions such that each of the N systems could have an arbitrarily prescribed value for ip . However, since t hese values do
not vary with t i me, it fo l lows, e.g. , that for a given value of E any
arb i trary value might be ass igned to ~ip. extended over all syst ems , t hrough
appropriate selection of in itial cond i t ions . On the other hand, D.p is
un iquely calcul able by the distribut ion of states , so t hat other dist ri but ions
of states correspond to other values of D.p . It is t hus clear that the exis-
tence of a second such integral ip would necessarily have t he consequence that the state distribution would not be determined by E alone but would necessari l y have to depend on the initial state of t he systems.
If 9 denotes an infinitesimally smal l region of al l state variables
Pt •·· ·Pn• q1, ... qn' which i s chosen such that E(p1. . . qn) lies between E and £+ 6£ when the state var iables bel ong to t he r egion g, t hen the distribution of stat es is characterized by an equation of t he form

(8)
where dN denotes the number of systems whose state variabl es belong to the r egion 9 at a given time. The equation expresses the condi t ion t hat the di stributi on i s stationary .
We now choose such an inf i nites imal region C. The number of systems whose state variables belong to the region C at a given t jme t = 0 is then

where the capi tal lett ers ind i cat e t hat t he dependent vari ables pertain t o
t ime t = 0.

We now l et elapse some arb itrary t ime t . If the systempossessed the

specific state variables specif ic state variables

P1, . .. qn
p1, ... ,qn

at t i me at ti me

t = 0, then it will possess the
t = t. Systems whose state

DOC . 3

33

variables belonged to the region C at t = O, and these systems only, will belong to a specific region g at time t = t, so that the fo l l owing equat ion applies

[9]
However, for each such system Liouville's theorem hol ds , which has the form

From the last three equations it fo l lows that

Thus , , is an invariant of the system, which from the above must have the for m ;( p1, ... qn ) = r*(£) . However, for all systems cons idered, ¢* (£) differs onl y infinites imally from ¢*(£) = const .• and our equation of stat e will then simpl y be

where A i s a quantity independent of the p's and q' s .

§S . On the (stationary) probability of the states of a s ys tem S that is mechanically linked wi th a system E whose energy is relat ivel y i nfi ni te

We again cons ider an infinite number (N) of mechanical systems whose

energy shall lie between two inf i nites imally different lim its E and E+ cE.

Let each such mechan i cal system be, again, a mechanical link between a system

S with state variables p1, ... qn and a system E wit h stat e variables

[11]

""t• · · ·Xn · The express ion for the total energy of both systems shall be con-

stituted such that those terms of the energy that accrue t hrough

1Cf . L. Boltzmann, Castheori e (Theory of gases] , Part 2, §32 and §37.

[10]

34

THEORY OF THERMAL EQUILIBRIL1ll

act ion of t he masses of one partial system on the masses of t he other partial system are negligi ble in compar ison wi th the energy E of the partial system [ 12 J S. Fur ther, the energy JI of the partial system E shall be infinitel y l arge compared with E. Up to t he infinites imal ly smal l of higher order, one might then put
E = H + E.
We now choose a r egion g that is infinitesimal ly smal l in al l state vari abl es p 1 ... qn' r 1 ... xn and is so const i tuted that E l ies bet ween the
const ant values E and E + 6E. The number dN of systems whose state
variables belong to t he region g i s then according to the results of the preced i ng sect ion

We note now that we are free to replace A with any continuous function of
the energy that assumes the value A for E = E, as t his will only
infinitesimally change our resul t . For this funct ion we choose A' .e-2hE,
where h denotes a const ant which i s arbitrary for the t ime being, and which we will specify soon. We wr ite , then,

We now ask: How many systems are in states in wh ich p1 is bet ween p 1 + dp 1 , and, respect i vely, p2 between p2 + dp2 .. . qn between qn and qn + dqn, but x1. .. xn have arbitrary val ues compatible with the condit ions of our system? If we cal l this number dN' , we obt ain
J d'N' = A' e-2hEdPi •. •dqn e -2h/ldr1 ... dXn •
The integration extends over those values of t he state variabl es for which H lies between E- E and E- E+ 6.E. Ye now claim that the value of h can

DOC. 3

35

be chosen in one and only one such way that the integral in our equation

becomes i ndependent of £.
J It is obvious that the integral e-2h8dr1... dxn' for which the l imits

of int egration may be determined by the limits E and E + aE, will for a

{14 ]

specif i c 6E be a function of E alone; let us call the l atter x(E) . The
int egral in the expression for dN' can then be written in the form

x(E- £)
Since E is infinitesimally small compared with E, thi s can be written, up
to quantities wh ich are infinitesimal l y small of higher order, in the form

x(E - E) = x(E) - Ex'(E)

The necessary and suffic ient cond i tion for this integral to be independent of E is hence

But t hen we can put

x' (E ) = o. x(E) = e- 2hE .w(E) ,

J where w(E) = dr1... dxn' extended over all values of the variables whose
energy function lies between E and E+ 6E. Hence the condition found for h assumes the form

e-2hE .w(E-) .{- 2h + -w'_ (E- )} = 0,
w(E) or
h = ½ w' (E) .
w(E)
Thus , there always exists one and only one value for h that satisf i es the cond i tions found. Further, since w(E) and w'( E) are al ways posi tive, as shal l be shown in the next section, h is also always a pos itive quantity .

36

THEORY OF THERMAL EQUI LIBRIUM

If we choose h i n t hi s way, the integral reduces to a quantity independent of £, so that we obtai n the fol l owi ng express ion for the number of syst ems whose variables p1. . . qn l ie within the indicated limits :

[15]

dN' = A" e- 2h£•dPt • • •dqn

Thus , also for a different meaning of A", this is the expression for the probability that the state variables of a system mechan i cal l y linked with a system of rel at ively infin ite energy l ie between infinitesimally close l imits [ 16) when the state has become stationary.

§4. Proof that th e quant i ty h is posi ti ve
Let cp( x) be a homogeneous quadrat i c f unct ion of t he var iables
x1. . . xn. We cons ider the quant ity z = J dx1... dxn' where the limits of
integration shall be determined by t he condition t hat cp( x ) lies between a certain val ue y and y+ fl, where fl is a constant . We assert t hat z , whi ch i s a fun ct ion of y onl y, always increases wit h increasing y wh en n > 2.
If we introduce the new variables x1 = ax1... xn = ox~, where
o = const., t hen we have

Furt her, we obt ain cp(x) = o2cp( x' ) .

Hence, t he lim i t s of i ntegration of the integral obtained for

are

.1L

and -y + - /J.

o2

o2 o2

cp(x ')

Further, i f we assume that /J. is infinites imally small, we obt ai n

[ 17 ) Here y' l ies between t he limits

DOC. 3

37

.1L and JL + A .

o2

o2

The above equation may also be written as

[y] z(y ) = on-2z

.

02

Hence, if we choose o to be positive and n > 2, we wil l always have

[18)

which is what had to be proved. We use this resu lt to prove that h is positive. We had found
where
and E lies between E and E+ 6E. By definition, w(E) is necessarily pos it ive, hence we have only to show that w1 (E) too is always posit ive .
'We choose E1 and E2 such that E2 >E1 and prove that w{E2) > w(E1) and resol ve w( E1) into infinitely many summands of the form
In the integral indicated, t he p' s have def inite values, which are such that V ~ E1. The l imits of integration of the integral are characterized by L lying between E1 - Y and E1 + 6E - V.
To each such infinitesimally small summand corresponds a term out of w(E2) of magnitude

38

THEORY OF THERMAL EQUILIBRIUM

where the p's and dp 's have the same values as in d[w(E1)] , but L l ies
between t he lim its E2 - V and E2 - V + 6E.
Thus , accord i ng to the propos it ion just proved,

Consequent l y,

where E has to be extended over all corresponding regions of t he p' s. However,

if the summation sign extends over all p's, so that
Further, we have
(19)
since the region of the p's , wh ich is determi ned by the equation

includes all of the region defi ned by the equat ion

§5. On the t emperature equ i l i brium
~e now choose a system S of a specific const itution and call it a thermometer. Let it interact mechanically with the system E whose energy is relatively infinit ely l arge . If the state of the ent ire system i s stat ionary, the state of the thermometer will be defined by the equat ion

DOC . 3

39

where dV is the probability that the values of the state var iables of the thermometer lie with i n the limits indicated. The constants A and h are related by the equation
where the integration extends over al l possibl e values of the state variables. The quantity h thus completely determines the state of the thermometer . We call h the temperature function, noting that , according to the aforesaid, each quant i ty H observable on the system S must be a function of h alone, as long as Y remains unchanged, which we have assumed. The quant ity
0
h, however, depends only on the state of the system E (§3), i .e., it does not depend on the way in which E i s thermally connected with S. Fromth is we i mmediately obtain the theorem: If a system E is connected wi th two i nfinitesimally small thermometers S and S' , the same value of h obtains for both thermometers . If S and S' are identical systems, t hen t hey wi l l also have identical values of the observable quantity H.
Ye now i ntroduce only ident i cal thermometers S and call H the observable measure of temperature. We thus arrive at the theorem : The measure of temperature B that is obser vable on S i s independent of the way i n which E is mechanically connected with S; the quantity H determines h, which in turn determines the energy E of the system E, and this in turn determines its state according to our assumption .
From what we have proved it follows immed iatel y that if two systems E1 and E2 are mechan i cally linked, then they cannot forma system that is in a stationary state unless the two thermometers S connected to themhave equal measures of temperature or , what amounts to the same, if they themselves have equal temperature functi ons . Since the state of the systems E1 and E2 is
completely defined by the quantities h1 and h2 or B1 and u2, it fol lows
that the temperature equi li brium can be determined on l y by the cond itions h1 = h2 or B1 = B2.
It now only remains to be shown that two systems t hat have t he same temperature function h (or the same measure of temperature H) can be

40

THEORY OF THERMAL EQUILIBRIUM

mechanically connected into one singl e system that has the same temperature

fu nc t i on.

