Walther Nernst, “Thermodynamic Calculation of Chemical Affinities” Excerpted from Berichte Der Deutschen Chemischen Gesellschaft, Vol

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Walther Nernst, “Thermodynamic Calculation of Chemical Affinities” Excerpted from Berichte der Deutschen Chemischen Gesellschaft, Vol. 47 (1914), pp. 608-635.1 General Comments phenomena of steam formation, fusion, and the transformation of various modifications, the Among the various changes in energy, same gap exists. those that are associated with the course of The principal fault in the ideas considered chemical processes have always been up to now seems to lie in the fact that, for conspicuous because of their magnitude. example, in the consideration of steam Therefore, the application of classical formation, one simply calculated the work to thermodynamics to processes of this type has be done in the dislocation of a molecule of met with particular success. Nowadays, no liquid from the interior of the liquid phase into intensive investigation of a chemical the gaseous space, using the theory of equilibrium is regarded as complete if it does potentials. This formulation is questionable, not make use of the two main laws of heat however, because even a small change in the theory. state of motion of the molecule can To be sure, a purely thermodynamic undoubtedly exert a great effect under certain approach will never be entirely satisfactory, if conditions; but we do not know how this is to only because the question of reaction velocity be taken into account—a fact which, of course, is entirely outside its field; and one will always cannot justify the silent neglect that has have to make sure that, in addition to generally been practiced up to now. thermodynamics, the principles of atomistics Of no lesser significance is a second are included. Atomistics has explained a circumstance that plays a decisive role number of processes in an entirely satisfactory specifically in the chemical processes manner, especially in the form of the kinetic themselves, for, as Planck has shown, the laws theory of gases, as, for example, the work of mechanics undergo a thorough done by a gas in its expansion, or the transfer transformation if one is dealing with the of heat through a gas. But if we ask about the motion of atoms about their resting position. accomplishments of atomistics in the Here, also, we can only say at this time that the mechanical explanation of chemical processes, neglect of Planck’s quantum theory (or we must admit openly that everything that has perhaps of any other, future theory which also so far been attempted in this field has not only leads to Planck’s radiation formula) had to remained incomplete but must be considered place the stamp of incompleteness on all basically faulty. previous attempts at explaining chemical Certainly the explanation of the law of processes mechanically. Certainly, we do not constant and multiple proportions and the know yet how the new views should be taken marvelous systematics, especially of organic into account in this matter; nevertheless, we compounds, are tremendous achievements of have made progress in being able to state with atomistics purely in the field of chemistry; but certainty now: Something mysterious is hidden these applications are not of a mechanical within the laws of atom mechanics which is nature and have hardly any relation to the explained in part by radiation theory, in part manner in which two atoms combine in a by the more recent studies of specific heats, compound, with the magnitude of the forces and which apparently must be thoroughly that enter in, and with the change in energy understood before a mechanical treatment of thus determined. Not only in chemical chemical processes will be possible. Thus, we processes, but also in the probably simpler know, for example, that the laws of motion of a double star are quite different from those of a U can be looked up in thermochemical diatomic gas, and we can at least indicate tables for room temperature, and we now want broadly in what sense the laws of pure to refer to the well-known law that U can be mechanics are modified in the second case. calculated for arbitrary other temperatures There is only one temperature point at from the specific heats. which we can probably use the laws of A can essentially be determined by two mechanics safely; namely, when the motion of methods, both of which were already used by the atoms has completely stopped, i.e., at Helmholtz; namely, by measurement of the absolute zero temperature. Without doubt the chemical equilibrium or of the electromotive laws of ordinary potential theory can be force. applied here; the heat formation that Of course, dA/dT can then be found by corresponds to the dislocation of the atoms measuring A at two slightly different from one state (e.g., in the form of free temperatures. elements) to another state (e.g., in the form of Historical Background a chemical combination) can be regarded as the equivalent of the forces exerted here; in Under such circumstances, the desire to other words, at absolute zero the chemical derive A thermodynamically beyond equation affinity must be equal to the heat formation. (1) appeared quite early. The first determined Now, the second heat law gives quite a effort in this direction came from Julius general relation between the maximum work Thomsen, if one disregards Helmholtz’ A, which we previously designated as the sum method of calculating the electromotive force of all forces produced in the chemical process of galvanic elements, which will be discussed under consideration and to which we shall further below. Thomsen, in his “Contributions refer below, as usual, briefly as “chemical to a Thermochemical System,” emphasized affinity,”2 and the heat developed, U: repeatedly as early as 1852 that strong dA manifestations of chemical affinity are A –U = T accompanied by intense heat formation and (1) dT that chemical processes associated with heat Since we observed above that the left side absorption occur only rarely. He therefore of the equation disappears at absolute zero, we arrived at the following conclusion: can write: “When a body falls, it develops a certain ⎛ dA ⎞ mechanical effect that is proportionate to its lim⎜T ⎟ = 0 ⎝ dT ⎠ for T = 0. weight and to the space traversed. In chemical But, according to this equation, even at T = processes that take place in the usual direction, 0, dA/dT (its negative value is also called a certain effect likewise appears; but it shows “entropy”) can still possess a finite value and itself in this instance as heat formation. The can even be infinitely large; it must, however, heat formation constitutes a measure of the be of less than first order. chemical force developed in the process.” Equation (1) contains the complete We have seen above that a chemical application of the two heat laws to chemical process may not be regarded—at least not processes; as has been shown especially by above absolute zero—as a phenomenon of Helmholtz, all that the older thermodynamics attraction, comparable to the falling of a stone; was able to teach can be demonstrated clearly but we shall not hold this against Thomsen, in by means of it. Therefore it will be useful to view of the fact that attempts to use this go into it a little more thoroughly. interpretation are still being made repeatedly today in spite of the kinetic theory of heat.

