Connecting thermodynamics to students' calculus

American Journal of Physics

Volumn 72, Issue 6, 2004, Pages 753-757

  Cannon, Joel W ^a  

^a WASHINGTON AND JEFFERSON COLLEGE   (United States)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 2642558559     PISSN: 00029505     EISSN: None     Source Type: Journal    
DOI: 10.1119/1.1648327     Document Type: Article

References (25)

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. 4. " 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. 4. " 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.
5. 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.
6. 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.
7. 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.
8. Reference 3, p. 68.
9. Reference 3, p. 246.
10. Reference 3, p. 262.
11. Daniel V. Schroeder, An Introduction to Thermal Physics (Addison-Wesley-Longman, Reading, MA, 2000), p. 150. Note that this definition assumes a quasistatic process.
12. Reference 9, p. 149.
13. 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.
14. 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.
15. 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.
16. Reference 4, Thomas and Finney, p. 862, features a section on "Partial Derivatives with Constrained Variables."
17. In a survey of commonly used calculus textbooks (Refs. 14-17). I found no problems involving partial derivatives where "all other independent" and "all other" variables were not equivalent.
18. James Stewart, Calculus (Brooks/Cole, Pacific Grove, CA, 1999), 4th ed., p. 840.
19. Dale Varberg and Edwin Purcell, Calculus with Analytical Geometry (Prentice-Hall, Englewood Cliffs, NJ, 1992), 6th ed., p. 687.
20. Earl W. Swokowski, Calculus with Analytical Geometry (PWS Kent, Boston, 1983), 2nd alternate ed., pp. 679-680.
21. Reference 4, Thomas and Finney, p. 840.
22. 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.
23. 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.
24. I have heard Professor Robert J. Hardy of the University of Nebraska express similar concerns.
25. A copy of a class handout that implements this procedure can be obtained by writing to the author.