Partition Function at a Fixed Pressure

[MathematicalPhysicist]

Homework Statement

I don't quite follow the solution to this problem (problem 2.11 in Bergersen's and Plischke's textbook), here are the quoted problem and its solution:

problem:

Consider a system of $N$ noninteracting molecules in a container of
cross-sectional area $A$. The bottom of the container (at $z = 0$) is rigid.
The top consists of an airtight piston of mass $M$ which slides without
friction.
(a) Construct the partition function $Z$ of the $(N + 1)$-particle system
($N$ molecules of mass $m$, one piston of mass $M$, cross-sectional area
$A$). You may neglect the effect of gravity on the gas molecules.
(b) Show that the thermodynamic potential $—k_BT\ln Z$ is, in the ther-
modynamic limit, identical to the Gibbs potential of an ideal gas of
$N$ molecules, subject to the pressure $P = Mg/A$.

solution:

(a) The Hamiltonian for the system consisting of $N$ particles plus the frictionless piston of mass $M$ is $$H = \sum_{i=1}^N p_i^2/(2m)+P_z^2/(2M) +Mgz$$

The partition function for a single gas molecule in a volume $V=Az$ is $Z_1 = \frac{Az}{\lambda^3}$ where $$\lambda = \sqrt{\frac{h^2}{2\pi m k_B T}}$$
The partition function for the complete system is then:
$$ Z = \frac{A^N}{N!\lambda^{3N}} \int_{-\infty}^\infty \frac{dP_z}{h}e^{\frac{-\beta P_z^2}{2M}}\int_0^\infty dz z^N e^{-\beta Mgz}$$
or
$$Z = \frac{A^N(\beta Mgz)^{N+1}}{\lambda^{3N}} \sqrt{\frac{2\pi M}{\beta h^2}}$$
We find $$(2.10) \ \ \ \ -k_B T \ln Z = - Nk_B T \ln \bigg(\frac{Ak_B T}{\lambda^3 Mg}\bigg)$$

(b) It was shown in the text that the chemical potential is given by $$ (2.11) \ \ \ \ \mu = k_B T \ln \bigg( \frac{N\lambda^3}{V} \bigg) = G/N$$

Identifying the pressure as $Mg/A$ and using the ideal gas law $PV=Nk_B T$ we see that $(2.10)$ and $(2.11)$ are equivalent.

Homework Equations

The Attempt at a Solution

My problem is with the solution to (a), it seems they plugged into the LHS of (2.10) $Z= Z_1^N$ where $k_B T = Mgz$ and not the expression $Z = \frac{A^N(\beta Mgz)^{N+1}}{\lambda^{3N}} \sqrt{\frac{2\pi M}{\beta h^2}}$, are they equivalent?

It doesn't look like that? what do you think?

 

[TSny]

My problem is with the solution to (a), it seems they plugged into the LHS of (2.10) $Z= Z_1^N$ where $k_B T = Mgz$ and not the expression $Z = \frac{A^N(\beta Mgz)^{N+1}}{\lambda^{3N}} \sqrt{\frac{2\pi M}{\beta h^2}}$, are they equivalent?

I don't understand the reason for the $z^N$ factor in the integrand below

[EDIT: Never mind, I see where the $z^N$ is coming from.]

With this expression for $Z$, then the next equation should read

where the exponent (N+1) should be -(N+1). Also, I don't think the $z$ in $(\beta Mgz)$ should be there. Of course, you should check this.

Then you get (2.10) if you assume N is very large and neglect certain small terms.