Comment on 'The Triple Product Rule and the Subtleties of the Mathematical Tools Used in Thermodynamics' by Felipe Américo Reyes Navarro and Rafael Edgardo Carlos Reye

Indian Journal of Advanced Mathematics (IJAM) ISSN: 2582-8932 (Online), Volume-5 Issue-2, October 2025 41 Retrieval Number:100.1/ijam.B122305021025 DOI: 10.54105/ijam.B1223.05021025 Journal Website: www.ijam.latticescipub.com Published By: Lattice Science Publication (LSP) © Copyright: All rights reserved. Comment on ‘The Triple Product Rule and the Subtleties of the Mathematical Tools Used in Thermodynamics’ by Felipe Américo Reyes Navarro and Rafael Edgardo Carlos Reye I.A. Stepanov Abstract: It is shown that much of the criticism of my paper ‘The Triple Product Rule is Incorrect’ Indian Journal of Advanced Mathematics (IJAM), Vol. 1, No. 2, 2021. 1 − 3, https://doi.org/10.35940/ijam.B1102.101221 presented in [1] is incorrect. Keywords: Triple Product Rule, Natural Variables, Work, Compression, Function of Stat. I. INRODUCTION The authors of [1] rebut my criticism in [2] of the derivation of the triple product rule, which was given in [3, 4, 6]. One can see that there are mistakes in their rebuttal. The derivation in [3, 4, 6] goes as follows: The total differential of z is: 𝑑𝑧=(𝜕𝑧 𝜕𝑥)𝑦𝑑𝑥+(𝜕𝑧 𝜕𝑦)𝑥𝑑𝑦 . … (1) We suppose that z is constant, which means that y = y(x) and 𝑑𝑦=(𝜕𝑦 𝜕𝑥)𝑧𝑑𝑥 . … (2) Introducing equation (2) into equation (1) with constant z, we obtain 0=(𝜕𝑧 𝜕𝑥)𝑦𝑑𝑥+(𝜕𝑧 𝜕𝑦)𝑥(𝜕𝑦 𝜕𝑥)𝑧𝑑𝑥 . … (3) Since equation (3) must be true for all dx, rearranging its terms gives (𝜕𝑧 𝜕𝑥)𝑦=−(𝜕𝑧 𝜕𝑦)𝑥. … (4) Manuscript received on 27 September 2025 | First Revised Manuscript received on 29 September 2025 | Second Revised Manuscript received on 09 October 2025 | Manuscript Accepted on 15 October 2025 | Manuscript published on 30 October 2025. *Correspondence Author(s) I.A. Stepanov*, Institute of Science and Innovative Technologies, Liepaja University, Liela 14, Liepaja, LV−3401, Latvia, Email ID: [email protected], ORCID ID: 0000-0002-5843-2346 © The Authors. Published by Lattice Science Publication (LSP). This is an open-access article under the CC-BY-NC-ND license http://creativecommons.org/licenses/by-nc-nd/4.0/ From this, the triple product rule follows: (𝜕𝑥 𝜕𝑦)𝑧(𝜕𝑦 𝜕𝑧)𝑥(𝜕𝑧 𝜕𝑥)𝑦=−1 .… (5) It can be seen that there are flaws in this derivation. For constant z, equation (1) becomes 0=(𝜕𝑧 𝜕𝑥)𝑦𝑧𝑑𝑥+(𝜕𝑧 𝜕𝑦)𝑥𝑧𝑑𝑦, … (6) and both terms on the right-hand side become zero. The authors of [1] disagree that both derivatives in equation (6) are zero. I consulted Prof. A. V. Ovchinnikov, one of Russia's leading mathematicians, who serves as Deputy Editor-in-Chief of the Russian Abstract Journal Matematika. He wrote to say that if z=const, then not only is dz zero, but also yz z x      = xz z y      =0 because a derivative of a constant is zero. Therefore, if one introduces the relation dd z y yx x   =   Into the previous formula, one can obtain only 0=0 and nothing else! The authors of [1] proposed a function: 𝑧=(25−𝑥2−𝑦2)𝑧𝑑𝑥. … (7) They perform a formal differentiation of the right-hand side of this for a constant z and obtain a derivative that is not equal to 0. However, in an exact differentiation, one has to consider that the right-hand side is also constant and that its derivative is 0. A stricter derivation of the triple product rule is given in [5] (p. 391), [6]. We can write 𝑑𝑥=𝑑𝑥|𝑧=const+d𝑥|𝑦=const=(𝜕𝑥 𝜕𝑦)𝑧𝑑𝑦 +(𝜕𝑥 𝜕𝑧)𝑦𝑑𝑧, … (8) and Comment on ‘The Triple Product Rule and the Subtleties of the Mathematical Tools Used in Thermodynamics’ by Felipe Américo Reyes Navarro and Rafael Edgardo Carlos Reye 42 Retrieval Number:100.1/ijam.B122305021025 DOI: 10.54105/ijam.B1223.05021025 Journal Website: www.ijam.latticescipub.com Published By: Lattice Science Publication (LSP) © Copyright: All rights reserved. 