Ideal gas in a cylindrical container

• CptXray
In summary, the conversation discusses the calculation of a partition function using integrals for momenta and position. The final result should be N-dimensional, but there is confusion about the inclusion of temperature and the effect of gravitational potential energy. The speaker also suggests using Cartesian coordinates for simplicity and mentions the need for limits on z.
CptXray
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##-dimentional. I'd really appreciate help, hints, literature that anybody could provide.

Last edited:
Looks N dimensional to me. What happened to Temperature?

The gravitational potential energy should be ## mgz ##, if I'm not mistaken.

I think it will be simpler to work in Cartesian coordinates rather than cylindrical coordinates.

Are you given any information about the height of the cylinder?

Abhishek11235
CptXray said:
the last part should be NNN times the volume of a cylinder (I guess):

It would if ##e^{-gz}## wasn't present. In that case you need limits on ##z##

What is an ideal gas?

An ideal gas is a theoretical gas that follows the ideal gas law, which describes the relationships between pressure, volume, temperature, and number of moles of gas particles. Ideal gases do not exist in real life, but they serve as a useful model for understanding the behavior of real gases.

What is a cylindrical container?

A cylindrical container is a three-dimensional shape with a circular base and straight sides that are perpendicular to the base. It is similar to a cylinder and is often used to store and transport gases and liquids.

What are the assumptions of an ideal gas in a cylindrical container?

The assumptions of an ideal gas in a cylindrical container include that the gas particles are in constant, random motion; they do not interact with each other; and the volume of the gas particles is negligible compared to the volume of the container. Additionally, the container is assumed to have perfectly rigid walls and the gas particles are assumed to have elastic collisions with the walls.

How does temperature affect an ideal gas in a cylindrical container?

According to the ideal gas law, increasing the temperature of an ideal gas in a cylindrical container will cause an increase in its volume, assuming the other variables remain constant. This is because the increased temperature causes the gas particles to move faster and collide with the walls of the container more frequently and with more force, resulting in an increase in pressure and volume.

What happens to an ideal gas in a cylindrical container when pressure is applied?

If pressure is applied to an ideal gas in a cylindrical container, the volume of the gas will decrease, assuming the other variables remain constant. This is because the pressure causes the gas particles to become more tightly packed together and the volume of the particles becomes a larger fraction of the total volume, resulting in a decrease in volume.

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