Let two mechanical systems t 1 and t 2 be merged into one system, but in such a way that the energy t erms that contain state variables of both

systems be inf initesimal ly small. Let ~l as well as
an infinit es imal ly small t hermometer S. The readings

tn21

be connected with
and n2 of t he

latter ar e cert ainly identical up to the infinitesimally smal l because t hey

refer only t o di fferent locat ions with in a s i ngle stationary state. The sam~

is of course true of the quant i ties h1 and h2. We now imagine that the energy terms common to bot h systems decrease infinitel y slowly toward zero .

Thereby the quantities H and h as well as the distributions of state of

t he two systems change infini t esimally because t hey are determined by t he

energy alone. If then t he complete mechanical separation of E1 and E2 is carried out. the relat ions

continue to hol d al l the same . and the distr ibut ion of states changes i nf in[ 20 ) itesi mal ly . H1 and h1, however, wi l l now pertai n only to E1, and H2 and
h2 only t o t 2. Our process i s str ictly revers ibl e. as it cons i sts of a sequence of stationary stat es . Ye t hus obtai n t he theorem:
Two systems hav ing t he same t emperature function h can be merged into a singl e system having t he t emperature fun ction h such t hat their distribut ion of states changes infin itesimal ly .
Equali t y of the quanti ties h i s t hus the necessary and suffic ient condit ion for t he stat ionary combinat ion (thermal equilibrium) of two systems. From this follows immediatel y: If the systems E1 and E2, as well as E1 and Ea • can be comb ined in a stat ionary fashion mechani cal ly (in thermal equ i librium), t hen so can E2 and Ea .
I would l ike to not e here that unti l now we have made use of t he assumpt ion that our systems are mechanical only inasmuch as we applied Liouvil le 's theorem and the energy pri nciple . Probably t he basi c laws of the theory of heat can be developed for systems t hat are def i ned in a much more general way. We wi l l not att empt to do this here, but wil l rely on the equations of

DOC . 3

41

mechan ics. We will not deal here with the i mportant quest ion as to how far t he trai n of t hought can be separated from the model employed and generalized .

§6. On tlie me chanical meaning of tlie quantit y h1

The kinet ic energy l of a syst em i s a homogeneous quadrati c f uncti on

of the quant i ties q. It is always possible to introduce variabl es r by a

linear subst itut ion such t hat the kinetic energy will appear in t he form

( 21 ]

and that

when t he int egral is extended over corresponding infinitesi mally small r egions. The quant i ties r are called momentoids by Boltzmann. The mean ki netic energy corresponding to one momentoid when the system together wi th one of much l arger energy fo rms a single system, assumes the form

( 22 ]

Thus , t he mean kinetic energy is the same fo r all momentoids of a system

and is equal to

u1 = nl ,

[24]

where L denotes the kinet i c energy of the system.

1Cf. L. Bol tzmann, Cas tlieorie, Part 2, §§33, 34, 42 .

[23]

42

THEORY OF THERMAL EQUILIBRIUM

§7. Ideal gases . Absolute temperature

The t heory we developed contai ns as a spec ial case Maxwell's di stribu-

tion of states fo r ideal gases. I.e . , if in §3 we under stand by the system S

one gas molecule and by :E the tot ality of all the ot hers , t hen t he expres-

sion for t he probability t hat t he values of t he variables p1.. ·Pn of S lie in a r egion g t hat i s infin it es imally small with r espect to all variables

wi ll be

I di/ = Ae-2hE g dp 1. . . dqn .

One can al so immediately realize f rom the expression f or t he quant i ty h found in §4 that , up to t he infinit esimally smal l , t he quantity h wi ll be the same for a gas molecule of another type occuring i n the syst em , s ince t he systems :E determin i ng h are ident ical for the two molecules up t o the infini t esimally small . This establ ishes t he general ized Maxwellian distr ibution of states for ideal gases. -
Further, it follows immed iately that t he mean ki net i c energy of mot ion of t he center of gravit y of a gas molecul e occurr ing in a syst em S has t he
¾ value h because it cor responds t o three momento ids. The kinetic t heory of
gases t eaches us t hat this quanti ty is proport ional to the gas pressure at constant vol ume . If, by defin it ion, this is t aken t o be proport ional to t he absolute temperature , one obt ains a rel ationship of the f orm

1 = ,.,. 1 = ½ w(E) •

ifli

w' (E)

[ 25 ] where K. denotes a un i versal constant. and w the function introduced in §3.

§8. The s econd law of the theo ry of heat as a consequence of the mechanical theory
We cons ider a given phys ical syst em S as a mechan ica] systemwit h coord inates Pi ··· Pn · As st ate variables of t he syst em we further i ntroduce t he quant ities

DOC. 3

43

dp1

dpn

at -- p1' • • ·ar -- pn'

P1... Pn shall be the external forces tend i ng to increase the coordinates of the system. Y. shall be the potential energy of the system. L its kinetic
i
energy, which is a homogeneous quadratic function of the p~s. For such a
system Lagrange's equations of motion assume the form

(v = 1, . .v = n) .

The external forces consist of two kinds of forces . The first kind, ,tt ), are the forces that represent the conditions of the system and can be derived from a potential that is a function of Pt · ·•Pn only {adiabatic walls, gravity, etc. ):
[26]

Since we have to consider processes which consist of states that infinitely

approximate stationary states, we have to assume that even though Ya

explicitly contains the time, the partial derivatives of the quant i ties

oVafOpv with respect to time are infinitesimall y small .

Pt The second kind of forces, 2) = "v' shall not be derivable from a

potential that depends on the Pv only. The forces n represent the forces

that mediate the influx of heat.

If one puts Ya + ri = Y, equations (1 ) become

[27]

n V

= -8-(wY-L;-)

+

d ,IT

{ oaiit;;}

•

The work supplied to the system by the forces nv during the time dt
represents then the amount of heat dq absorbed during dt by the system S, which we will measure in mechanical units.

44

THEORY OF THERMAL EQUILIBRIIDf

(28 ]
However, since
(29 ]
and, further,
[30 ]
we have
(31]
Since , further
[ 32]
we wi ll have

Jkl dQ =

dpv + dl

1,

=

1
4iJi

=

L
nK.

•

[33] (1 )
We will now concern ourselves with the express ion

This represents the increase of pot ential energy in the system t hat woul d t ake place duri ng t ime dt if Y were not expl icitly dependent on time . The t ime element dt shall be chosen so large that the sum i ndicat ed above can be replaced by i t s average value for infinitel y many systems S of equal temperatur e , and at the same ti me so small that the expl i cit changes of h and Y [34 ] wi th time be infini tes imally smal l .
Suppose that infinitel y many systems S in a st at ionary state, all of whi ch have identical h and Ya' change to new stat ionary systems which are characterized by val ues h+ 8h, Y+ 81' common to al l. Generally, "811 shall denote the change of a quantity during trans it ion of the systemto a new state; the symbol "d" shall no longer denote the change with time but di ffer ent ials of defin i te int egrals . -

DOC . 3

45

The number of systems whose state variables lie in the infinites imally small region g before the change is gi ven by the formula

(35 ]

here we are free to choose the arbitrary constant in V for each given h
and Va such that A wi l l equal unity . We shal l do this to simplify t he
calculat ion and shal l cal l th i s more precisely defined function Y-. . It can easily be seen that the value of the quant i ty we seek will be

(2)

[36 ]

where the integrat ion shoul d extend over all values of the variables, because this express ion represents the increase of the mean potential energy of the system that would t ake effect if t he di stribution of st at es would change in conformity wi th 6J"I' and 6h, but V would not change explicitly.
Further, we obtai n

(3 )

Here and in the fol l owing the int egrations have to be extended over al l poss i ble val ues of the variables. Further, it should be kept i n mind that the number of systems under consideration does not change . This yi elds the equat ion

or
( 37}

or

(4)

I4NK: e- 2h ( Y"+l )u1:(hY)dp 1.. . dqn + 4K-l6h = 0 .

