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