Statistical physics assignment 1
1. Fugacity:
Work out the fugacity of air at 290K (recall that the average molecular weight is 29!). Do we
need to worry about quantum statistics?
2. State counting:
(a) A Bose-Einstein system has k = 2 states a; b with ga = 2; gb = 1 and N = 3 . By
drawing energy level diagrams and assigning identical balls to represent particles, verify
the theoretical formulae for
f3;0g ,
f0;3g ,
f2;1g , and
f1;2g in the BE case. (Hint: to
aid visualisation, draw the energy levels as 2 lines side by side for the lower two degenerate
levels, with one line above them for the upper level.)
(b) Suppose in some related system, b becomes equal to a . Now check that your total count
in (a), now agrees with
f3g for the new system.
(c) Repeat the calculation of each case in (a), this time for a Maxwell-Boltzmann system, by
allocating distinguishable balls to the levels, and check the formula in each case.
(d) Draw the single configuration possible for a Fermi-Dirac system with N = 3 , and k = 2
states a; b with ga = 2; gb = 1 .
3. Schottky defects:
Atoms displaced from lattice sites inside a crystalline solid can migrate by thermal creeping
and eventually arrive at the crystal surface. Consider an crystal with N atoms altogether, of
which there are n Schottky defects (you may have to assume n  N ). Suppose the energy of
formation of a single Schottky defect is w.
(a) Show that the entropy of the system is
S(n) = kB ln
N!
n!(N 􀀀 n)!
:
(b) Using this, and maximising S at equilibrium, show that at temperature T the defect density
is
n
N 􀀀 n
=
1
ew=kBT 􀀀 1
:
(c) Estimate this density if w = 1 eV and T = 290K or 103K

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