Proof of the Nernst theorem

Eur. Phys. J. Plus (2025) 140:528 https://doi.org/10.1140/epjp/s13360-025-06503-w Regular Article Proof of the Nernst theorem José-María Martín-Olallaa Departamento Física de la Materia Condensada, Universidad de Sevilla, Avenida de la Reina Mercedes, 41012 Sevilla, Andalucía, Spain Received: 19 March 2024 / Accepted: 1 June 2025 © The Author(s) 2025, corrected publication 2025 Abstract The Nernst heat theorem is proven from purely thermodynamic arguments connected with the second law of thermodynamics. The proof stipulates that T0 is formalized by a Carnot thermometer, and is independent of the vanishing of the specific heats, or the unattainability of the zeroth isotherm. With this proof, the second law of thermodynamics would extend its applicability, and the third postulate of thermodynamics would be narrowed to the fact that the entropy of a finite density, chemically homogeneous body must not be negative. 1 Introduction In classical thermodynamics, the Nernst theorem or Nernst heat theorem states that the isothermal change of entropy (S)T associated with a process between two states of a system in internal equilibrium vanishes as the temperature vanishes [1–4]. It is one of the two general properties of matter in the vicinity of the zeroth isotherm. Formally, the theorem sets the vanishing of (∂S/∂ X)t(∂2A/∂T∂X)whenT→0+,whereXis a suitable mechanical parameter such as the pressure, volume, or magnetic field, and A(T,X) is a suitable thermodynamic potential like the free energy or the free enthalpy. Note that the formulation of the theorem excludes mixing processes, and the variation of entropy associated with them. Nernst presented his theorem in 1905 after empirical evidence related to chemical equilibrium and to the expansion coefficients at very low temperatures [5]. Soon afterward, he collected evidence on the vanishing of the specific heats as the temperature vanishes [6], a second general property of matter at very low temperatures—related to (∂2A/∂T2), further completed by Simon [7]. Eventually Nernst proposed the unattainability of the zeroth isotherm as a summary of his evidences and presented a proof by contradiction [6]. Nernst maintained that since a Carnot engine operating at the zeroth isotherm would negate the second law, the unattainability of the zeroth isotherm is then deduced. Einstein refuted the proof noting that at T0 every irreversibility, no matter how small, would throw the system away from the zero isotherm, paving the way for the third law of thermodynamics [8, p. 293]. Later, Epstein refined the rebuttal saying that at T0, the isothermal process becomes also adiabatic, and therefore, no practical procedure would be able to accomplish this process. Hence, the engine would not operate and could not possibly challenge the second law [1,4,9,10]. No other attempt to probe the theorem is known. The isothermal change of entropy plays a crucial role in reversible Carnot engines, which on their own play a crucial role in the second law of thermodynamics. A reversible Carnot engine extracts an equivalent heat Qh/Th(sans-serif letters will be used to identify exchanges in reversible processes) from a heater at temperature Thand pours it into the cooler. For that to occur the substance undergoing the cycle must be able to sustain an isothermal change of entropy (S)TQh/That the temperature of the heater and at the temperature of the cooler Tc, both conducted by an isothermal change of the mechanical parameter X. The engine produces work W(S)T×(Th−Tc). This work presents a proof of the Nernst theorem purely based on thermodynamic arguments connected with the fact that the Carnot thermometer must be able to operate at T0. The proof does not require the vanishing of the specific heat. It is not a proof by contradiction, like the proof presented by Nernst, but of consistency of the second law. 2 Background Planck’s statement of the second law of thermodynamics reads it is impossible to construct an engine which will work in a complete cycle, and produce no effect except the raising of a weight and the cooling of a heat reservoir [11]. It formalizes an observation ae-mail: [email protected] (corresponding author) 0123456789().: V,-vol 123 528 Page 2 of 4 Eur. Phys. J. Plus (2025) 140:528 by Carnot [12]: “the production of heat alone is not sufficient to give birth to the impelling power; it is necessary that there should also be cold; without it, the heat would be useless.” Consequently, an engine cannot operate with a solo reservoir (unary engines); instead, it must heat, at least, a second, distinct reservoir, thus setting a binary engine. The statement can be formalized as: P=⇒(W>0⇒Qc<0), (1a) ¬P⇐=(W>0∧Qc≥0).(1b) It reads: Planck’s statement P implies that if a weight is lifted (W>0, antecedent), then a reservoir must be heated (Qc<0, consequent); if a weight is lifted and (∧) no reservoir is heated, then Planck’s statement is negated (¬P). This logical structure provides a basis for proving Carnot’s theorem and Clausius’s theorem. The very notion of temperature is alien to the conservation of energy (in this context WQh+Qc,whereQh>0 is the heat supplied by the heat reservoir which is cooled) and to Planck’s statement [11]. Instead, Carnot’s theorem shows that in reversible binary engines (Carnot engines), the ratio of the exchanged heats is independent of the engine itself, and can be expressed as the ratio of a quantity that depends only on the two reservoirs: Tc Th −Qc Qh ,(2) Because the heat exchanges point in opposition, the minus sign in equation (2) ensures that the ratio Tc/This positive. With W>0 energy conservation mandates Qh>−Qc, therefore Th>Tc, which aligns with the intuitive idea of hotness and coldness. Equation (2) formally introduces the concept of temperature and provides a universal thermometer: the Carnot thermometer [13]. 3 The proof In the context presented in Sect. 2,ifT0 is assigned to a finite density substance, then a reversible Carnot engine operating with a zero temperature “cooler” and a heater must be conceptually admitted. Otherwise, the assignment T0 would be external to the framework of the second law that brings Planck’s statement. This reversible Carnot engine must exchange no heat with the cold reservoir since, from equation (2), if Tc0, then Qc0. It then follows that the engine would negate Planck’s statement unless no heat were exchanged with the heater: Qh0, which would result in W0. Hence, since Th 0, we have Qh/Th0. Therefore, from equation (2), we arrive at Qc/Tc0, for Tc0. Because of the universality of Carnot theorem, this result must apply irrespective of the body of finite density that undergoes the cycle and, given a body, irrespective of the specific mechanical or chemical configuration under which the engine operates. Because Q/Texchanged by the reversible engine matches with (S)Tin the substance undergoing the cycle, it then follows that every (S)Tat T0 must be zero for every body of finite density, irrespective of the change in the configuration of the body. Continuity then requires that (S)Tvanishes as the temperature vanishes, which is the Nernst theorem. Alternatively, a literal reading of Planck’s statement may consider that the other effect might not be energetic of necessity. As an example, a reversible Carnot engine lifting a weight and operating at T0—where Qc0—would not negate the statement because Qh/Thwould be poured into the T0 reservoir, being that the required additional effect. This reading is implying that the isothermal change of entropy at T0 can be nonzero. Consequently, it fails to align with the empirical observations underpinning the Nernst theorem and is, therefore, discarded. 4 Corollaries Two well-known corollaries of the Nernst theorem would now follow from the second law. First, S(T0, X) is unique, irrespective of X, should limT→0S(T,X) exist. It would be the only circumstance in which the second law points toward absolute S, instead of S, albeit incompletely. Second, irrespective of limT→0S(T,X), it is the case that S(T,X)>S(0, X) for any nonzero T. Since the entropy must not decrease adiabatically, no adiabatic cooling can bring the temperature of a finite density system arbitrarily close to T0. This is the statement of unattainability of the zeroth isotherm. 5 Remark on the absolute value of entropy The most comprehensive summary of Nernst’s empirical observations is due to Planck, who presented the following statement: as the temperature diminishes indefinitely, the entropy of a chemical homogeneous body of finite density approaches indefinitely near to the value zero [3,4,14] This concerns absolute entropy, not entropy change. 123 Eur. Phys. J. Plus (2025) 140:528 Page 3 of 4 528 Fig. 1 Any reversible Carnot engine operating at T0must consist of a round trip from a given equilibrium state E1to E0at T0. Following the vanishing of cx, the entropy at T0 is placed at a finite value of S While the statement is presented as one law (the third law of thermodynamics), it encompasses the two distinct, general properties of matter in the vicinity of the zeroth isotherm: (1) the entropy is unique (as deduced from the vanishing of (S)T), and (2) the entropy has a floor, which can be set at zero, as deduced from the fast vanishing of the specific heats cxas the temperature vanishes in view that the integrand in ScxdT/Tremains bounded. The fast vanishing of the specific heats is unrelated to the arguments presented in the proof of the Nernst theorem. Therefore, it is the one and only general property of finite density systems that remain independent from the second law. It can be summarized by the following statement: the entropy of a finite density, chemically homogeneous body is not negative. 