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**1. Homework Statement**

Pathria 6.8: Evaluate the partition function of a classical ideal gas consisting of N molecules of mass m confined to a cylinder of vertical height L, which is in a state of thermal equilibrium at constant T in a uniform gravitational field of acceleration g. Calculate the specific heat and explain why it is larger than the free space value. (The last part is my problem).

**2. Homework Equations**

PV=nrT

partition function Z = sum (e^(-BE))

**3. The Attempt at a Solution**

I made it pretty far fr the first part, although I think I made a mistake somewhere. I'm really looking for help on why the specific heat is larger than the free space value. I am wondering if it has to do with pressure due to gravity.

I used the barometric Boltzmann distribution, taking an arbitrary cross-sectional area A.

So a cross-section of the column has a volume Adh and mass m=ro(h) Adh

So the cross-section exerts a force mg ro(h) n the gass below it

So the excess pressure at height h over the pressure at h+dh is: P(h) -P(h+dh) = -dp = mg ro(h) dh

pV=nRT

ro = N/V

n=N/N(0)

so P=ro k t

dro(h)/dh = -(mg/kt) ro(h)

ro(h)= ro(h(0)) exp[(-mg(h-h0)/kT]

epsilon=mgh

sum(n eps) = E

sum (n) = N

general partition function is: Q=sum [exp(-BE)]

E=sum(n(sub h) mgh)

assume h(0)=0

Z=sum [exp(-B n(sub h) mgh)] = exp(-BmgNL^2/2) --> is this right??

to get the specific heat:

U = N partial ln Q / partial B = N exp (-BmgNL^2/2) (-mgNL^2/2)

= -(mgN^2L^2/2) exp(-BmgNL^2/2)

Cv = dU/dT = (-mgN^2L^2/2T)^2 (1/k) exp(-BmgNL^2/2) --> somehow I think this is wrong, it looks like it has the wrong units

now how do I explain why the specific heat with the gravitational field (this answer) is larger than the free space value?

**1. Homework Statement**

**2. Homework Equations**

**3. The Attempt at a Solution**