46

THEORY OF THERMAL EQUILIBRIUM

Y and Z denote t he mean values of t he potent ial and kinet ic energi es
of the N systems . Adding (3) and (4), one obtains

or, because
[38 ]

h = !!_ ,
4l

6h = - ~ - 6L,
4L2

If we substit ute t hi s f ormula i n (1) , we obt ain

[39 )
Thus , dQ/ T i s a complete different ial . Since

one may also set

il = nK

Thus , apart from an ar bitrary addit ive const ant , '£'F / T is t he express ion for the ent ropy of the system , where we have put I:"" = ~ + L. The second law thus appears as a necessary consequence of t he mechanistic wor l d picture.
§9 . Calcula t i on of th e ent ropy
The expression f = E"' / T that we obtained for the ent ropy f onl y
appears t o be simpl e , because E"' remains t o be calculated from t he conditions of t he mechanical syst em. I.e . , we have
E"'- = E + E0 ,

DOC . 3

47

where E is given directly, but £0 has to be determined as a function of E and • h from the condition

I e-2h(E-E0 )dP1 •·· dqn -_ N •

[40]

In this way, one obtains

+ canst.

[41]

I n the expression thus obtained, the arbitrary constant that has to be added

to the quantity E does not affect the result, and the third term, denoted

"canst. , 11 is independent of V and T.

The expression for the entropy f is strange, because it depends solely

on E and T, but no longer reveals the special form of E as the sum of

potential and kinetic energy. This fact suggests that our resu l ts are more

general than the mechanica] model used, the more so as the express ion for h

found in §3 shows the same property .

[ 42 ]

§10. Extension of the second law

No assumptions had to be made about the nature of the forces that corre-

spond to the potential Va' not even that such forces occur in nature. Thus ,

t he mechanical theory of heat requires that we arrive at correct results if we

apply Carnot's principle to ideal processes, which can be produced from the

observed processes by intrnduc ing arbitrarily chosen Va's. Of course, the

resul ts obtained from the theoretical cons ideration of those processes have a

real meaning only when the ideal auxi l iary forces Va no longer appear in

them.

[43]

Bern, June 1902. (Received on 26 June 1902)

48

FOUNDATIONS OF THERMODYNA IICS

Doc . 4 ATHEORY OF THE FOUNDATIONS OF THERMODYNAmcs
by A. Ei nstein [Annalen der Physik 11 (1903 ): 170-187]

(1)

In a recently publi shed paper I showed t hat the l aws of t hermal equi -

librium and the concept of ent ropy can be derived with the help of the kinet ic

theory of heat . The question that then arises natural ly i s whether t he

kinetic theory i s r eal ly necessary for the derivation of t he above foundat ions

of the theory of heat, or whether perhaps assumpt ions of a more general nature

may suffice . In th i s article it shall be demonstrated that the latter is t he

[2) case, and it shall be shown by what kind of reasoning one can reach t he goal.

§1 . On a general math emat i cal representation of the proc esses i n isolated physi cal sys t ems

Let the stat e of some physi cal system t hat we consider be un iquely determined by very many (n) scal ar quant ities p1,p2. . . pn, wh ich we call [3 ) stat e variab l es . The change of t he system in a t ime element dt is then determined by the changes dp 1,dp2... dpn t hat the st ate variables undergo during that ti me element .
Let the system be isolated, i.e., t he system cons idered shoul d not interact with ot her syst ems. It i s then clear that the state of t he system at a given instant of time un iquely determ ines the change of the system in the next t i me element dt, i.e., the quantit ies dp1,dp2.. . dpn . Th is statement i s equ i valent t o a system of equations of the form

( 1)

Tidp . = cpi( pl • • · Pn )

(i = 1 . .. i = n) ,

where t he cp' s are un ique funct ions of the ir arguments . In general, for such a system of linear differential equations there
does not exi st an inte_gral of the form

DOC . 4

49

which does not contain the t ime explicitly. However, for a system of equations that represents the changes of a physical system closed to t he outs ide, we must assume that at least one such equat ion exists , namely the energy equation

At the same t i me, we assume that no further integral of this kind that i s

independent of the above equation is present.

[4]

§2. On the stationary distr ibution of state of infinitely many isolated phys ical systems of almost equal energies

Experience shows that after a certain t ime an i solated syst em assumes a

state i n wh i ch no perceptible quantity of the system undergoes any further

changes with time; we call this stat e the stat ionary state . Hence it wi l l

obviously be necessary for the functions t.pi to fulfill a certain condition

so that equations (1) may represent such a physical system .

If we now assume that a perceptible quant ity i s always represent ed by a

t ime average of a certain function of the state variables p1. . . pn' and that these state variables p1 . . . pn keep on assuming the same systems of values with al ways the same unchanging frequency, then it necessari l y follows from

this condition, which we shall elevate to a postulate, that the averages of

al l functions of the quantities Pt···Pn must be constant; hence, in

accordance with the above, all perceptible quantities must also be constant.

We wil l specify this postulate precisely . Starting at an arb itrary

point of time and throughout time T, we consider a physical system t hat is

represented by equat ions (1) and has the energy £. If we imagine having
chosen some arbitrary region r of the state var iables Pi · · ·Pn• then at a

given instant of t ime T within t he chosen region

the values of
r or outside

the it;

variables p1. .. pn wil l hence, during a fract ion

lie of the

time T, which we shall call r, they will lie in t he chosen region r . Our

condition then reads as follows : If Pi ···Pn are stat e variabl es of a

50

FOUNDATIONS OF THERMODYNAMICS

physi cal system, i. e. , of a system that assumes a stationary st ate, then for

each region r t he quantity r/ T has a def i nite li miting value fo r T = w.

For any infi nitesimally small region this l imiting value is infini t es i mally

small.

1'hc foll owing consideration can be based on th is postulate. Let t here

be very many (N) i ndependent physical systems , all of which arc r epr esented

by t he same system of cquat ions (1). 'We select an arbitrary instant t and

inquire after t he distr i but ion of the possi bl e states among t hese N systems ,

assuming t hat the energy E of all systems lies bet ween t"" and the
infinitesimally close val ue r + 6V. Fromthe postulat e int roduced above,

i t follows immediat ely that t he probabi l ity t hat t he stat e variables of a
system randomly selected from among N syst ems will lie wi t hi n the region r

at t ime t has the value

1I. m
T = w

T
7

=

const.

The number of systems whose st ate var iables l ie wi th in t he reg ion r at time

is thus

N•

1I.m

T
7j',

T = ro

i.e . , a quant ity independen t of t ime . If g denot es a region of t he coordi nates Pi ··· Pn that i s infinitesi mally small in al l variables , t hen t he number of syst ems whosP stat e var iables fill up an arb itrar ily chosen inf ini tesimal l y smal l region g at an arbitrary time will be

(5 ) (2 )

The function f i s obt ained by express ing in symbols t he condition t hat t he dist ribution of states expressed by equation (2) is a stat ionary one. Speci f ical l y, the region g shal l hf> chosen such t hat p1 shall l ie bet ween the defin ite val ues p1 and p1 + dp1, p2 between p2 and p2 + dp2, .. .pn between pn and pn + dpn ; t hPn we have at t he t ime t

DOC. 4

51

where the subscr i pt of dN denotes the time . Taking into account equation (1), one obtai ns furthermore at time t + dt and t he same region of t he state variables

However, s i nce dNt = dNt+dt' because t he distribution i s stat ionary, we have

[6 ]

Thi s yields

, O<pv _ \ B(log t) . _ \ B(log t) . dpv _ d(l og l )

- l ~ - l llpV

<PV - l llpV

dt - dt '

where d(log l )/ dt denotes the change of t he function log l with respect to

t ime for an indi vidual system, taking into account the changes with time of

the quantities Pv·

One obtains further

-J !f- v=n a

dt 2

+ ; ( E)

f =e

v=l v

= e- m+t,(E)

The unknown function ¢ i s the time- independent integration constant which
may depend on the variables Pi · · ·Pn• but can contain them, according to t he assumptions made in §1, only in the combination in wh i ch t hey appear in the energy E.
However, since t,(E) = ¢( £"') = const. for all N systems considered, t he expression for c reduces in our case to

V=n {J

-I dt l ::-

l = const . e

v=l v

const. e-m

According to the above we now have

52

FOUNDATIONS m· THERMODYNAMI CS

For t he sake of simpl ic ity we now i ntroduce new state variables for the system cons idered; they shal l be denoted by 7v. We then have

where t he symbol D denot es the f unctional determi nant . - We now want t o choose t he new coord inates such that
This equat i on can be satisf i ed in infinitely many ways , e.g., by setting

Using t he new variables , we t hus obtai n
[7] Henceforth we wil l al ways suppose t hat such variables have been introduced.
§9 . On the dis tribution of state of a s ys tem i n contact wi th a sys tem of relati vely infi nitely la rge energy.
We now assume that each of the N isol ated systems i s composed of two
partial systems ~ and u i n int eraction. Let the state of the partial

DOC. 4

53

system E be determi ned by the values of the variabl es D1. . .DA, and that of t he system u by the values of the var iabl es x1. . . xl . Further, let the energy £, which for each system shal l lie between the val ues er- and c/1",
i. e . , shal1 equal r up to the inf i nites imally small, be composed of two
terms , of which the first, H, shal l be determined only by the values of the
state variables of t, and the second, n, only by the state variabl es of u,
so that, except for the relatively infinitesi mally small, one has

E = R+1J .

