- #1
CptXray
- 22
- 3
- Homework Statement
- For an ideal gas of N particles in a cylindrical container with constant gravitational potential ##\vec{g}=g\hat{e}_{z}##: find system's internal energy, entropy and specific heat in constant volume. The system is in thermal equilibrium with surroundings of temperature ##T##.
- Relevant Equations
- Partition function:
$$Q = \frac{1}{N! h^{3N}} \int \exp{(-H\beta)} d \Gamma \textrm{,}$$
where ##d\Gamma = d\vec{r}_{1}...d\vec{r}_{N}d\vec{p}_{1}...d\vec{p}_{N}##, ##\beta=\frac{1}{k_{B}T}##, ##k_{B}## - Boltzmann constant, ##d\vec{r}_{i}=r_{i}dr_{i}d\phi_{i} dz_{i}##.
Hamiltonian in cylindrical coordinates:
$$\frac{1}{2m}(p^2_{r}+p^2_{\phi}/r^2+p^2_{z})+gz$$
Helmholtz free energy:
$$F=-k_{B}T\ln(Q)$$
Entropy:
$$S=-\frac{\partial F}{\partial T}$$
Specific heat at constant volume:
$$C_{V}=T\frac{\partial S}{\partial T}$$
It looks more like a computational obstacle, but here we go.
Plugging all of these to the partition function:
$$Q = \frac{1}{N! h^{3N}} \int -\exp(\frac{1}{2m}(p^2_{r}+p^2_{\phi}/r^2+p^2_{z})+gz)d\Gamma=$$
$$= \frac{1}{N! h^{3N}} \int \exp{(\frac{-1}{2m}p^2_{r})}dp_{r_{1}}...dp_{r_{N}}
\int \exp{(\frac{-1}{2m}p^2_{z}-gz)}dp_{z_{1}}...dp_{z_{N}} \cdot \\
\cdot \int \exp(\frac{-1}{2m}p^2_{\phi}/r^2)dp_{\phi_{1}}...dp_{\phi_{N}}d\vec{r}_{1}r_{1}...d\vec{r}_{N}r_{N}\textrm{.}$$
I know how to calculate the first integral with respect to momenta ##p_{r}## and the one with ##p_{\phi}## momenta.
$$\int \exp{(\frac{-1}{2m}p^2_{r})}dp_{r_{1}}...dp_{r_{N}} = (2m\pi)^{N/2} \textrm{,}$$
$$\int \exp(\frac{-1}{2m}p^2_{\phi}/r^2)dp_{\phi_{1}}...dp_{\phi_{N}} = (2m\pi)^{N/2}r^N \textrm{.}$$
What I'm left with is:
$$
Q = \frac{1}{N! h^{3N}} (2m\pi)^{N}r^N \int \exp{(\frac{-1}{2m}p^2_{z}-gz)}dp_{z_{1}}...dp_{z_{N}}\int d\vec{r}_{1}r_{1}...d\vec{r}_{N}r_{N}
\textrm{.}
$$
the last part should be ##N## times the volume of a cylinder (I guess):
$$
Q = \frac{1}{N! h^{3N}} (2m)^{N}r^{3N}\pi^{2N} z^{N} \int \exp{(\frac{-1}{2m}p^2_{z}-gz)}dp_{z_{1}}...dp_{z_{N}}
\textrm{.}
$$
But I'm quite certain it's wrong at this point because ##Q## should be ##N##-dimensional. I'd really appreciate help, hints, literature that anybody could provide.
Plugging all of these to the partition function:
$$Q = \frac{1}{N! h^{3N}} \int -\exp(\frac{1}{2m}(p^2_{r}+p^2_{\phi}/r^2+p^2_{z})+gz)d\Gamma=$$
$$= \frac{1}{N! h^{3N}} \int \exp{(\frac{-1}{2m}p^2_{r})}dp_{r_{1}}...dp_{r_{N}}
\int \exp{(\frac{-1}{2m}p^2_{z}-gz)}dp_{z_{1}}...dp_{z_{N}} \cdot \\
\cdot \int \exp(\frac{-1}{2m}p^2_{\phi}/r^2)dp_{\phi_{1}}...dp_{\phi_{N}}d\vec{r}_{1}r_{1}...d\vec{r}_{N}r_{N}\textrm{.}$$
I know how to calculate the first integral with respect to momenta ##p_{r}## and the one with ##p_{\phi}## momenta.
$$\int \exp{(\frac{-1}{2m}p^2_{r})}dp_{r_{1}}...dp_{r_{N}} = (2m\pi)^{N/2} \textrm{,}$$
$$\int \exp(\frac{-1}{2m}p^2_{\phi}/r^2)dp_{\phi_{1}}...dp_{\phi_{N}} = (2m\pi)^{N/2}r^N \textrm{.}$$
What I'm left with is:
$$
Q = \frac{1}{N! h^{3N}} (2m\pi)^{N}r^N \int \exp{(\frac{-1}{2m}p^2_{z}-gz)}dp_{z_{1}}...dp_{z_{N}}\int d\vec{r}_{1}r_{1}...d\vec{r}_{N}r_{N}
\textrm{.}
$$
the last part should be ##N## times the volume of a cylinder (I guess):
$$
Q = \frac{1}{N! h^{3N}} (2m)^{N}r^{3N}\pi^{2N} z^{N} \int \exp{(\frac{-1}{2m}p^2_{z}-gz)}dp_{z_{1}}...dp_{z_{N}}
\textrm{.}
$$
But I'm quite certain it's wrong at this point because ##Q## should be ##N##-dimensional. I'd really appreciate help, hints, literature that anybody could provide.
Last edited: