This paper offers some qualitative understanding of the chemical potential, a topic that students invariably find difficult. Three “meanings” for the chemical potential are stated and then supported by analytical development. Two substantial applications—depression of the melting point and batteries—illustrate the chemical potential in action. The origin of the term “chemical potential” has its surprises, and a sketch of the history concludes the paper.

1.
Ralph Baierlein, Thermal Physics (Cambridge U. P., New York, 1999), pp. 148–155.
2.
The partition function Z is the sum of Boltzmann factors taken over a complete, orthogonal set of energy eigenstates. Consequently, both Z and F depend on temperature, the number of particles of each species that is present, and the system’s external parameters. If more than one species is present, the chemical potential μi for species i is the derivative of F with respect to Ni, computed while the numbers of all other particles are kept fixed and while temperature and all external parameters are held fixed.  Equation (6) indicates that the chemical potential may be computed by applying calculus to a function of N or by forming the finite difference associated with adding one particle. When N is large, these two methods yield results that are the same for all practical purposes. Convenience alone determines the choice of method.
3.
Charles Kittel and Herbert Kroemer, Thermal Physics, 2nd ed. (Freeman, New York, 1980), pp. 118–125.
4.
F. Mandl, Statistical Physics (Wiley, New York, 1971), pp. 222–224.
5.
For examples, see Ref. 1, p. 154, and Ref. 3, pp. 118–120.
6.
Martin Bailyn, A Survey of Thermodynamics (AIP, New York, 1994), p. 213.
7.
Reference 1, pp. 233–234. Nuclear spin systems may be in thermal equilibrium at negative absolute temperatures. In such a case, the Helmholtz free energy attains a maximum. For more about negative absolute temperatures, see Ref. 1, pp. 343–347 and the references on pp. 352–353.
8.
Reference 6, pp. 212–213; Ref. 1, p. 228.
9.
J. Willard Gibbs, The Scientific Papers of J. Willard Gibbs: Vol. I, Thermodynamics (Ox Bow, Woodbridge, CT, 1993), pp. 56, 64, and 65, for examples.
10.
Relation (21) can be seen as a consequence of a remarkable mathematical identity, most often met in thermodynamics: if the variables {x,y,z} are mutually dependent, then (∂x/∂y)z(∂y/∂z)x(∂z/∂x)y=−1. To understand why the minus sign arises, note that not all of the variables can be increasing functions of the others. For a derivation, see Herbert B. Callen, Thermodynamics (Wiley, New York, 1960), pp. 312–313. The correspondence {S,N,E}={x,y,z} yields Eq. (21), given Eq. (20) and the expression (∂S/∂E)N,V=1/T for absolute temperature.
11.
A classic reference is S. R. De Groot and P. Mazur, Non-Equilibrium Thermodynamics (North-Holland, Amsterdam, 1962), Chaps. 3, 4 and 11.
James A. McLennan provides a succinct development in his Introduction to Nonequilibrium Statistical Mechanics (Prentice–Hall, Englewood Cliffs, NJ, 1989), pp. 17–25.
H. B. G.
Casimir
develops an illuminating example in his article, “
On Onsager’s Principle of Microscopic Reversibility
,”
Rev. Mod. Phys.
17
,
343
350
(
1945
)—but beware of typographical errors.
A corrected (but less lucid) version is given by H. J. Kreuzer, Nonequilibrium Thermodynamics and its Statistical Foundations (Oxford U. P., New York, 1981), pp. 60–62.
12.
As used here, the term “multiplicity” denotes the number of microstates associated with a given macrostate. Entropy is then Boltzmann’s constant times the logarithm of the multiplicity.
13.
Reference 3, pp. 134–137. Francis W. Sears and Gerhard L. Salinger, Thermodynamics, Kinetic Theory, and Statistical Thermodynamics, 3rd ed. (Addison–Wesley, Reading, MA, 1975), pp. 327–331.
14.
C. B. P. Finn, Thermal Physics (Routledge and K. Paul, Boston, 1986), pp. 190–193. Also Ref. 4, pp. 224–225.
15.
Here are two more characterizations that offer little help. Some authors describe the chemical potential as a “driving force” (e.g., Ref. 3, p. 120). Others call μ or a difference in μ a “generalized force” [e.g., Herbert B. Callen, Thermodynamics (Wiley, New York, 1960), pp. 45–46]. These strike me as unfortunate uses of the word “force.” In mechanics, students learn that a “force” is a push or a pull. But no such push or pull drives the statistical process of diffusion. The terms “driving force” and “generalized force” are more likely to confuse than to enlighten.
16.
