Pathria: Specific Heat in a gravitational field

[howin]

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).

Homework Equations

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

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?

 

[Jhero]

Thank you for your post. It seems like you have made some good progress on evaluating the partition function for a classical ideal gas in a gravitational field. However, I believe there are a few mistakes in your calculations that may have led to your confusion about the specific heat being larger than the free space value.

First, in your expression for the excess pressure at height h, you have used the barometric Boltzmann distribution, which is only valid in the case of a constant gravitational field. In this problem, the gravitational field is not constant, as it varies with height. Therefore, you cannot use this distribution and instead you should use the more general Boltzmann distribution:

P(h) = P(0) exp(-mgh/kT)

where P(0) is the pressure at the bottom of the cylinder.

Next, in your expression for the partition function, you have assumed that the height of the cylinder is L. However, the problem statement mentions a "vertical height L," which implies that the cylinder may have a different length in the horizontal direction. Therefore, the partition function should be:

Z = sum[exp(-nBmgh)]

where n is the number of molecules in the cylinder and B is the Boltzmann factor (1/kT).

With these corrections, the partition function can be evaluated to be:

Z = (1 - exp(-nBmgL))/ (1 - exp(-BmgL))

The specific heat can then be calculated as:

Cv = (kT)^2 (d^2 ln Z/ dT^2)

= (kT)^2 (d/dT) [(nBmgL)/(1 - exp(-BmgL))]

= (kT)^2 [(nBmgL)^2 exp(-BmgL)]/[(1 - exp(-BmgL))^2]

This expression for the specific heat is indeed larger than the free space value, which can be obtained by taking the limit as L goes to infinity. This is because in the presence of a gravitational field, the potential energy of the molecules is not constant, but varies with height. Therefore, the average energy of the molecules will also vary with height, leading to a larger specific heat compared to the case of free space, where the potential energy is constant.

I hope this explanation helps you understand why the specific heat is larger in the presence of a gravitational field. Keep up the good work on your calculations