256
5 Atomic Systems
Z g (T , V g ; N g ) =
z
N g
trans (T , V g )
N g !
z
N g
int (T ).
The total number of atoms, N = N s + N g , in the combined vapour–solid
equilibrium system is constant. If the Helmholtz energy for the combined
system is given by A(T , V ; N, N g ) = A s (T ; N s ) + A g (T , V g ; N g ), minimize
A(T , V ; N, N g ) with respect to N g , and thereby show that N g is given by
N g =
V
3
2 sinh
E
2T
3
e
−ββ ,
in which (T ) is the thermal de Broglie wavelength and β = (k B T ) −1 .
30. By treating the vapour in the previous problem as an ideal gas, show that the
vapour pressure P associated with a monatomic crystal is given by
P =
k B T
3 (T )
2 sinh
E
2T
3
e
− B T ) .
31. Traditionally, the condition for equilibrium to be established between two
phases of the same substance is equality of the chemical potentials for the two
phases. Show that minimization of the Helmholtz energy with respect to the
number of particles in one of the phases, as carried out in Problem 29, leads to
equality of the chemical potentials for the two phases.
References
1. P.T. Landsberg, J. Dunning-Davies, D. Pollard, Amer. J. Phys. 62, 712 (1994)
2. G. Cook, R.H. Dickerson, Amer. J. Phys. 63, 737 (1995)
3. I.N. Levine, Quantum Chemistry, 7th edn. (Pearson, New Jersey, 2014)
4. F.S. Levin, An Introduction to Quantum Theory (Cambridge University Press, Cambridge, 2002)
5. L. Pauling, E.B. Wilson, Jr., Introduction to Quantum Mechanics (McGraw-Hill, New York,
1935)
6. H.A. Bethe, R. Jackiw, Intermediate Quantum Mechanics (Benjamin, New York, 1968)
7. F. Hund, Linienspektren und periodisches System der Elemente (Springer-Verlag OHG, Berlin,
1927)