A concise rebuttal on a concise proof of the equivalence of the Nernst theorem and the heat capacity statement of the third law of thermodynamics

This is the submitted version of an article accepted at Modern Physics Letters A © 2024 World Scientific Publishing The Version of Record will be available at doi:10.1142/S0217732324750038 article number 2475003 (3 pages) A concise rebuttal on a concise proof of the equivalence of the Nernst theorem and the heat capacity statement of the third law of thermodynamics Jos´e-Mar´ıa Mart´ın-Olalla Universidad de Sevilla, Facultad de F´ısica, Departamento de F´ısica de la Materia Condensada, ES41012 Sevilla, Spain∗ The interplay between adiabatic cooling and isothermal ordering is analyzed to rebut the proof of the equivalence of the Nernst theorem and the heat capacity statement of the third law presented by Su and Chen. File: mainInitial.tex Encoding: utf8 Words in text: 568 Words in headers: 27 Words outside text (captions, etc.): 48 Number of headers: 3 Number of floats/tables/figures: 1 Number of math inlines: 41 Number of math displayed: 2 Subcounts: text+headers+captions (#headers/#floats/#inlines/#displayed) 549+26+48 (2/1/41/2) _top_ 19+1+0 (1/0/0/0) Section: Acknowledgments PACS numbers: 05.70.-a; 65; 65.40.Ba; 65.60.7a; 82.60.-d Keywords: specific heat; unattainability statement; Planck statement Recently Su and Chen[1] addressed the relationship between the Nernst theorem and the vanishing of the specific heat and presented a “concise proof” of their equivalence. The issue had raised some confusion and clarification earlier.[2; 3; 4; 5; 6] In classical thermodynamics the Nernst theorem states that “the isotermal change of entropy (∆S)Tvanishes as the temperature vanishes”. It follows evidences extracted from chemical equilibrium, phase equilibrium and the vanishing of thermal expansion coefficients. Formally the theorem sets the vanishing of (∂S/∂X)T= (∂Y/∂T)X= (∂2A/∂T∂X), where X, Y are suitable mechanical parameters such as the pressure/volume or the magnetic field/magnetization, and A is a suitable thermodynamic potential like the free energy or the free enthalpy. The vanishing of the specific heats is another kind of evidence in itself. It refers to the fact that the susceptibility Cx/T = (∂S/∂T )X=−(∂2A/∂T2) has a ceil —ie does not become exceedingly large— as the temperature vanishes. The domain of (∂S/∂T)Xis restricted by stability: it must be positive. The domain of (∂S/∂X)Tis not restricted by stability: it can be positive, negative or zero at any given temperature. In their concise argument Su and Chen analyze the interplay between the isothermal change of entropy (∆S)T, the adiabatic change of temperature (∆T)s, and the iso- Xchange of entropy (∆S)X. The interplay between this three quantities is given by Euler’s chain rule: ∂S ∂T X∂T ∂X S∂X ∂S T =−1,(1) which is geometrically shown in figure 1. The physical interpretation of figure 1 is of interest here. Let us suppose that E1is located at the lowest temperature ever achieved. In order to further cool the sample we need to peform adiabatic cooling, as noted by Su and Chen, but prior to that we need to reduce isothermically the entropy of the sample. The point is that E1→E3is not possible through the X=X1line. Instead, the path E1→E2→E3is mandatory. From that the susceptibility is given as: ∗[email protected]; https://orcid.org/0000-0002-3750-9113; https://ror.org/03yxnpp24 Typeset by REVT EX 2 X1 X2=X1+ ∆X (∆S)T (∆T)s E1 E2 E3 T3T1 s2 s1 T s Figure 1 A two stroke process in which Xis restored. First an isothermal decrease of the entropy (∆S)T, followed by an adiabatic cooling (∆T)S. The susceptibility (∂S/∂T )X(>0) is given by the ratio (∆S)T/(∆T)swhen ∆X→0. With the equilibrium state E1at the lowest available temperature T1, this is the only way to step further into lower temperatures T3. ∂S ∂T X = lim ∆X→0 (∆S)T (∆T)s =−lim ∆X→0 S(T, X + ∆X)−S(T, X) T(S, X + ∆X)−T(S, X).(2) Equation (2) is similar to Equation (7) by Su and Chen only that, Cx/T is expressed as (∂S/∂T)X. In Equation (2) the numerator vanishes as the temperature vanishes following the Nernst theorem. The denominator vanishes as the temperature vanishes as per the unattainability statement, which, in fact, follows from the Nernst theorem. However, there is no way to elucidate which of the two in Equation (2) —(∆T)Sand (∆S)T— vanishes faster. With Cx=T(∂S/∂T)x=T(∆S)T/(∆T)s, there is no way to elucidate whether heat capacities vanish or not. In their argument Su and Chen set −(∆T)s< T, which is correct, and make (∆T)S/T finite from which they get Cxvanishes. But the premise is only possible if (∆T)S vanishes as fast as T. No proof of that is given. Indeed no proof of that exists until further evidence —such as the vanishing of CX— is introduced. The bottom line is that in S(T, X) the dependence on Xis independent from the dependence on T. Therefore, the condition limT→0(∂S/∂X)T= 0 does not force any further condition on limT→0(∂S/∂T)X.[6; 7; 8] ACKNOWLEDGMENTS The author declares no competing interest. This work was not funded. This version is published in zenodo doi: 10.5281/zenodo.11064906 REFERENCES [1] S. Su and J. Chen, A concise proof of the equivalence of the Nernst theorem and the heat capacity statement of the third law of thermodynamics, Modern Physics Letters A37, 2250246 (2022). [2] Z. Yan and J. Chen, An equivalent theorem of the Nernst theorem, Journal of Physics A: Mathematical and General 21, L707 (1988). [3] P. T. Landsberg, A comment on Nernst’s theorem, Journal of Physics A: Mathematical and General 22, 139 (1989). [4] I. Oppenheim, Comment on ’an equivalent theorem of the Nernst theorem’, Journal of Physics A: Mathematical and General 22, 143 (1989). [5] A. J. Kox, Confusion and clarification: Albert Einstein and Walther Nernst’s heat theorem, 1911–1916, Studies in History and Philosophy of Science Part B: Studies in History and Philosophy of Modern Physics 37, 101 (2006). [6] A. Y. Klimenko, Teaching the third law of thermodynamics, The Open Thermodynamics Journal 6, 1 (2012). [7] M. Planck, Treatise on Thermodynamics (Longmans, 1927 (Repub Dover 1990)). [8] J. M. Mart´ın-Olalla and A. Rey de Luna, Universal restrictions to the conversion of heat into work derived from the analysis of the Nernst theorem as a uniform limit, Journal of Physics A: Mathematical and General 36, 10.1088/0305-4470/36/29/303 (2003).