As used in this paper, the term “semi-classical” means that the thermal de Broglie wavelength is much smaller than the average interparticle separation. Hence quantum mechanics has no substantial direct effect on dynamics. Nonetheless, vestiges of the indistinguishability of identical particles persist (as in division by N! in the partition function), and Planck’s constant remains in the most explicit expressions for entropy, partition function, and chemical potential. A distinction between fermions and bosons, however, has become irrelevant.  The analysis in Sec. III treated the helium atoms as a semi-classical ideal gas.
17.
See Ref. 12.
18.
David L. Goodstein provides similar (and complementary) reasoning in his States of Matter (Prentice-Hall, Englewood Cliffs, NJ, 1975), p. 18.
19.
G.
Cook
and
R. H.
Dickerson
, “
Understanding the chemical potential
,”
Am. J. Phys.
63
,
737
742
(
1995
).
20.
Sherwood R.
Paul
, “
Question #56. Ice cream making
,”
Am. J. Phys.
65
,
11
(
1997
).
21.
The most germane of the answers is that by
F.
Herrmann
, “
Answer to Question #56
,”
Am. J. Phys.
65
,
1135
1136
(
1997
).
The other answers are
Allen
Kropf
,
Am. J. Phys.
65
,
463
(
1997
);
M. A.
van Dijk
,
Am. J. Phys.
65
,
463
464
(
1997
); and
Jonathan
Mitschele
,
Am. J. Phys.
65
,
1136
1137
(
1997
).
22.
Reference 1, pp. 279–280.
23.
Although the entropy per particle is usually higher in a liquid than in the corresponding solid, exceptions occur for the helium isotopes 3He and 4He on an interval along their melting curves. See J. Wilks and D. S. Betts, An Introduction to Liquid Helium, 2nd ed. (Oxford U. P., New York, 1987), pp. 15–16.
24.
Frank C. Andrews, Thermodynamics: Principles and Applications (Wiley, New York, 1971), pp. 211–223.
Also,
Frank C.
Andrews
, “
Colligative Properties of Simple Solutions
,”
Science
194
,
567
571
(
1976
).
For a more recent presentation, see Daniel V. Schroeder, An Introduction to Thermal Physics (Addison–Wesley, Reading, MA, 2000), pp. 200–208.
25.
Recall that particles diffuse toward lower chemical potential. If the concentration increases locally, only an increase in μint will tend to restore the original state and hence provide stability. For a detailed derivation, see Ref. 6, pp. 232, 239, and 240.
26.
One might wonder, does charging the electrodes significantly alter the electrons’ intrinsic chemical potentials? To produce a potential difference of 2 V, say, on macroscopic electrodes whose separation is of millimeter size requires only a relatively tiny change in the number of conduction electrons present. Thus the answer is “no.”
27.
Reference 1, pp. 246–257.
28.
Some of the energy for dc operation may come from the environment (as an energy flow by thermal conduction that maintains the battery at ambient temperature). Such energy shows up formally as a term TΔS in ΔG [seen most easily in Eq. (B2)]. Measurements of dΔφ/dT show that this contribution is typically 10% of the energy budget (in order of magnitude). Gibbs was the first to point out the role of entropy changes in determining cell voltage (Ref. 9, pp. 339–349).
29.
Dana
Roberts
, “
How batteries work: A gravitational analog
,”
Am. J. Phys.
51
,
829
831
(
1983
).
30.
Wayne
Saslow
, “
Voltaic cells for physicists: Two surface pumps and an internal resistance
,”
Am. J. Phys.
67
,
574
583
(
1999
).
31.
Jerry Goodisman, Electrochemistry: Theoretical Foundations (Wiley, New York, 1987).
32.
Reference 9, pp. 63–65 and 93–95.
33.
Reference 9, pp. 146 and 332.
34.
A. Ya. Kipnis, “J. W. Gibbs and Chemical Thermodynamics,” in Thermodynamics: History and Philosophy, edited by K. Martinás, L. Ropolyi, and P. Szegedi (World Scientific, Singapore, 1991), p. 499.
35.
The letters are listed in L. P. Wheeler, Josiah Willard Gibbs: The History of a Great Mind, 2nd ed. (Yale U. P., New Haven, 1962), pp. 230–231.
36.
Reference 9, p. 425.
37.
E. A.
Guggenheim
introduced the phrase “electrochemical potential” and made the distinction in “
The conceptions of electrical potential difference between two phases and the individual activities of ions
,”
J. Phys. Chem.
33
,
842
849
(
1929
).
38.
Reference 3, pp. 124–125.
39.
Reference 9, p. 95–96.
40.
Reference 1, pp. 262–264.
41.
Elizabeth Garber, Stephen G. Brush, and C. W. F. Everitt, Maxwell on Heat and Statistical Mechanics: On “Avoiding All Personal Enquiries” of Molecules (Lehigh U. P., Bethlehem, PA, 1995), pp. 50 and 250–265.
42.
Reference 41, p. 259.
This content is only available via PDF.
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.