# A monatomic gas in a 2d Universe - multiplicity

• TFM
In summary, when trying to solve for the entropy of a monatomic ideal gas in a two-dimensional universe, it is helpful to use the results from the three-dimensional case to determine the multiplicity of the gas. There is also an expression for the entropy of the gas in terms of U, A and N. Finally, for a gas where all particles are indistinguishable, the entropy is simply the total multiplicity of the gas.
TFM

## Homework Statement

Consider a monatomic ideal gas that lives in a two-dimensional universe (“flatland”), occupying an area A instead of a volume V.

(a)

By following the logic of the derivation for the three-dimensional case, show that the multiplicity of this gas can be written:

$$\Omega = \frac{1}{N!}\frac{A^N}{h^{2N}}\frac{\pi^N}{N!}(2MU)^N$$

(b)

Find an expression for the entropy of this two-dimensional gas. Express your result in terms of U, A and N.

N/A

## The Attempt at a Solution

I have practically finished part a, with one small exception. for the pi fraction, I have 2pi instead of pi?

Some of my working out:

The space multiplicity:

$$\Omega_{space} = (\frac{A}{(\Delta x)^2})^N$$

The multiplicity of momentum:

$$\Omega_{mom} = (\frac{A_{hypercircle}}{(\Delta p_x)})^{2N}$$

$$\Omega_{mom} = (\frac{A_{hypercircle} ^{2N}}{(\Delta p_x^{2N})})$$

The total multiplicity is the multiple:

$$\Omega = \Omega_{space}\Omega_{mom}$$

$$\Omega = ((\frac{A}{(\Delta x)^2})^N )(\frac{A_{hypercircle} ^{2N}}{(\Delta p_x^{2N})} )$$

This can be rearranged, and using Heisenberg's Uncertainty princeple, $$\Delta x \Delta p_x \approx h$$

this gives:
$$\Omega = \frac{A^N A_{hypercircle}}{h^{2N}}$$

For indistinguishable particles:

$$\Omega = \frac{1}{N!}\frac{A^N}{h^{2N}}A_{hypercircle}$$

Area of hypercircle:

$$\frac{2\pi^N}{(N - 1)!}{\sqrt{2mu}^{2N-1}}$$

and using approximations:

$$\Omega = \frac{1}{N!}\frac{A^N}{h^{2N}}\frac{2\pi^N}{N!}{2mu^N}$$

As you can see, close, just that annoying 2...

Any ideas how to get rid of it?

TFM

I have attached the entre workings out. any idewa how to get rid iof that pesky 2...?

TFM

#### Attachments

• Workings Out.jpg
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Sorry for 2 year late reply. You're probably graduated by now. But according to Schroeder's book, for part (a),

you can simply throw away the 2 because the multiplicity is very large compared to the 2, and thus it won't really make a difference. He then apologizes below for sloppy working. But... oh well that's what I just read.

## 1. What is a monatomic gas?

A monatomic gas is a type of gas composed of single atoms, as opposed to diatomic gases which are composed of two atoms. Examples of monatomic gases include helium, neon, and argon.

## 2. What is a 2d Universe?

A 2d Universe is a hypothetical universe with only two spatial dimensions. This means that objects within this universe can only move in two directions, rather than the three dimensions we are familiar with in our own universe.

## 3. What is multiplicity in relation to a monatomic gas in a 2d Universe?

Multiplicity is a measure of the number of microstates that a system can have while maintaining the same macroscopic properties. In the case of a monatomic gas in a 2d Universe, multiplicity refers to the number of ways the atoms in the gas can be arranged within the two dimensions while still maintaining the same overall properties.

## 4. How does the multiplicity of a monatomic gas in a 2d Universe compare to that of a monatomic gas in a 3d Universe?

The multiplicity of a monatomic gas in a 2d Universe is generally lower than that of a monatomic gas in a 3d Universe. This is because the 2d Universe has fewer dimensions for the atoms to move in, resulting in fewer possible arrangements and therefore a lower multiplicity.

## 5. Why is studying a monatomic gas in a 2d Universe important?

Studying a monatomic gas in a 2d Universe can provide insights into the behavior of gases in different dimensions and help us better understand fundamental physical principles. It can also have practical applications in fields such as materials science and nanotechnology.

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