2 Furthermore, Thomsen himself already paper by Helmholtz on the conservation of recognized the untenability of the above energy (1847). The method of calculation, concept in the early seventies, probably which was only indicated there, was later influenced chiefly by the results of his very carried out by William Thomson for several ingenious method for determining the affinity examples. More intensive study has shown, in between acids and bases. agreement with our earlier observations, that We know that the same law was one can indeed frequently calculate the formulated in 1869 by the second master of electromotive force of galvanic elements very thermochemistry, Berthelot, and energetically accurately from the heat formed, especially in defended by him for a long time. Berthelot’s cases where the affinity is strong, but that one formulation is the following: may in no way speak of a strict law. “Every chemical transformation which It is evident that all further progress must takes place without the interposition of a be tied to equation (1); a relationship must be foreign energy aims toward the production of found that is independent of the special nature that substance or that system of substances of the reaction under consideration, if the which develops the most heat.” uncertainty inherent in equation (1) is to be Both formulations—the older one by overcome. Thomsen, as well as the later one by Berthelot- For a certain class of reactions-namely, lead one to set A = U in formula (1) for all those where a gas is formed from one or more temperatures. It is unnecessary to give the solid substances—Le Châtelier, Matignon, and reasons for the inadmissibility of this equation Forcrand found the following approximate again in more detail, but a reference to a relationship: if Q designates the heat formed at remark by Horstmann will be useful for further constant pressure, and Tʹ, the absolute clarification. According to him, the proof of temperature at which the dissociation pressure chemical equilibrium or, what amounts to the of the gas being formed equals atmospheric same thing, of a reversible reaction was pressure, then: sufficient for refuting Berthelot’s principle. Q Since the reaction takes place in one or the = approx. 32 Tʹ . other sense, depending on the ratio of In this case, disregarding the variation of Q quantities of the reacting components, on one with temperature, the second heat law gives or the other side of the equilibrium, the Q reaction in the vicinity of chemical equilibrium ln p = − + const must proceed in one instance according to RT . Berthelot’s principle with heat formation and We recognize immediately that Le Châtelier- in the other, certainly in opposition to that Matignon’s rule gives a value of principle, with heat absorption. approximately 32 for the undetermined We have emphasized above that the integration constant multiplied by R. This rule electromotive force of a galvanic element is is only approximately valid; nevertheless, it proportional to the affinity of the process provides an important cue, and it probably providing the current. The Thomsen-Berthelot deserved more attention than it was given principle, then, can also be expressed in such a formerly. We shall become acquainted with a way that the electromotive force of galvanic more precise formulation later on. elements would have to be proportional to the Van’t Hoff set up an equation in 1904 that heat formed per electrochemical gram element. was hardly satisfactory. If one wants to satisfy It is of historical interest to emphasize that this the effect of temperature on U by the formulation is found already in the famous (seemingly!) simplest equation:

