Statistical Mechanics - Physics at Oregon State University

48 CHAPTER 3. VARIABLE NUMBER OF PARTICLES
is an ideal gas. This is again an idealization. The internal chemical potential
for an ideal gas with density n and temperature T is
ˆµ(n, T ) = kBT log(n) + µ0(T ) (3.18)
which only depends on n via the logarithm. The gas in this column in the
atmosphere is in diffusive contact, and equilibrium requires that the chemical
potential is the same everywhere. Therefore we find:
which leads to
kBT log(n(h)) + mgh = kBT log(n(0)) (3.19)
mgh
− k n(h) = n(0)e B T (3.20)
Because the gas inside a small slice is ideal, we can use the ideal gas equation
of state, pV = NkT , to obtain the isothermal atmospheric pressure relation
mgh
− k p(h) = p(0)e B T (3.21)
3.3 Differential relations and grand potential.
The six most common basic variables describing a system are the extensive parameters
entropy S, volume V, number of particles N, and the intensive parameters
temperature T, pressure p, and chemical potential µ. Every new measurable
quantity adds one extensive parameter and one intensive parameter to this list.
The free energy is measures by the internal energy U or of a quantity derived
from the internal energy by a Legendre transformation. Of the six basic variables
only three are independent, the other three can be derived. As we have
seen before we need at least one extensive parameter in the set of independent
variables, otherwise the size of the system would be unknown.
If our choice of basic variables is T,V, and N, the way to construct the other
functions is via the partition function and the Helmholtz free energy, using
partial derivatives to get S,P, and µ and using U = F + T S to get U. If our
choice of basic variables is U,V, and N, we calculate the entropy first. Remember
that
and
1
T =
p
T =
∂S
∂U V,N
∂S
∂V U,N
(3.22)
(3.23)
The third partial derivative can be obtained in the following way. A general
change in the entropy as a result of changes in the basic variables is