(8 1

Two systems in interaction wh i ch satisfy this condition wi l l be called two
systems in contact . We al so assume that fJ is infin i tesimal ly smal l compared
wi th //.
For the number dN1 of the N-systems whose state var iabl es rr1.. .nA
and x1... r1 lie between n1 and n1 + dn1, n2 and n2 + dn2, . . .DA and
DA+ dilA, and r1 and r1 + dr1, r2 and r2 + dr2, . . . rt and r1 + dr i, we get the express ion

where C can be a f unct ion of £ = H + q. However, since according to the above assumpt ion the energy of each of
the systems cons idered up to t he infinites imal ly small has the value r, wP
can replace C by const . e-2h/1" = const .e-2h(H+f] ) without causing any changes in the resul t, where h is a constant still to be def ined prec isely. Hence, the expression for dN1 becomes

The number of systems whose state variables r lie between the indi -
cated limits, while the values of the variabl es n are not subj ected to any
restrictive condition, may t hus be represented in the form

54

FOUNDATIONS OF THERMODYNAMICS

where the i ntegral is to be extended over all val ues of Il to which correspond values of the energy // ly i ng between E"' - 1J and V + 6c+"- - 'l · Had t he i ntegration been carried out, we would have fo und t he distribution of the stat e of t he systems u. Th is i s i n fact poss ibl e.
We put

where t he integral on thP l eft -hand side is t o be extended over al l val ues of the variables for which // lies between the defin i te values £ and I,' + 6V. The i ntegral t hat appears in the expression dN2 t hen assumes t he form
x<E"' - 1J ) ,

or, s i nce 1J i s infini tesimally smal l compared wi th L~,

x(C" ) - x '( C" )·TJ .

Thus , if h ran be chosen such that x '( E"' ) = 0, the i ntegral reduces
to a quanti t y that i s independent of the state of u. It i s poss ible to put , up to the inf ini t es imally smal l ,
I x(E) = e-2h£ dl11... dll~ = e-2hE-w(E) ,
[9] where t he int egration limits are the same as above, and where w denot es a
new funct ion of £.
The condi tion for h now assumes t he form

consequently :

x '( l"") ,,- 2hE"'·{ w' (£.f< ) - 2hw( E"')} = O

If h is chosen in t his way, t he expression for dN2 will assume t he form

DOC. 4

55

(3)

With suitable choice of the constant this expression represents the proba-

(10]

bility that the state variables of a system in contact with another syst em of

relativel y infinitel y large energy will lie within the indicated limits. The

quantity h depends only on the state of the above system E of relativel y

infinitely large energy .

§4. On absolute temperature and thermal equilibri um

Thus , the state of the system u depends only on the quantity h, and (11 ]

the latter only on the state of the system I:. We cal l the quantity

1/ 4hK- = T the absolute temperature of the system I:, where K- denotes a

un i versal constant.

[ 12)

If we call the system u "thermometer, " then we can i mmediatel y advance

the fol l ow i ng proposit ions:

1. The state of the thermometer depends onl y on the absolute tempera-

ture of the systemI:, and not on the ki nd of contact of the systems I: and u.

2. If i n case of contact two systems I:1 and I:2 impart the same state to a thermometer u, then they have the same absolute temperature and

wi l l also impart the same state to another thermometer u' in case of

contact.

Further, suppose t wo systems I:1 and I:2 are in contact and I:1 i s also in contact with a thermometer u. The distribution of states of u

depends then only on the energy of th<' syst em (I:1 + I:2) , i. e . , on th<> quan-

tity h1, 2· If the interact ion between i nfinitely slowly, t his does not change

I:1 the

exapndresIs:2ionisforimtahgei n<e'dnetrgoydencr1e,a2se

of the system (I:1 + I:2). which can be read i ly seen from our defin ition of

contact and the expression for the quantity h that we formulated in the last

sect ion . Finally, if the interaction had ceased completely, t he di str ibution

of states of u, which does not change during the separation of I:1 and I:2, will now depend on I:1, i .e., on the quantity h1, where the index denotes association with the system I:1 alone. Hence we have

56

FOUNDATIONS OF THERMODYNA\UCS

By an analogous line of drgument, one could have obt ained
hence
or, in words : If one separat es two systems E1 and E2 in contact which fo rm an isolated system (E1 + E2) of absol ute t emperature T, t hen the now [13] isol ated systems E1 and E2 wi l l have the same t emperature after separat ion . We imagine a given system i n contact with an ideal gas. This gas shall be complet ely descri babl e in t erms of t he kinetic theory of gases. As t he system u we consi der a si ngle monoatomi c gas molecu l e of mass µ whose state shall be compl etely determi ned by its orthogonal coordinates x, y, z and t he velocities {, n, ( . In accordance with §3, we obtain for the probabili t y that t he state variab les of t his molecule lie bet ween the lim its x and x + dx, ... ( and ( + d( t he well -known Maxwellian expression

By integration, one obtai ns from this fo r the mean kinetic energy of this molecule
[1 4 ]
However, the kinet i c theory of gases teaches t hat at constant vol ume of the gas t his quantity is proport ional to the pressure exerted by t he gas . The l at t er is by defin it ion proportional t o the quant ity designated in physics as absolute temperature . Thus t he quanti t y we designated as absolute temperature i s nothing else but the t empcraturr of a systemmf'asu red by t he gas t hermometer .

DOC . 4

57

§5 . On infi nit ely slow processes
Until now we have only considered systems that are in a stationary state. Now we are also going to investigat e changes of stationary states, though only those that proceed so slowly that the distribution of stat es exi sting at an arbitrary instant differs onl y infinitesimal ly from the stationary distribution; or, more preci sely, t hat, up to the infinitesimal ly smal l, the probability that the state variables l i e in a certain region C can be represented at any moment by the formula found above . We call such a change an infinites imal ly slow process.
If the funct ions <pv (equat ion (1)) and the energy E of a system are specified, then, accord i ng to the above, its stationary state distribution i s also specifi ed . An infin i tely slow process will thus be specif i ed eit her by a changi ng £, or by the funct ions cpv containing the t ime explicitl y, or by bot h circumstances s imult aneously, but in such a way t hat the corresponding differential quotients with respect to time are very small.
We assumed that t he state variables of an isolated system change according to equations (1). However, conversely, i f there exi st s a system of equat ions (1 ) according to which the state variables of a system are changing , thi s system does not al ways have to be an i solated one. For it can happen that a system under cons ideration is influenced by other systems in such a way that th is influence depends onl y on such function s of the variabl e coor di nates of the influencing systems which do not change when the di stri bution of states of the influencing system i s constant . In this case the change of the coordi nates Pv of the system cons ider ed can also be represented by a system havi ng the form of equations (1). However, the functions i.pv wi l l then depend not only on t he physical nature of the system in quest ion, but also on certai n constants that are defined through the influencing systems and their distribu tions of states. Thi s kind of i nfluence on the system under consideration we call adiabatic. It i s easy to see that as long as the di stribut ions of state of the adiabatically influencing systems do not change, there exi st s an energy equation for the equat ions (1) in this case as wel l. I f the states of the ad iabatical ly influenc ing systems do change, then the funct ions cpv of the [15) systems cons idered change explicit ly with time, with equat ions (1 ) maintai ning

58

FOill\DATIONS OF THERMODYNAMICS

their validity at all ti mes . Such a. change of the distribution of states of the system under cons ideration we cal l an adiabat ic onP.
We now consider a second kind of changes of the state of a system E. Consider a syst em t hat can be infl uenced adi abat ically . WE' assume that at time t = 0 the syst em E enters into such an interaction with a system P of a di fferent t emperatu re that we call ed "in contact" above, and we remove t he syst em P after the t ime necessary for the equalizat ion of the temperat ures of E and P. The energy of E has t hen changed . The equations (1) of E are inval id duri ng t he process but vali d before and aft er it, while the fun ct ions cpv are t he samP before and after t he process . Such [16] a process we cal1 11 i sopycni c" and the energy supplied to E, "heat suppl ied."
It is evident that, up to the inf initesimal l y smal l, it is possi bl e to construct each inf initely slow process f rom a succession of infi ni t esimal l y smal l ad iabatic and isopycnic processes , so t hat in order to get a general overv iew we have to study the latter ones only.