3 (2) U = U0 + a T, form, may at some time again become integrating (1) gives important.” It was particularly noticeable that for solid substances, affinity and heat (3) A = U0 + aT + a T ln T, where a is the constant of integration. Van’t formation frequently coincide. It was clear Hoff assumed that a was small. This from the beginning, on the other hand, that the hypothesis is not only arbitrary but also identification of these two magnitudes actually evidently inaccurate. For, even if we assume becomes meaningless for gaseous systems, for the case that a equals zero, we only need to the maximum work depends on the initial and alter the temperature scale—i.e., divide the final concentrations of the reacting gases, space between the melting and boiling points while the heat formation is entirely of water into a million instead of a hundred independent of these. Thus, the question arose parts—and we immediately have a finite and whether a relationship between heat formation even sizable value for a. It is hardly likely that and chemical equilibrium could be found the natural laws are guided by the fact that empirically, at least for comparable reactions Celsius divided the above-mentioned such as: temperature interval into a hundred parts and Cl2 + H2 = 2HCl that he happened to choose water as the 2NO = N2 + O2 standard substance. or: The earlier attempts to go beyond equation 2H2 + O2 = 2H2O (1) thus were unsuccessful; but at least the 2CO + O2 = 2CO2 problem had been sharply formulated. The 3O2 = 2O3 . clearest position taken, next to Helmholtz, was Therefore, together with a large number of probably that of Le Châtelier as early as in collaborators, and guided by such 1888. I want to reproduce his words here:3 considerations, about ten years ago I “It is very probable that the integration undertook the determination of equilibriums in constant, like the other coefficients of the gases, on which only very little, and usually differential equation, is a definite function of uncertain, observed material was previously certain physical properties of the reacting available. substances. The determination of the nature of Thermodynamic Considerations the function would lead to complete knowledge of the laws of equilibrium. In its application to gaseous systems, the Independently of new experimental data, it second heat law leads to the following result. would determine a priori the complete Experience teaches us that the specific heats equilibrium conditions which correspond to a vary only slowly with temperature, so that it is given chemical reaction; up to now, it has not suitable and convenient, to begin with, to been possible to determine the exact nature of assume an expression of the form this constant.” (4) c = c0 + aT + bT2 + ... If I may now discuss my part in the to be valid; c0 would thus be the specific heat solution of the problem, it seemed noteworthy at very low temperatures. to me from the beginning that, for an Furthermore, since, according to a law by erroneous law of nature, Berthelot’s rule still is Kirchhoff, the temperature dependence of the too frequently applicable to be ignored heat of reaction U is determined by the entirely, and therefore I had already specific heats of the substances participating in emphasized in the first edition of my textbook the reaction, we can also set of theoretical chemistry (1893) “that it is quite 2 3 (5) U = U0 + aT + bT + gT + ... , possible that Berthelot’s principle, in a clearer

