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The Ideal Gas
-------------
The basic equations:
Pi = (1/Z)exp(-βEi)
Z = Σie-βEi
E = -∂lnZ/∂β
KBT = 1/β
S = βE + lnZ
A = -KBTlnZ
Consider a box with volume, V, and N identical non-interacting
particles. The states are the continous collection of the position
and momentum states for each particle. Compute the partition
function.
Z = ∫dx3Ndp3Nexp(-βΣn(pn2/2m))
= (VN/N!)[∫exp(-βp2/2m)dp]3N
The N! is to compensate for the fact that the states in classical
mechanics are distinguishable.
Let u2 = βp2/2m ∴ p = u√(2m/β) and dp = √(2m/β)du. The ∫
becomes:
Z = (VN/N!)[√(2m/β)∫exp(-u2)du]3N
= (VN/N!)(2mπ/β)3N/2
Now,
lnZ = -(3/2)Nlnβ + constant
E = ∂lnZ/∂β
= (3/2)NKBT where β = 1/KBT
This is the familiar term associated with the energy of an ideal
gas.
Pressure
--------
Consider a piston of area, A that is moved a distance, dx, by
the gas (the gas is doing work on the piston).
dE = -PAdx (work = force x distance)
= PdV
∴ P = -∂E/∂V
Consider an adiabatic process. The change in entropy associated with
an adiabatic process is 0 (ΔS = 0). Thus, we can write:
P = -∂E/∂V|S
Theorem:
∂E/∂V|S = ∂E/∂V|T - (∂E/∂S|V)(∂S/∂V|T)
Proof:
Consider following diagram:
T
| \
| \
| \
| \
| \
| \ constant S (adiabatic)
|
------------------------------ V
How does E change along a line of constant S?
ΔE/ΔV|T = (∂E/∂V|T)ΔV + (∂E/∂V|V)ΔT
ΔE = (∂E/∂V|T)ΔV + (∂E/∂V|V)ΔT
ΔE/ΔV = ∂E/∂V|T + (∂E/∂V|V)(ΔT/ΔV)
= ∂E/∂V|T + (∂S/∂T|V)(∂E/∂S|V)(ΔT/ΔV)
Along the S curve dS = 0. Therefore,
dS = (∂S/∂V|T)ΔV + (∂S/∂T|V)ΔT = 0
Therefore,
ΔT/ΔV = -(∂S/∂V|T)/(∂S/∂T|V)
Substituting back into the ΔE/ΔV equation gives:
ΔE/ΔV ≡ ∂E/∂V|S = ∂E/∂V|T - (∂S/∂T|V)(∂E/∂S|V)(∂S/∂V|T)/(∂S/∂T|V)
= ∂E/∂V|T - (∂E/∂S|V)(∂S/∂V|T) Q.E.D.
Now T = ∂E/∂S|V. Therefore,
∂E/∂V|S = ∂E/∂V|T - T∂S/∂V|T
= ∂(E - TS)/∂V|T
= ∂A/∂V|T where A is the Helmholtz Free energy
P = -∂E/∂V|S
= -∂E/∂V|T + (∂E/∂S|V)(∂S/∂V|T)
= -∂(E - TS)/∂V|T
= -∂A/∂V|T
= KBT∂lnZ/∂V|T since A = -KBTlnZ
From the ideal gas we found:
Z = (VN/N!)(2mπ/β)3N/2
Therefore,
lnZ = NlnV + (3N/2)ln(2mπ/β) - lnN!
This leads to:
P = KBT∂lnZ/∂V|T = KBTN/V
Or,
PV = NKBT