§6. On th e concept of en tropy

Let t here be a phys ical systemwhose instantaneous state shall be

completel y det ermined by t he values of t hP stat e var iables Pi··· Pn • Let t his

system undergo a small , inf initel y slow process , in which t he syst ems that

i nfluence thi s syst em adiabati cally experience an infin i t esimally small change

of state, and energy i s being suppli ed to the system considered by systems in

contact. We take account of the adiabat ical l y i nf l uencing systems by

st ipulat ing t hat in addit ion t o the p1... pn' t he energy E of t he system considered shall also depend on some parameters J1,J 2. .. , whose val ues shall be determined by t he distr ibutions of st ates of the systems that i nfluence

adiabatical l y the system considered. In purely ad iabat ic processes there

holds at any instant a syst em of equations (1) whose fun ct ions cpv depend

not only on th~ coordi nates pv but al so on the slowly changing quanti ties

J; for ad iahat.ic processes t oo, there wi l l hol d at any i nstant t he energy

equation, whose fo rm is

~
l

~ at

cpv

=

O

•

II

DOC . 4

59

We now i nvestigate the energy increase of t he systemdur ing an arbitrary infinit esimal l y smal l, infin i tely slow process .
For each time el ement dt of the process we have

(4)

dE = l\' oIJXl' d). + l\' 1IfJiE";, dpv •

For an infinites imally smal l isopycnic process, all d). van ish i n each timP element, and thus the fi rst term of the right -hand side vanishes too. However, since according to the previous section, in an isopycnic process dE i s to be considered as heat supp lied, for such a process the heat suppl ied dQ is represented by the express ion

However, for an adiabatic process , during wh ich equations (1) are always valid, we have, according to the energy equation,

On the other hand, according to the previous section, dQ = 0 for an adiabatic process, so that one can put

dQ ='loIJpE dpv
II

for an adiabatic process as well . Hence, this equation must be considered as

valid for any arbitrary process duri ng each t ime element . Thus equation (4)

becomes

(4 I )

l dE = Md). + dQ.

This expression represents the energy change of the system occur ring during t he whole i nfinitesi mal ly smal l process at changed values of d). and dQ as well .

60

FOUNDATIONS OF THERMODYNAMICS

At the beginn ing and t he end of t he process , t he distribut ion of states

of t he system considered is stationary, and when the system is i n contact with

a system of r el at ively infinit ely large energy before and after t he process ,

t his assumption having formal s ign if icance only , this distribution is defined

by the equat ion having the form

d/1 = const.

e -

2hE

•dp1.

.

.

d p 11

ec- 2hE•dP1 •• •dPn '

where dV denot es the probabi lity that the val ues of the system' s state variables lie with in t he limits indicated at any arbitrar ily chosen moment . The const ant c is defined by t he equat ion

(5 )

Jec-2hE•dP1 .. . dPn = 1 ,

where t he i ntegrat ion has t o be extended over al l values of t he variables. Specif ically, if equation (5) holds before the process under
cons ideration, then af t erwards we have

(5 I)
and t he t wo last equat ions yi eld

or, since the expression in parentheses can be t aken as a constant during

i ntegration because t he system's energy E never differs markedly from a

fixed average val ue before and aft er the process , and taking into account

equation (5 ),

(5 ")

l de - 2Edh - 2h M d~ = 0

However , accordi ng t o equation (4 1 ) we have

DOC. 4

61

l -2hdE + 2h Md~+ 2hdQ = 0,

and by adding these two equat ions one obtains

or, s i nce 1/ 4h = K. T,

2h ·dQ = d(2hE - c) ,

This equation states that dQ/ T is a total differential of a quantity that we wi ll cal l the entropy S of the system. Tak ing into account equation (5) , one obtains

where the integrat ion has to be extended over al l val ues of the variables .
§7. On the probab i l i ty of distribut i ons of states
In order to derive the second law in its most general form, we have to
i nvestigate the probabil i ty of distributions of states. lrle consider a very large number (N) of i sol ated systems , all of wh ich
can be represented by the same system of equations (1) , and whose energi es coincide up to the i nfin itesi mally small. The distribution of states of these
N systems can then be represented by an equation of the form
{2 I)
where in general f depends explicitly on the state variabl es p1. . .pn and also on t i me . Here the function f compl etely characterizes the distribut i on of states.
It fo l lows from §2 that when the di stribut ion of states i s constant, which, according to our assumpt ions , i s always the case at very l arge values of t, we must have f = const., so that for a stat ionary distribution of states we wi l l have

62

FOliNUATIONS OF THER\IODYNAMICS

From th is it follows immediately t hat t he expression for the probabilit y dfl for t he values of the state variables of a. system r andoml y chosen from among t he N syst ems to li e in t he infinitesimally small r egion 9 of the state var iables located within the assumed energy limits is given by

Thi s proposition can al so be formulat ed as follows : If the whole per t inent region of state variables that is det ermined by the assumed energy limi t s i s divided into € partial r egions 91.92... 9€ such t hat

and i f one denotes by v1, v2, et c., the probabil ities that t he val ues of the
state variables of the arbit rarily chosen syst em lie wi th in 91,92.. . at a certain instant, t hen
The probab il ity that at a given moment the system considered will belong to a spec ific regi on from among t hese g1. .. g1 regions is thus just as great as t he probabi lity t hat i t will belong t o any other of t hese r egions .
The pr obability that , at a randomly chosen t ime, f l of l'/ systems considered wil l belong to the region g1, f 2 t o region g2, .. . tf to regi on 91,• is hence
or al so, s i nce f 1,t2... fn are to be thought of as very large numbers: t=l
l log fl = const. - f l og t .
f =l

DOC. 4

63

If f, i s suff iciently large , one can put wi thout noticeable error

log fr = const. - J f l og f dp 1... dpn .

In t his equation V denotes the probabi lity that a given di stribution of

states , which is expressed by t he numbers t: 1,t:2. . . fl, or , else, by a speci fic funct ion f of p1··· Pn according to equation {2'), prevai ls at a given t i me.

If in this equation f were const ant, i .e. , independent of the pv' s

within the energy l imits cons idered, then the di stribution of states con-

si dered would be stat ionary, and, as can easily be proved, t he expression for

t he probability f of the di str ibution of st ates wou ld be a maximum. If f

depends on the values of t he pv ' s, t hen it can be shown that t he expression

for log V for t he distribution of states considered does not have a~

extremum, i .e . , that there ex ist distributions of states differing

infin it esimally from the considered one for which fr i s l arger.

If we follow the N systems considered fo r an arb i trary t ime i nterval,

t he distribution of states , and t hus al so V, wi ll cont inual ly change wi t h

time, and we will have to assume that always mor e probabl e distributions of

st ates will follow upon improbabl e ones , i. e . , t hat II increases until t he

di stribut ion of states has become constant and f a maximum.

[ 17]

It will be shown in the following sections that the second law of

t hermodynam ics can be deduced from this proposition .

First of all, we have

where the function f determines the distri bution of states of the N systems at a certain time t, t he function f 1 determi nes t he distribut ion of states at a certain later t ime t', and the integration on both sides i s t o be extended over all val ues of the variables. Further, if the quant i t i es log t: and log t: 1 of the ind i vidual systems fromamong the N systems do not differ markedly from each ot her, then, since

64

FOUNDATIONS OF THERMODYNAMICS

J J (dp 1... dpn = f ' dp1.. .dpn =N,
the last equat ion becomes

(6)

- log ( 1 ~ - log ( .

§8. Appl icat i on of the results obtained to a part i cular case
We consider a finit e number of physical systems u1, u2. .. that together form an isolated syst em , which we shall call total system. The systems u1,u2... shall not interact marked l y with each other thermally, but t hey might affect each other adiabat i cal l y. The distribution of states of each of the systems u1,u2. . . , which we shall call part ial systems , shall be stationary up to the i nfin ites imal l y smal l . The absol ute t emperatures of the part ial systems may be arbitrary and different from each other .
The distribut ion of states of the system u1 wi l l not be markedl y different from the di stribut ion of states t hat wou l d hold if u1 were i n cont act wit h a physical system of the same temperature. We can t herefore represent its distribution of states by the equation

where the indices (1) indicat e aff iliat ion with the part ial system u1. Anal ogous equat ions hold for the other partial systems. Since the
i nstantaneous values of the state variables of the individual part ial systems are independent of those of the other systems , we obtain for t he distribution of states of the total system an equat ion of t he form
[ 18] (7)
where t he summat ion is to be extended over all systems , and the integration over the arbitrary region 9, whi ch is infi nitesimal l y small in all the variables of the total system .

DOC. 4

65

Ye now assume that after some time the partial systems u1,u2. . . enter into some arbitrary interaction with each other, but that during that process the total system always remains an isolated one . After the lapse of a certain time there shal l arise a state of the total system in which the partial systems u1,u2. . . do not affect each other thermally and, up to the infinitesimally small, exist in a stationary state.
Then an equat ion compl etely analogous to that holding before the process wil l hold for the distribution of states of the total system:

(7 ' )
We now cons ider N such total systems . Up to the inf i nites imally smal l , equation (7 ) shall hold for each of these systems at t ime t, and equation (7' ) at time t' . Then the distribution of states of the N total systems considered at times t and t' wil l be given by the equat ions

To these two di stributions of states we now appl y the resul ts of the prev ious section . Neither the

nor the

E'

=

~
N•el

(

c'
v

-

2h ' E' ) v v

for the individual systems among the N systems are here markedly different, so that we can apply equation (6), which yields
l l (2h'£' - c ') ~ (2h£ - c) ,

66

FOUNDATIONS OF THERMODYNAMICS

or, noting that accor ding to §6 t he quantities 2/i1E1 - c1, 2li2E2 - c2, . . . are
identical with t he entropies s1,s2.. . of the partial syst ems up to a
un i versal constant ,
(8)
i .e., t he sum of the cntrnpi es of the part ial systems of an i solated syst em after some ar bitrary process is equal t o or larger t han t he sum of the entrop ies of the par tial systems befo re the process.