4 where U0 is the heat of reaction near absolute dA dU lim = lim zero. (8) dT dT (for T = 0); Substituting in the equation of the reaction but it is to be noted that the above equation is isochore applicable, to begin with, only to pure solid or d ln K U = RT 2 liquid substances; at absolute zero, gases stop (6) dT being capable of existing, and the behavior of and integrating, we easily find solutions must still be investigated more closely. U0 α β γ 2 ln K = − + lnT + T + T +...+ I We recognize further that equation (8) (7) RT R R 2R combined with (1) gives where I is the constant of integration. dA dU The right-hand side of equation (7) thus lim = 0 lim = 0 contains, besides the constant of integration, dT , dT (for T = 0). only pure thermal magnitudes (heat of The relationship reaction, specific heat, or, respectively, its dU lim = 0 temperature coefficients); but the second heat dT (for T = 0) law says nothing at all about the integration teaches us that the atomic heats of the constant itself. elements and compounds must be strictly At very low temperatures, the effect of all additive at low temperatures; already in the other members disappears, and we obtain first practical applications of my theorem I RT ln K = –U0, was led to the conjecture that all of them must i.e., we can here calculate the equilibrium from converge toward very small values at low the heat of reaction U0; at finite temperatures, temperatures. The experimental and theoretical however, first the effect of aT ln T and RTI, studies of recent times have, as we know, not and then also that of the remaining members only confirmed this conjecture but even made will be noticeable. it into a certainty that the specific heats of all From the experimental point of view, we solid substances converge toward zero at low arrive at the problem that the behavior of gases temperatures. cannot be measured at lower temperatures We shall make use of this below, and we because they cease to remain capable of shall find the following equation to be strictly existing in significant concentrations. valid for low temperatures, if we include the If we now consider the opposite of gaseous very important result that was confirmed equilibrium namely, a reaction between solid theoretically by Debye and experimentally by substances only—a limiting transition to Eucken and, quite recently, also by Schwers absolute zero seems entirely possible in and myself, that the specific heat changes in theoretical as well as experimental respects. proportion to the third power of the absolute And here it is noticeable that Berthelot’s temperature at low temperatures: principle frequently fits quite well, especially δ 4 if one is dealing with relatively great reaction A =U0 − T (14) U = U + δT4, 3 . heats. Thus, the conjecture forced itself upon 0 me (1906) that this is a matter of a limiting law In the discussion of the following of such a kind that A and U not only become examples, we shall limit ourselves to a graphic equal at absolute zero but approach each other representation, and I shall give references to asymptotically. Thus we should have: the appropriate publications for the specific numerical material. It is even quite possible and, in many cases, practicable, to determine

5 the relationship between affinity and heat of curve out of the entire set of curves. In other reaction by purely graphic means. words, every arbitrary value of the affinity A is We shall assume, for example, that the compatible with any experimentally given heat of reaction was measured for a single shape of the heat formation, so that the second arbitrary temperature and that we know the heat law abandons us here to a large extent. It specific heats of the reacting solid substances gives us a precise answer only for absolute down to very low temperatures. By assuming zero, since the curves for heat formation and the T3 law for the very low temperatures that affinity intersect here so that both quantities are inaccessible to measurement, we are then become identical, as Berthelot had assumed to in a position to draw the heat of reaction as a be the case for all temperatures. function of temperature with great accuracy But if we now include the new heat law, down to absolute zero. the A curve must run parallel to the U curve at The integral of the equation absolute zero; in other words, from the infinite dA set of A curves one, and only one, is fixed as A –U = T being possible. (1) dT If we wish to determine it, not by is calculation from equation (15), in which c T U A T dT cT must be set equal to zero according to the new = − ∫ 2 + (15) 0 T ; law, but by purely graphic means, we must for T = 0, we have first, starting at absolute zero, draw it parallel A = U0; to the U curve; the further direction is given at i.e., as was already mentioned, Berthelot’s law every point by the equation is valid here without restriction. However, for dA A −U tanα = = higher temperatures the value of the dT T . integration constant c becomes decisive, and With some practice, the A curve can be drawn the second heat law leaves this value quickly and with sufficient accuracy in this undetermined. way.