6 Discussion The burden of the proof lies in the meaning of the assignment T0. Earlier in the eighteenth century, the absolute zero was empirically conceived as a scenario where the volume or the pressure of a gas vanished. In the argument between Nernst and Einstein, the absolute zero meant p0[8, page 294]. From a classical kinematic point of view, T0 is conceived as the state where motion, kinetic energy, vanishes. In some axiomatic presentations of the thermodynamics, T0 is conceived as a scenario where (∂U/∂ S)Xvanishes (Uis the energy of the system) [15,16]. Yet, in the classical framework derived from the second law, equation (2) provides a natural zero for the temperature, associated with Qc0. This is a formal assignment of general validity, beyond the empirical p0orV0, and prior to the definition of the entropy in the classical presentation of thermodynamics. The above proof commits to this association and makes a consistent use of that. It aligns with the proof of the theorems that are deduced from the second law of thermodynamics, such as Carnot theorem or Clausius theorem, linking general properties of cycles with general properties of substances undergoing the cycle. In contrast, under Einstein’s rebuttal of Nernst’s proof—a Carnot engine operating at T0 cannot be constructed—it follows that T0 cannot be determined by equation (2), which is paramount for defining the temperature, but externally, i.e., empirically. In summary, T0 would be assigned to a finite system only if the reversible Carnot engine that attempts to assess the temperature of the system consists of a round trip connecting a given state E1of the working fluid with the state E0,seeFig.1. The loop, purely conceptual, will enclose no area, no work will be produced, and no entropy will be exchanged.1 The interplay between Planck’s statement of the second law and the Nernst theorem (N) is also worthy of discussion. The Nernst theorem can be described by quantities related to reversible Carnot engine. First, the limit T→0 is the limit Qc→0−,asperEq.(2). Second, the observation (S)T→0 is translated into Qh→0+and into W→0+. Therefore, the Nernst theorem expresses the following relations: N=⇒((Qc→0−)⇒(W→0+)),(3a) ¬N⇐=((Qc→0−)∧W>0).(3b) Comparing with the right material implication in (1a), the right material implication in (3a) is saying: The consequent (Qc) vanishes only if the antecedent (W) vanishes as well. On the other hand, the left material implications in (1b)andin(3b) differ slightly. In (1b), Qcis seen as a categorical variable: if Qc0andW>0, then the statement is negated; if Qcis negative, then the statement survives, no matter how close to zero Qc might be. Einstein’s and Epstein’s refutations of Nernst’s proof are in line with this view: It is impossible to construct an engine 1The Carnot engine is usually shown as a rectangle in a ST chart. Actually a Carnot engine—an engine operating with two distinct heat sources—requires two isothermal processes and two parallel paths shifted by an entropy . In the limit →0, the paths coalesce in one round trip path as shown in Fig. 1 [17]. Note that, conceptually, the engine would keep operating with Tc0. It would only cease to deliver any work. 123 528 Page 4 of 4 Eur. Phys. J. Plus (2025) 140:528 operating at T0 but, for any T>0, no matter how close to zero it might be, the engine could be constructed as the categorical argument presented by them no longer sustains; nonetheless, Planck’s statement would sustain as long as Qc<0.2 In contrast, proposition (3b) brings Qcas a continuous variable. In this line, the Nernst theorem acknowledges that the other energetic effectQcthat appears in the conversion of heat into work is not categorical but, like any other mandatory effect, a measurable quantity, larger than a value prescribed by nature through Qh/Th(the entropy carried by the engine) and the thermophysical properties of the substance undergoing the cycle [17]. Acknowledgements APC were covered by Universidad de Sevilla (RoR: 03yxnpp24) as a member of the CRUE-CSIC Alliance and following the 2025-2027 agreeement of the Alliance with Springer Nature. Funding Funding for open access publishing: Universidad de Sevilla/CBUA. Data Availability Statement No data associated in the manuscript. Declarations Conflict of interest The author declares no conflict of interest. Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/. References 1. P. Epstein, A Textbook of Thermodynamics. Wiley (1937) 2. A. Pippard, Classical Thermodynamics. Cambridge Univ. Press (1957) 3. G.N. 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At any Tc 0, the argument no longer applies because the isotherm is now diathermal, and the entropy of the substance can indeed decrease isothermally, irrespective of how close to zero Tcmight be. 123