I describe subtle calculus ideas that are essential for thermodynamics, but are typically not encountered by students in calculus or prior physics classes. I argue that these previously unencountered subtleties are a substantial cause of the difficulty that many students encounter in learning thermodynamics and that thermodynamics can be taught more effectively by introducing the subtleties within an environment with which students are familiar rather than insisting that students learn them at the same time that they encounter new physics concepts such as entropy and thermodynamic potentials. I show how Legendre transforms can be used to illustrate the important calculus concepts and the nature of thermodynamics calculations. An added advantage of this approach is that it provides a coherent picture of the thermodynamic potentials.

Topics
Entropy, Thermodynamics, Convex function, Education
1.
Here E is the internal energy, F the Helmholtz free energy, H the enthalpy, and G the Gibbs free energy.
2.
Craig F. Bohren argues that this notation should be adopted in thermodynamics. See Craig F. Bohren and Bruce A. Albrecht, Atmospheric Thermodynamics (Oxford U. P., New York, 1998).
3.
Charles Kittel and Herbert Kroemer, Thermal Physics (Freeman, San Francisco, 1980), 2nd ed., p. 40. The authors write that “The notation (∂g1/∂U1)N1 means that N1 is held constant in the differentiation of g1(N1,U1) with respect to U1. That is, the partial derivative with respect to U1 is defined as ∂g1/∂U1=limΔU1→0[g1(N1,U1+ΔU1)−g1(N1,U1)]/ΔU1. For example, if g(x,y)=3x4y, then (∂g/∂x)y=12x3y and (∂g/∂y)x=3x4.” Note that the description the authors give is identical to the partial derivative definition found in calculus books where this subscript is unnecessary. See, for example, James Stewart, Calculus (Brooks/Cole, Pacific Grove, CA, 1999), 4th ed., p. 931.
4.
George B. Thomas and Ross L. Finney, Calculus and Analytical Geometry (Addison–Wesley, Reading, MA, 1992), 8th ed., p. 834;
Dale Varberg and Edwin Purcell, Calculus with Analytical Geometry (Prentice–Hall, Englewood Cliffs, NJ, 1992), 6th ed., p. 684. I thank Professor Dean Morrow of the Washington and Jefferson College Mathematics Department for alerting me to how mathematics texts use subscripts.
5.
Because they were introduced by different people at different times, they did in some sense fall “from the sky as random and unrelated drops of rain.” However, there is no need for the students to relive this history when the concepts can be introduced in a unified manner.
6.
Reference 3, p. 68.
7.
Reference 3, p. 246.
8.
Reference 3, p. 262.
9.
Daniel V. Schroeder, An Introduction to Thermal Physics (Addison–Wesley–Longman, Reading, MA, 2000), p. 150. Note that this definition assumes a quasistatic process.
10.
Reference 9, p. 149.
11.
Reference 3, p. 68. I have replaced Kittel’s τ, σ, and U with T, S, and E. The title for the equation varies. Reif calls it the “fundamental thermodynamic relation.” See F. Reif, Fundamentals of Statistical and Thermal Physics (McGraw–Hill, New York, 1965), p. 161. Schroeder calls it the “thermodynamic identity,” see Ref. 9, p. 111.
12.
Reference 4, Thomas and Finney, p. 862, features a section on “Partial Derivatives with Constrained Variables.”
13.
In a survey of commonly used calculus textbooks (Refs. 14,15,16 17). I found no problems involving partial derivatives where “all other independent” and “all other” variables were not equivalent.
14.
James Stewart, Calculus (Brooks/Cole, Pacific Grove, CA, 1999), 4th ed., p. 840.
15.
Dale Varberg and Edwin Purcell, Calculus with Analytical Geometry (Prentice–Hall, Englewood Cliffs, NJ, 1992), 6th ed., p. 687.
16.
Earl W. Swokowski, Calculus with Analytical Geometry (PWS Kent, Boston, 1983), 2nd alternate ed., pp. 679–680.
17.
Reference 4, Thomas and Finney, p. 840.
18.
Ashley Carter and Ralph Baierlein explicitly introduce and discuss the properties of Legendre transforms in their texts (all texts implicitly introduce Legendre transforms, when they define F, G, and H)⁠. Similar to what I propose here Carter illustrates the Legendre transform’s use with abstract variables (rather than E, S, and F, etc.). What I suggest in this paper is that these introductions can be better targeted to deal with misunderstandings that have not been previously dealt with, and to confront the student with the reason for and the meaning of thermodynamic notation.
Ashley H. Carter, Classical and Statistical Thermodynamics (Prentice–Hall, Upper Saddle River, NJ, 2001), p. 130. Ralph Baierlein, Thermal Physics (Cambridge U. P., Cambridge, 1999), p. 225.
19.
I have heard Professor Robert J. Hardy of the University of Nebraska express similar concerns.
20.
A copy of a class handout that implements this procedure can be obtained by writing to the author.
This content is only available via PDF.
© 2004 American Association of Physics Teachers.
2004
American Association of Physics Teachers
AAPT members receive access to the American Journal of Physics and The Physics Teacher as a member benefit. To learn more about this member benefit and becoming an AAPT member, visit the Joining AAPT page.