Figure 1 shows this. The solid curve U represents the dependence of the heat formed Examples of Condensed Systems on the absolute temperature; thus U0 is the In condensed systems, also, Berthelot’s value this quantity assumes at absolute zero; principle sometimes fails completely; in then each of the dotted curves A is a solution particular, at the melting point and at the of the above equation, and one sees at once transformation point the affinity equals zero that there is no point, and therefore no value because the two phases in question are here in for A, through which we could not draw an A equilibrium, while the heat formed (heat of

6 fusion or, respectively, heat of transformation) be considered definite, and it is this alone that can even have considerable values. The we are concerned with, that the two curves are application of the new heat law thus leads in tangent to each other, in agreement with the this instance to especially characteristic new heat law. consequences, the influence of the specific Combination of Water of Crystallization heats, which had formerly not been In recent times,5 the reaction sufficiently recognized, proving decisive for CuSO4 + H2O = CuSO4,H2O the position of the melting or transformation has been investigated very intensively. The point. As an example, let us look at the quantities measured were the heat of hydration transformation of sulphur. with liquid water, the dissociation potentials at Transformation of Sulphur higher temperatures, and the specific heats of Different authors have measured the heat the two salts and of ice down to very low of transformation of rhombic into temperatures (Figure 4). monosymmetric sulphur, the maximum work to be obtained in this and the specific heats for both modifications; in addition, the temperature of the transformation point is known exactly.4 It turned out that, with the aid of the simple formulas U = 1.57 + 1.15.10–5T2 A = 1.57 – 1.15.10–5T2, all observations can be reproduced almost within the accuracy of the errors of observation.

With the aid of the second heat law, it was then possible to calculate the dissociation potential p also for the ordinary zero The accompanying curve (Figure 3) gives temperature point; if p is the vapor pressure of a picture that approaches reality still more ice at this temperature, it is found that closely; here, the U curve is drawn from the p available thermal measurements and the A A = RT ln = 4,415 cal curve is determined graphically as described π . above. The latter is in agreement with the On the other hand, with the aid of the new heat available measurements to the extent allowed law, using the heat formed in this reaction at by the accuracy of the available thermal the same temperature (4,910 cal) and using the measurements; since the very small difference specific heats, the result was: of the specific heats of the two modifications A = 4,475 cal, of sulphur determines the course of the U in satisfactory agreement with the value above. curve, it can, of course, be indicated only Finally, for absolute zero, it was calculated within a certain accuracy. But it can probably that

7 A0 = U0 = 4,680 cal, Thus, we may probably say in summary and it can be seen (as seems to be most that it has been possible to test the new heat frequently the case) that U increases with law on a very extensive and varied group of temperature while A decreases; but the latter facts, numerous chemical equilibriums having quantity would cross the temperature axis only been calculated from thermal data or from the at such high temperatures that the ice would combination of thermal data and vapor- have long ceased to exist and that, in practice, pressure measurements. a point of transformation is therefore not Independent of this, it may also be derived, present. as I was unable to do in detail here, from a fact Such a process will always occur if the established by very many measurements done molecular heat of the water of crystallization is in recent times according to which the specific smaller than that of ice; potassium heats of solid and liquid substances assume ferrocyanide, which crystallizes with three vanishingly small values at very low moles of water, is an example of the reverse temperatures. case. The accompanying diagram (Figure 5)6 Third, as Mr. Planck recently explained shows the energy relationships, which are very here, it is closely related to the theory of peculiar here; one sees that U becomes 0 at T = energy quanta, and thus even the phenomena 160, and that U becomes negative at higher of heat radiation, strange as this may sound at temperatures, while A remains positive and first, give support to our law from an entirely even increases. different direction.

1[Copied from Eduard Farber, Ed., Milestones 3Les équilibres chimiques (Paris, 1888), p. 184. of Modern Chemistry, Basic Books, New 4For details cf. Nernst, Theoretische Chemie, 7. York, 1966, pp 186-202. Translation by Aufl. (1913), pp. 736-737. Elisabeth F. Lanzl. —CJG] 5A. Siggel, Zeitschrift für Elektrochemie, Vol. 2Undoubtedly it is more practical to define 19 (1913), p. 341. chemical affinity by the so called 6Schottky, Zeitschrift für physikalische “thermodynamic potential,” but in our Chemie, Vol. 64 (1908), p. 441; W. Nernst, considerations this does not make any Berichte der Berliner Akademie (1910), p. noticeable difference. 277.