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The Triple Product Rule Is Incorrect Indian Journal of Advanced Mathematics (IJAM) ISSN: 2582-8932 (Online), Volume-1 Issue-2, October 2021 The Triple Product Rule is Incorrect Igor A. Stepanov zz Abstract: The triple product rule, also known as the cyclic dz d x d y . (2) chain rule, cyclic relation, cyclical rule or Euler's chain rule, xyy x relates the partial derivatives of three interdependent variables, and often finds application in thermodynamics. It is shown here that its derivation is wrong, and that this rule is not correct; hence, We suppose that z is constant, meaning that y = y(x) and the Mayer's relation and the heat capacity ratio, which describe y the difference between isobaric and isochoric heat capacities, are ddyx . (3) also untrue. Also, the relationship linking thermal expansion and x z isothermal compressibility is wrong. These results are confirmed by many experiments and by the previous theoretical findings of Introducing (3) into (2) at constant z, we obtain the author. Keywords: Cyclic chain rule; Cyclic relation; Mayer's z z y relation; Heat capacity ratio; Isochoric heat capacity; 0 dxx d . (4) Compressibility x yz yx x I. INTRODUCTION Since (4) must be true for all dx, rearranging its terms gives Functions theory contains the triple product rule, also known as the cyclical relation, cyclic chain rule or Euler's chain rule [14]. Suppose that there is a function f(x, y, z) = 0 z z y . (5) and this implicit equation can be solved for each of the x y x variables x, y, and z as a differentiable function of the other yzx two. The triple product rule is a formula that relates the partial derivatives of these functions, and is given by From this, the triple product rule follows. x y z It can be seen that there are flaws in this derivation. For 1. (1) constant z, (2) becomes yz z xy x zz 0 dxy d , (6) It is often used in thermodynamics, for example in the xy derivation of formulae for the difference between isobaric and yz xz isochoric heat capacities. It can be shown that there is a mistake in the derivation of the triple product rule, and that and both terms on the right-hand side become zero. this rule is invalid. As a consequence, these thermodynamic Furthermore, the differential dy in the second term on the formulae are also wrong. The results obtained are confirmed right-hand side of (2) is taken at constant x: dy = dy(x = by many experimental results and previous outcomes reported const). Therefore, at constant z, we may introduce (3) into (2) by the author. only when dy in (3) is also taken at constant x: y II. THEORY dy ( x const) d x , (7) x zx Let us perform a non-strict derivation of (1) [1,2,4]. The total differential of z is: which does not make sense. Let us perform a stricter derivation of (1) [3,4]. We can write Manuscript received on April 01, 2021. xx Revised Manuscript received on April 06, 2021. dx d xzyconst +d x const d y d z (8) Manuscript published on October 10, 2021. yzz y * Correspondence Author Igor Stepanov*, Scholar/Institute of Science and Innovative Technologies, Liepaja University, Liepaja, Latvia, LV-3401. Email: and [email protected] © 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/) Retrieval Number: 100.1/ijam.B1102101121 Published By: DOI: 10.35940/ijam.B1102.101221 1 Lattice Science Publication Journal Website: www.ijam.latticescipub.com © Copyright: All rights reserved. The Triple Product Rule is Incorrect yy 1 y , z x , z x , y . (18) dy d x d z . (9) 1 2 1 xz zx In the usual (admittedly sometimes confusing) notation, this is Substituting (9) into (8), we obtain (1).” Really, only two cases are possible on the left-hand side y y x y x y x of (14): x or x . In the former one, the last dx d x d z . (10) z y y x y z z zz z x y term in (14) becomes zero. One can show that this derivation is wrong. If on the The first term on the right-hand side equals dx, and hence left-hand side of (14) y is constant, then y is constant also on x y x its right-hand side. Consequently, both terms on the 0. (11) right-hand side of (14) turn to zero. Furthermore, as the first yz z xy z term on the right-hand side of (14) is zero, one may not introduce (16) into (14) because the last factor in (16) is not From this, the triple product rule can easily be obtained. zero. The flaw in this derivation is as follows. Equation (9) A. Discussion and Conclusions cannot be substituted into (8), since dy in (8) is taken at The triple product rule is used in the derivations of the constant z: dy = dy(z = const), while dy in (9) is for varying z. Mayer's relation and the heat capacity ratio [5,6], which relate A specialist in mathematical physics sent me the following the isobaric and isochoric heat capacities C and C . “mathematically rigorous proof” of (1): P V According to the Mayer's relation, C C > 0, and from the “Assume there are an open set Ω of (x, y, z) and three P V heat capacity ratio, C /C > 1. In [5,6], it was shown that the differentiable functions , , and such that on Ω, the P V calculation of heat capacities with the Mayer's relation and the equation f(x, y, z) = 0 is equivalent to heat capacity ratio gives wrong results; for example, the x y,,,,, z y x z z x y (12) isochoric heat capacity becomes greater than the isobaric one [5]. In thermodynamics there is a relation that connects the Since all three equations hold on Ω, each one can be thermal expansion coefficient and the isothermal substituted in the others. So compressibility [6]: y x,, x y (13) and for a constant y, differentiation with respect to x gives VVP . (19) TPT PTV 0 1x , x , y 2 x , x , y 1 x , y , It follows from (1). Let us check (19) for a gas. Let us heat a gas by the heat exchange (introducing a quantity of heat q (14) into it). Its temperature and volume will then increase and the derivative on the left-hand side of (19) will be positive. The where 1 and 2 are the derivatives with respect to the first increase in temperature also causes an increase in pressure and second argument of the and therefore both derivatives on the right-hand side of (19) function, respectively. Moreover are positive. (The volume in the numerator of the derivative on the right-hand side of (19) increases because of heating.) x x,, z z , (15) This is a contradiction, since both sides of (19) must have equal signs. Now let us heat the gas with compression. In this case, one so differentiation with respect to x gives can write the following equation: 1 11 x , z , z x , z . (16) VTV (20) PPT TVP Here, on the left-hand side, the derivative is negative, the first Combining (14) and (16) gives derivative on the right-hand side is positive, and the second derivative is negative because the decrease in volume leads to the temperature increase. Here, it is necessary to note that in 1 1 x , z , z 2 x , x , y 1 x , y . (17) this derivative, and thus Retrieval Number: 100.1/ijam.B1102101121 Published By: DOI: 10.35940/ijam.B1102.101221 2 Lattice Science Publication Journal Website: www.ijam.latticescipub.com © Copyright: All rights reserved. Indian Journal of Advanced Mathematics (IJAM) ISSN: 2582-8932 (Online), Volume-1 Issue-2, October 2021 the volume in the numerator is the argument and the 11. I.A. Stepanov, “Exact calculation of the internal energy of the ideal temperature in the denominator is the function [7]. gas in statistical mechanics,” Phys. Sci. Int. J. vol. 14(2), Febr. 2017, pp.15. Free online. The change in the volume causes the change in the 12. I.A. Stepanov, “Determination of the isobaric heat capacity of gases temperature. Again, both sides of (20) have different signs, heated by compression using the Clément-Desormes method,” J. which is a clear contradiction. Chem. Biol. Phys. Sci. Section C vol. 10, May 2020, pp.108116. The authors of [8] performed an experimental check of (1) Free online. and (19), assuming that x = F was the tension of a rubber band, y = L was its length, and z = T was the absolute AUTHORS PROFILE temperature, and found that the right-hand side of (1) was Igor Stepanov, got MSc in physics from the equal to 0.88. In [6], an analysis of their work was given and University of Latvia in 1982 and PhD from the a mistake was found: from their experiment, it follows that the University of Sheffield in 2012. Currently I work as a researcher at the Liepaja University, Latvia. Some of right-hand side must be positive. Its absolute value is still not my important findings are: clear, and the conclusion in [6] that it is equal to 0.88 is premature, although plausible. The signs of the partial derivatives in (19) obtained in [8] have been confirmed by The 3-dimensional Ising model was solved exactly.
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