§9. /Jeriva t ion of llll' second law
Let there be an iso l ated total system whose parti al systems shall be called /I, JI, and E1,E2.... Let t he system II, which we shall call heat reser voir , have an energy t hat is in-finitely large compared with the system JI (engine). Simi larl y, the energy of the systems E1,E2.. . , wh ich interact adiabatically with each other, shall be infinit ely large compared wi th that of JI. We assume that all the partial systems JI, II, E1, E2... are in a stat ionary st ate .
Suppose that the engine It passes through a cyclic process dur ing which i t changes the di str ibutions of states o-f t he syst Pms E1,E2... infi nitely slow ly through ad iabatic influence, i. e. , perfo nns work, and receives the amount of hrat Q fromthe system fl. The rec i pr ocal adiabatic influence of the systems E1,E2... at t he end of the process will t hen differ from that before the process. We say that t he engine J/ has convert ed the amount of heat Q into worl<.
We now calculate the incr ease in ent ropy of t he individual partial systems duri ng t he process considered. Accordi ng to the result s of §6 t he entropy incr ease of t he heat reservoir II equals -Q/ 1' if T denotes the absolute temperature. The entropy of JI is the same before and af ter the process because the system f/ has undergone a cyclic process. The systems E1,E2... do not change thei r entropies during the process at all because these systems onl y exper ience an adiabatic infl uence that is infinitely sl ow. Hence t he ent ropy increase S' - S of t he total syst em has the value

DOC . 4

67

S' - S = - j .
Since according to the results of the last section this quant ity S' - S is always ~ 0, it follows that

This equation expresses the impossibility of the existence of a perpetuum mobile of t he second ki nd.
Bern, January 1903. (Received on 26 January 1903)

68

GENERAL MOLECULAR THEORY OF HEAT

Doc. 5
ON THE GENERAL MOLECULAR THEORY OF HEAT
by A. Einstein [Ann al en der Physik 14 (1904): 354-362]

In t he foll owing T present a few addenda t o an article I published last year. 1
When I refer to t he "general molecular t heory of heat," I mean a theory that i s essentially based on the assumpt ions put forth in §1 of the articl e cited. In order t o avo id unnecessary repet itions, I assume t hat the r eader i s familiar wit h that article and use the same notations I have used there .
First , I derive an express ion for the entropy of a syst em , which i s [ 2] completely analogous to the express ion found by Boltzmann for ideal gases and [3] assumed by Pl anck in his t heory of rad iat ion. Then I give a simple derivation
of t he second l aw. After t hat I exam ine t he meaning of a universal constant, [4 ] which plays an import ant role in the general molecular theory of heat. I
conclude with an application of the t heory t o black-body radiation , wh ich yields a most i nt erest ing relationship between t he above-mentioned universal [ S] constant , which is determi ned by the magnitudes of t he element ary quant a of matter and electricity, and the order of magnitude of t he radiation wavelengths , without recourse t o special hypotheses .
§1. On the express i on for ent ropy
For a system t hat can absorb energy only i n the form of heat, or, in other words, for a system not affected adiabat i cally by other systems , t he fo l l owing equation holds between the absol ute t emperature T and the energy E, accord ing to §3 and §4, loc.c i t.:
[1] 1A. Ei nste in, Ann. d. Phys . 11 (1903) : 170.

DOC . 5
(1)
where K denotes an absolut e constant and w is defined (sl ightly differently than in the articl e cited) by the equation

69
[6] [7]

The i ntegral on the right i s to be extended over all values of t he state variables that completely and uniquely define the instantaneous state of t he syst em, and to which correspond val ues of the energy that l i e between E and E + 6£.
From equation (1) it fo l lows that
Jf S = = 2Klog [w(E)] .

Omitting the arbitrary integrat ion constant, the expression thus represents

the entropy of the system. Thi s expression for the entropy of a system holds

not only for systems that experience purely thermal changes of state, but also

for systems that pass through arbitrary adiabatic and isopycnic changes of

[8]

st ate.

The proof can be deduced from the last equation of §6, loc. cit.; I omit

it because here I do not intend to present any application of the l aw in its

general signif i cance.

§2. Deri vation of the second law
If a system is located in an environment of a given constant temperature 10 and is in thermal interaction ("contact") with this environment, then, as experi ence shows, it too assumes the temperature T0 and maintai ns this temperature T0 for all times.
However, according t o t he molecular theory of heat, this law does not hold strictly, but rather in a certain approximation - even t hough this approxi mat ion is very good for all systems access ible t o direct investigat ion.

70

GENERAL MOLECULAR THEORY OF llEAT

I f the system cons idered has been in t hat env i ronment for an inf i nitely long t ime , the probabi lity W that the val ue of the system's energy lies between the limits E and £+ 1 at an ar bitrarily chosen instant (§3, loc . cit. ) will be

where C is a constant. This value is different fromzero for every £ but has a maximum for a certai n I, and - at least for all systems access ibl e to direct investigat ion - is very smal l for any nppr ec iabl y larger or smaller E. We call t he system 11beat reservoir" and assert in bri ef: the above expr ession represents the probab ilit y that thf' energy of the heat reservoir in question wil l have the value £ in the environment menti oned . Using the resul t of t he previous sect ion, we can also write
1-(s - £]
It' = Ce2" "'T; ,
where S denotes the entropy of the heat reservo i r. Let t here bP a number of heat reservoirs , all of them i n the environment
{10 ] at temperat ure T0 • The probabil ity that the energy of the first reservoir wi ll have the valuf' £1, t he serond the val ue £2... , and the last the value El, is , then, in an eas ily understood notation,

{11 ] (a)
Let these reservoirs enter into interact ion with an engine that passes through a cyclic process. Assume that during this process no heat exchange takes place eit her b<>tween the heat reservoir and the environment. or between the engine and thP environment. Aft er the process cons idered, l et the energies and entro1>i es of the systems be, respect ively,

DOC. 5

71

and

The probability of the tot al state of the heat reservoir defined by t hese

[ 12 ]

values will be

(b )

Neither t he state of the environment nor thr state of the engi n£' has

changed during the process , because the latter underwent a cyclic process.

If we now assume that less probable states never follow the more

probable ones , we have

W' ~w.

[13]

But we also have, accordi ng to the energy principle,

If we t ake this into account, t hen i t fol l ows from equations (a) and (b) that

S. On th e meaning of the constant K i n the kinet i c theory of atoms

[l4]

Let us consider a physical system whose instantaneousstate i s completel y determined by t he values of the state variabl es

72

GENERAL MOLECULAR THEORY OF HEAT

If the system considered is in "contact" with a system of r elatively i nf in itel y large energy and of abso lute t emperature 10 , then its di stribut ion of states i s deter mined by t he equation

In this equat ion K. is a universal constant whose meaning shall now be examined .
On the basis of the ki net ic t heory of atoms , one arrives at an interpretation of t his const ant in t he following way, famili ar from Boltzmann' s [15) works on the theory of gases.
Let t he Pv' s be t he ort hogonal coord inates x1y1z1,x2y2. . . ,xnynzn, and {1~1(1,{2q2... ,{nqn(n the veloc it i es of the i nd iv idual atoms (cons idered to be point like) of the system. One can choose these state variables because
l [1 6) they satisfy the condi tion Or.pvf 8pv =0 ( loc. cit . , §2 ). One has then :
lnm
E =t (x1·· ·zn) + f({i + n! + (! ) ,
1
where the first summand denotes the potent ial energy and the second the kinetic energy of t he system. Let now an i nfinitesimal l y small region dx1.. . dzn be given . We fi nd the mean value of t he quantity
corresponding to this region:

DOC . 5

73 (17 ]

This quantity i s thus i ndependent of the choice of the region and the choice of the atom, and hence is in general the mean value for the atom at t he absolute temperature 10 . The quantity 3K equals the quotient of the mean kinetic energy of an atom and the absolute temperature . 1
Further, the constant K is closely connected with the number N of
true mol ecul es contained in one molecule as the chemists understand it (equivalent we ight based on 1 g hydrogen as unit).
It is well known that for such a quant ity of an ideal gas, and with gram and centimeter used as units, we have

pv = BT, where B = 8.31 x 107 .

(19]

Accord ing to t he kinetic theory of gases, however,

pv = '23 NL- ,

1Cf . L. Bol t zmann, Yorl . uber Castheori e [Lectures on the theory of gases]

(18]

£ (1898): §42.

74

GENERAL MOLECULAR THEORY OF HEAT

where l denot es t he mean value of the kinetic energy of motion of the center
of gravity of a molecule . If one also takes int o account t hat

one obtains

l, = l,
V

Hence t he constant 211: equals t he quot ient of the constant R and t he number

of molecul es contained in one equival ent.

[ 20]

If. i n accordance w.ith O. E. Meyer. one sets N = 6.4 x 1023 , one gets

( 21 ] l'i, = 6. 5 )( 10-11.

