LOGISTICS: Brian Turrell will take the Friday lecture this week, as I will be in Toronto at a CIAR meeting.
HANDOUTS: Assignment 4
READING: Finish up Ch. 3.
This might look a little dangerous: if
< F, we could get P
> 1!
This can never happen, though, since Z is dominated by the smallest
min:
Z = exp(-F/
) >
exp(-
min/
)
so F <
min
and therefore no
can ever be smaller than F.
A long skinny box (cross sectional area A and length x, so
volume V = A x) contains a particle in a certain quantum state
| > for which there are nx
half-wavelengths in the length x and therefore a momentum in the x
direction given by px = h/
.
This produces a pressure p on the ends of the box. If one end moves
a distance dx under the force pA the work done on the particle
is dW = - p A dx = - p dV. In this process the wavelength gets longer
but the particle stays in the same state (though the state itself changes).
Thus there is no change of entropy in this process. All that changes
is the volume and the energy. Mathematically, dU =
(
U/
V)
dV for such an adiabatic (isentropic) process.
Comparing this with the work done on the particle (or a gas of independent,
non-interacting particles obeying the same rules) implies that p = -
(U/
V)
which is (one of) the formal definition(s) of p:
the volume rate of change of energy at constant entropy.
If we specify constant entropy processes
(d = 0)
and recall the original definition of temperature
as well as the definition of pressure derived above, then we find
p =
(
/
V)U,
an alternate definition of pressure.
dU =
d
- p dV
Taking the exact differential yields
dF = dU - d
-
d
=
d
- p dV -
d
-
d
or dF = -
d
- p dV. That is, the free energy is constant in any
reversible process at constant temperature and volume.
F is always minimum in thermal equilibrium.
dF =
- d
- p dV
Note that the ``formula'' for the free energy,
F = - log
,
can be derived from the above definition of the entropy and the
thermodynamic definition of F using the expression for
the energy U in terms of a temperature derivative of log Z.
Check it for yourself.
(/
V)
= (
p/
)V
p = -(U/
V)
+
(
/
V)
We have Z = 1 + e-/
giving F = -
log(1 + e-
/
)
From this and the definition of the entropy as the temperature derivative of F
at constant volume, we obtain (after a little calculus)
= log(1 + e-
/
)
+ (
/
) /
(e
/
) + 1).
We have Z = 1/(1 -
e-/
)
giving F =
log(1 - e-
/
).
Thus
= - log(1 -
e-
/
)
+ (
/
) /
(e
/
) - 1).
p = -(U /
V)
=
(
/
V)U
= -(
F /
V)
What we need now to do any practical calculations is a
specific result - a formula for p in terms of
N, V and , for instance.
Such a relationship is called an equation of state.
On Friday Brian Turrell will introduce the most important one of all,
that for an ideal gas.