𝑑𝑦=(𝜕𝑦 𝜕𝑥)𝑧𝑑𝑥+(𝜕𝑦 𝜕𝑧)𝑥𝑑𝑧. … (9) Substituting equation (9) into equation (8), we obtain 𝑑𝑥=(𝜕𝑥 𝜕𝑦)𝑧(𝜕𝑦 𝜕𝑥)𝑧𝑑𝑥+[(𝜕𝑥 𝜕𝑦)𝑧(𝜕𝑦 𝜕𝑧)𝑥+(𝜕𝑥 𝜕𝑧)𝑦]𝑑𝑧. … (10) The first term on the right-hand side equals dx, and hence (𝜕𝑥 𝜕𝑦)𝑧(𝜕𝑦 𝜕𝑧)𝑥+(𝜕𝑥 𝜕𝑧)𝑦=0. … (11) From this, the triple product rule in Equation (5) can be easily obtained. The flaw in this derivation is as follows. Equation (9) cannot be substituted into equation (8), since dy in equation (8) is taken at constant z: dy = dy(z = const), while dy in equation (9) is for varying z. The differential dy in equation (9) at constant z will look like: 𝑑𝑦=(𝜕𝑦 𝜕𝑥)𝑧𝑑𝑥 … (12) and its substitution into equation (8) will not lead to success. Finally, equation (S20) in [1] describes the compression of a gas. It contains the derivative: (𝜕𝑉 𝜕𝑇)𝑃. … (13) At a compression, the temperature of a gas increases, but its volume decreases; hence, this derivative must be negative. Nevertheless, the authors of [1] use another derivative, namely: (𝜕𝑉 𝜕𝑇)𝑃=𝛼𝑉 … (14) where the thermal expansion coefficient is involved, one cannot use equation (14) to describe mechanical compression. The derivative must be rederived. II. FINDING AND DISCUSSION It is shown that the derivation of the triple product rule given in [3, 4, 6] is indeed wrong. Equation (20) in [2] is demonstrated to be unsuitable for describing the compression process. III. CONCLUSION The thermodynamic identity, Eq. (19) in [2], which results from the triple product rule, is not always correct. Before applying Eq. (19), it is necessary to check every derivative in it. Here is that identity: (𝜕𝑉 𝜕𝑇)𝑃=−(𝜕𝑉 𝜕𝑃)𝑇(𝜕𝑃 𝜕𝑇)𝑉. … (19) DECLARATION STATEMENT Some of the references cited are older, noted explicitly as [3], [4], and [5]. However, these works remain significant for the current study, as they are pioneering in their fields. I must verify the accuracy of the following information as the article's author. ▪ Conflicts of Interest/ Competing Interests: Based on my understanding, this article has no conflicts of interest. ▪ Funding Support: This article has not been funded by any organizations or agencies. This independence ensures that the research is conducted with objectivity and without any external influence. ▪ Ethical Approval and Consent to Participate: The content of this article does not necessitate ethical approval or consent to participate with supporting documentation. ▪ Data Access Statement and Material Availability: The adequate resources of this article are publicly accessible. ▪ Author’s Contributions: The authorship of this article is contributed solely. REFERENCES 1. Felipe Américo Reyes Navarro and Rafael Edgardo Carlos Reye, ‘The Triple Product Rule and the Subtleties of the Mathematical Tools Used in Thermodynamics’, Indian Journal of Advanced Mathematics, (IJAM), Vol. 4, No. 1, 2024, 4 − 8, DOI: https://doi.org/10.54105/ijam.A1164.04010424 2. I. Stepanov, ‘The Triple Product Rule is Incorrect’, Indian Journal of Advanced Mathematics, (IJAM), Vol. 1, No. 2, 2021. 1 − 3, DOI: https://doi.org/10.35940/ijam.B1102.101221 3. J. R. Elliott and C. T. Lira, Introductory Chemical Engineering Thermodynamics, (2nd ed.), Prentice Hall, 2012. The works remain significant, https://www.eng.uc.edu/~beaucag/Classes/ChEThermoBeaucage/Book s, see the declaration 4. S. M. Blinder, ‘Mathematical Methods in Elementary Thermodynamics”, Journal of Chemical Education, Vol. 43, No.2, 85 − 92 (1966). DOI: https://doi.org/10.1021/ed043p85, works remain significant, see the declaration 5. A. H. Carter, Classical and Statistical Thermodynamics, Prentice Hall, 2001. The works remain significant, DOI: https://doi.org/10.3390/e4060165, see the declaration 6. Triple product rule, available from the World Wide Web: https://en.wikipedia.org/wiki/Triple_product_rule [Accessed 27 September 2025]. AUTHOR’S PROFILE Igor Stepanov got an MSc in physics from the University of Latvia in 1982 and an MPhil from the University of Sheffield in 2012. Currently, I work as a researcher at Liepaja University, Latvia. Some of my critical findings are: ▪ I solved the 3-dimensional Ising model ▪ This solution is published in the JP Journal of Heat and Mass Transfer. 2023. Vol 33. No 1. 51 − 70. Disclaimer/Publisher’s Note: The statements, opinions and data contained in all publications are solely those of the individual author(s) and contributor(s) and not of the Lattice Science Publication (LSP)/ journal and/ or the editor(s). The Lattice Science Publication (LSP)/ journal and/or the editor(s) disclaim responsibility for any injury to people or property resulting from any ideas, methods, instructions, or products referred to in the content.