§4 . Th e general s ign i f i cance of the const an t K

Let a given system be in contact with a systemof rel at i vel y infi nitely large energy and temperature T. The probability dll t hat t he value of i ts energy wi ll lie between £ and £ + d£ at an arb itrari l y chosen instant is

E

(2 2 ]

dll

=

C;

2 K

1wEdE

For t he mean value £ of E one obtains

Si nce , f urther ,

we get

Different iation of t his equation with r espect to T yields

DOC . 5

75

[23)
This equation states that the mean value of the bracketed expression vani shes, and hence

2K'f2 adT£ = P- . - E--l

[24)

I n general, the inst antaneous value E of the energy differs from £ by a
certain amount, wh ich we call "energy fluctuation" ; we put

lt'e then obt ain

,r:>,- _ EE- = .£...~.. = 21,,P. adET

[25)

The quantity ~ is a measure of the thermal stabil i ty of the system; the
larger the ~, the less this stability. Thus the absolute constant K determi nes the thermal stab i lity of t he
system . The rel ationship just found is interesting because it no longer contains any quantity remi ni scent of the assumptions on which the theory is based.
The magnitudes of '?',?,etc. can be calculated by successive differentiat ions wi thout any difficulty .

§5. Appl i cation to radiat ion
The last-found equat ion would allow an exact determination of the uni versal constant K if it were possibl e to determine the mean val ue of t he square of the energy fluct uation of a system; however, at the present state of our knowledge this is not the case. In fact, there i s only a single ki nd

76

GENERAL MOLECULAR THEORY OF HEAT

of physical system f or which we can surm ise from experience t hat it possesses energy fl uct uation: th is is empty space f i l l Pd wi th t emperature radiat ion .
That i s , if the l inear dimensions of a space fi l led wi th temperat ure radiation are very large in comparison with the wavelength corresponding to the maximum energy of t he radiation at the t emperature in quest ion, t hen the mean energy fluct uati on wil l obviously be very small in comparison with t he mean radiation energy of that space. In contrast, if the radiat ion space i s of the same order of magnitude as t hat wavelength, t hen the energy f l uctuation will be of t he same order of magni tude as t he energy of t he radiation of the radiat ion space .
Of course, one can object that we are not permitted to assert that a radiati on space should be viewed as a sys t em of the kind we have assumed, not even if t he appl i cabi lity of t he general molecular theory is conceded. Perhaps one wou ld have t o assume , fo r example, t hat the boundar ies of the space vary with i t s elect romagnet ic states . However, t hese circumstances need not be considered, as we are dealing with orders of magn i tude only .
If, then , in the equation obt ained in the last section, we set
[2 6)
and accord i ng to t he Stefan-Boltzmann law

[27]

E = cvT4 ,

where v denotes the volume i n cm3 and c the constant of this law, then
we must obtain for 3,fv a value of the order of magnitude of the wavelength of the maximal radiation energy that corresponds to t he temperature in quest ion.
One obtains

where we have used for K t he val ue obta ined fromt he kinetic theory of [28] gases, and 7.06 x 10-15 for c.

DOC. 5

77

If Jm i s the wavelength of the energy max imum of the rad iation, then experiment yields
[29)

One can see that both the kind of dependence on the temperature and the

order of magnitude of Jm can be correctly determined from the general

mol ecular theory of heat, and cons ider ing the broad general ity of our assump-

t ions, I believe that th i s agreement must not be ascribed to chance.

(30 )

Bern , 27 March 1904 . (Recei ved on 29 March 1904)

78

REVIEW OF BELLUZZO

Doc . 6
Rev iew of G. DELI,UZZO, "Principles of Graphic Thermodynamics"
("Pr i ncipi di t ermod i nami ca grafi ca, " Il Nuo vo Cimento 8 (1904 ): 196- 222, 241 - 263)
[Deiblatt er zu den Annal en der Physik 29 (1905 ): 235]

This articl e , which is obviously meant for engineers , i s di vided into fou r sections , t he fi r st of whi ch treat s graphically the changes of state of arbitrary fl uids. Thus , the fam i l iar areal const ruction of t he work performed (l ) by t he body, of the energy increase (6£) , and of t he heat absorbed {C) are given in the pv-plane in §3, while in §1 and §5 t he increase of entropy for an arbitrary change of state is presented as an area with G and T (the absol ute t emperature) , and wi t h C and 1/ T, respectivel y, as coordi nates. Th is i s followed by the theory of cyclic processes and t he def i nition of revers ib ility and irreversibi lit y of the processes . A process is considered to be reversi ble or irrevers ib l e, respectively, drpending on whether the pressure exerted on the fluid duri ng t he process does or does not equal the i nner pressure of the fl uid; th is sti pu l at ion, which , by the way, is irrelevant fo r what fo llows , does not make sense , because then the princi pl e of t he equalit y of action and r eaction would not be satisfied in any i rreversi ble process . Thr second section of the article contains t he application of the theory to ideal gases; examined ar e the changes of state at constant volume , const ant pressure, and constant temperature, as well as t he adi abati c and polyt ropic cha.nge of state. fhe last section deal s with the efflux of gases through pipes; t he hypot hesis of Saint-Venant and Wantzel i s [l] replaced by (already known) theoret ical considerat ions . The t hird and fourth sect ions of t he articl e contai n t he t heory of the saturated and the superheated water vapor, which ar e t reated in a corresponding way, with special cons ideration gi ven to t he theory of t he efflux of water vapor t hrough pipes and to t he t heory of improving the ef fi ciency of steam engines by superheating. For the equat ion of stat e for wat er vapor, p ( v+ const .) = [ 2 ] const. T i s used, following Battell i and Tumlirz.

DOCS. 7 & 8

79

Doc. 7
Revi ew of A. FLIEGNER, "On Claus ius's Law of Entropy" ("Uber den
Claus ius' schen Entropi esatz, " Naturforsch ende Cese l lschaf t in Zu ri ch . Yierte l jahrsschrift 48 (1903) : 1-48 )
[Beiblat t er zu den Anna len der Physik 29 (1905): 236]

The aut hor examines t he entropy changes of a system during a process

presumed t o be st r ictly discont i nuous (di scontinuous mcpansion of a fl uid) and

concludes f rom his calcu lations t hat the entropy decreases at t he beginning of

the sudden expansion. Cons iderati ons concerning i rrevers ibl e chemical

processes l ead t he author to t he conclusion t hat t he equat ion dQ/ T S dS

holds onl y for exother mi c but not for endothermic processes. Similarly, the

equation i s not supposed to hold for cool ing mixtures. It i s t herefore

underst andabl e t hat the aut hor closes wi th t he following sentence: "Thus , t he

question of whether the entropy of the universe does change at al l, and if it

does , then in which sense , cannot yet be answered at all at present , and wi ll

pr obably r emain undecided forever."

[ l]

Doc. 8 Review of W. McFadden ORR , "On Clausius' Theorem for Irrevers i ble Cycles ,
and on the Tncrcase of Entropy" (Philosoph ica l Jfaga zi ne and Journal of Science 8 (Series 6) (1904): 509-527)
[Beib la t t er zu den Annalen der Physik 29 (1905): 237]

The author shows t hat in t he Vorlesungen uber The rmodynamik [Treatise on

Tliermodynamics] Planck appl ies t he concepts "revers ible" and "irreversible" in [l]

a sense somewhat different from that in which he defines them. Then he

advances a series of objections t hat may be raised against various ways of

(2 )

representing the foundations of t hermodynami cs ; especially notewortl1y among

t hese objections i s t hat by Bertrand , i. e. , that t he pressure, temperat ur e,

[3]

and entropy a.r e defi ned only for t he case tl1at at least suff icieutly small

80

REVIEW OF BRYAN

parts of a system can be regarded as being in equilibrium; a simi lar objection i s rai sed with respect to the heat supplied.

Doc . 9 Review of G. H. BRYAN, "The Law of Degradation of Energy as the Fundamental Pr i nci ple of Thermodynamics" ("Das Gesetz von der Entwertung der EnPrgie als Fundament al prinzip der Thermodynamik, 11 in lleyer , S. , ed . , Fes tschrift.
Ludwig Boltzmann gewidme t zum sechzigsten Ceburts tage 20 . Feb ruar 1904 . (Leipzig: J.A. Barth, 1904) : 123- 136)
[Be i blatter zu den Annal en der Phys ik 29 (1905 ): 237]
The author starts out f rom the energy principl e as wel l as the pri ncipl e [ 1] of t he decrease of free energy. The free energy (available energy) of a
system i s defined as the max imal mechanical work that the system can perform during changes compatible with the ext ernal conditions. This is fol lowed by the defin ition of heat suppli ed to the system. Then the concept of t hermal equilibrium, t he second law, the concept of absolute temperature, and the concept of energy arP developed from the st ated fundamental princ iples in an elegant way, and, finally, the equations of thermodynamic equilibri umare der ived.

