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 Thermork 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 [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 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 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 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 quan- tity h1, 2· If the interact ion between i nfinitely slowly, t his does not change I:1 the exapndresIs:2ionisforimtahgei ntween 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