DOC . 10

81

Doc. 10 Review of N. N. SCHILLER, "Some Concerns Regarding the Theory of Entropy Increase Due to the Diffusion of Gases Where the Initial Pressures of the
Latter Are Equal " ("Einige Bedenken betreffend die Theorie der Entropievermehrung durch Diffusion der Gase bei einander gleichen Anfangsspannungen der letzteren, 11 in Meyer, S., ed., Festschrift. Ludwig Boltzmann gewidmet zum sechzigsten Ceburtstage 20. februar 1904.
(Lei pzig: J.A. Barth, 1904): 350-366)
[Beiblatter zu den Annalen der Physik 29 (1905) : 237]

First it is shown that a homogeneous gas can be reduced i sothermally to an n-time smaller volume wi thout suppl y of work and heat if one assumes the existence of wal l s that are permeable by a part of the mass of a gas but not by the rest of the mass of the gas; according to the author, this assumption [I] does not contain any contradiction. Then it is demonstrated that the expression for the entropy of a system consisting of spatially separated gases of equal temperature and pressure has the form
[2)
the entropy of the system after diffusion can be represented by the same formula. From this it is concluded that the entropy is the same before and after diffusion . The author arrives at the same result by a line of reasoning [3] that cannot be reproduced here. In this line of reasoning one operates with a surface that separates a chemically homogeneous gas into two parts such that in thermal and mechanical equilibrium the gas pressure in the two parts is different; it is (implicitly) assumed that during the passing of the gas through this surface no work is transferred to the gas by the latter.

82

REVIEWS OF WEYRAUCH AND OF VAN 'T HOFF

Doc . 11 Review of J . J. WEYRAUCH , "On t he Specific Heats of Superheated Water Vapor"
("Uber di e spezif ischen Warmen des iiberhitzten Wasserdampfes ," Zeit schri f t [l] des Vereines deut scher Ingeni eure 48 (1904 ): 21-28, 50-54. Reprint , 9 pp. )
[Beibla tt er zu den Anna len der Ph ysik 29 (1905) : 240]

Determinat ions made thus far of the specific heat cp are presented and [2] compared (I). Equations of state for water vapor sui t able for practical [3) appli cation are presented and discussed (II ) and, using t hose by Zeuner, cp
and cv for sat urated steam (I I I) and cp and cv for arbi t rarily superheated steam are der ived t hermodynamical ly . Then t he total heat and the steam heat are det ermined (V). In (VI) and (VII) there follow the fundamental equat ions of the theory of heat for superheated steam , t he ir application to special cases , and several numerical examp les.

Doc. 12 Review of J . IL van't HOFF, "fhe Influence of t he Changes in Specif ic Heat
on the Work of Conversion" ("Einfluss der Anderungen der spezifi schen Warme auf die Umwandlungsarbeit, 11 in Meyer, S. , ed., Fest schrift. Ludwig Bolt zmann gewidmet zum sechzigst en Geburt s tage 20. Februar 1904 . (Leipzig:
J .A . Barth, 1904) : 233-241) [Bei blatt er zu den Anna l en der Phys i k 29 (1905 ): 240]
The author shows by way of t hermodynamics t hat the work of convers ion E (supplied to the surroundings) of a syst em A into a system B (e.g., by melt ing) in isothermal convers ion can be represented i n t he form
£ = Eo + AT - ST l g T.

DOC. 12

83

(A i s a constant, T the absol ute temperature, S = SA - S0 the difference between the specific heats , wh ich are assumed to be independent of T. For reasons of analogy (because during isothermal expansion of a gas E = AT = 2T [ l]
lg (v0/ vA)) , AT is regarded as determined by change of concentration .
The equation is applied to experiments of Richards, who for convers ions [2 ] of the kind

\lg+ ZnSO4.aq = Zn + MgSO4 .aq
(where the initial ZnSO4 and the .MgSO4 formed have the same concentration) by the electric method found that

ndE =- KS,

where K is approximatel y t he same for all convers ions exam ined . Omitt i ng
the term AT, the author obt ains from t he above equat ion

adEl = - S(1 + lg T) - 6. 7 S .

[3]

Mean val ues of observat ions yielded :

[4]

Reaction
Mg+ ZnSO4 }lg + CuS04 ~lg + Ni SO4 Mg + FeSO4 Zn+ FeSO4

[#] ! (-S)
5 5.4 5.9 6.3 7.3

Reaction
Zn + NiS04 Fe + CuSO4 Ni+ CuSO4 Zn+ CuSO4 Fe+ NiSO4

[#] /(-S)
8 7. 5 7 7.4 7.1

The equation for E, applied to fus ion as wel l as to conversion of allot ropic element s and pol ymorph ic compounds (again neglecting t he term AT) fur ther yiel ds the proposition: The form which i s stable at the higher t emperature (e.g., liqu id} has the higher specif ic heat. This conclus ion i s almost always conf irmed by experiment. Finally, i t is concl uded from the

84

REVIEW OF GIAMMARCO

[5] equation that the Thomson-Berthelot rule must be valid at low temperatures, but t hat at higher t emperatures the term -ST lg T may cause deviations when
SA> s8•

Doc. 13

Review of A. GIAMMARCO, "A Case of Corresponding States in Thermodynamics"

("Un caso di corrispondenza in termodinamica, " Il Nv.ovo Cimento 5 (5 )

(1] [2]

(1903 ): 377-391)

[Beiblatter zv. den Annal en der Physik 29 (1905 ) : 246]

If one has a liquid (volume v) in a closed cyl indrical tube and above it its saturated vapor (volume v'), and one plot s v/v ' as a funct i on of the absolute temperature T in orthogonal coordinates, one obtains , depending on the amount of the enclosed substance, a curve that has a maximum (v/v')max' or a curve that i s convex toward the abscissa, or one (as the limiting case) that approaches the critical t emperature l inearly . The author investigated ether, alcohol and chl oroform in this way and finds that the above maxima (v/v ')max l i e on a straight line. Accord ing to t he law of corresponding states , two temperatures T and T' at whi ch two different substances have the same (v/v' )max must be correspond i ng temperatures (the method for the determination of corresponding temperatures) , hence T/ Tc = T'/T~ . Using the (absolute ) critical t emperatures of ether (467° ) , alcohol (517°) , chloroform [3] (541° ) (Bureau des Longitudes , 1902) , the author f inds from his observations:

DOC. 13

85

Corresp. abs . temperat ures

( Y/Y' )max

T
~

Ether Alcohol Chloroform Ether Alcohol Chlorofor m

387° 428.07

447.09 0.320 0.320

0.330

0. 828

391 432.8

452.8

0.340 0.340

0.350

0.837

394 435.8

456

0.355 0.356

0.360

0.843

404 447

467.9

0.395 0.400

0.409

0 . 865

C4 l

414 456 .5

478

0.440 0.440

0.448

0 . 883

423 468 .2 427 472 . 7 437 485.3 458 506 .6 467 517

489 .6 494.4 505.8 530 541

0.490 0 . 510 0. 556 0 . 655 0.695

0.490 0.510 0.556 0.652 0.698

0.495 0 . 511 0 . 556 0.652 0.698

0.905 0.914 0 . 93 5 0 . 981 1

Examining the curve that const i tutes the l imi ting case, the author f i nds that the disappearance of the men i scus dur ing heat ing and i ts appearance dur i ng cooling occur at the same temperature (the crit i cal temperature ).

86 [ 1]

HEURISTIC VIEW OF LIGll1
Doc. 14 ON A HEUR ISTIC POINT OF VIEW CONCERNmG THE PRODUCTION
AND TRANSFORMATION OF LIGHT by A. Einst ein
[Annalen de r Phys ik 17 (1905) : 132- 148]

There exists a profound formal di fference between the t hPoret ical concept ions phys ic i sts have formed about gases and ot her pondcrable bodies , and Maxwell 's t heory of el ectromagnetic processes in so-cal led empt y space. While we conceive of the stat e of a body as being completely determ ined by the pos it ions and vel ocit ies of a vPry large but nevertheless finite number of atoms and electrons , we use conti nuous spatial f unctions to determine the electromagnetic state of a space, so that a fin ite number of quantit ies cannot be cons idered as sufficient for the complete descr ipt ion of the el ectromagnetic state of a space . According t o \laxwell 's theory, energy i s to be cons idered as a cont i nuous spatial fu nct ion for al l pure l y electromagnetic phenomena, hence al so for light, while according to the current concept ions of phys icists t he energy of a ponderable body is to be described as a sum [2 ] extending over the atoms and electrons . The energy of a ponderable body cannot be broken up into arbitrar i ly many, arbitrarily small parts , while accord i ng to Maxwe ll' s theory (or, more generally, accord i ng to any wave theory) the energy of a light ray emitted from a po int source of light spreads continuously over a steadily i ncreasing volume .
The wave t heory of light, whi ch operates with continuous spatial f unc[3) t ions, bas proved it self spl end idly in describing purely opt ical phenomrna and
wi ll probably never be replaced by another theory . One tshould keep in mi nd, however, t hat opt ical observat ions apply to ti me averages and not t o momPnt ary values, and i t i s conceivable that despite the complete conf irmat ion of the theories of diffract ion, reflect ion, r efract ion, dispersion, et c. , by experiment, the theory of li gh t, wh ich operatrs with continuous spat ial fu nctions , may l ead to cont radict ions with experi ence when it is applied to t he phenomena of product ion and t ransformation of 1ight .
Indeed, it seems to me that the observations regarding "black- body [41 rad iation," photol uminescence, production of cathode rays by ultraviolet