Density of states in the ideal gas

• I
• rabbed
In summary, Boltzmann's counting of states did not give the same (extensive) entropy as the entropy of Clausius. Gibbs then fixed this (Gibbs paradox) by considering indistinguishable particles.

rabbed

The MB energy distribution is: MB_PDF(E, T) = 2*sqrt(E/pi) * 1/(kB*T)^(3/2) * e^(-E/(kB*T))
How do I arrive at the density of states which hides inside the expression 2*sqrt(E/pi) * 1/(kB*T)^(3/2) ? I've only seen DOS that have "h" in them.. I want it to contain only E, pi, kB and T.. This is how far I've gotten (using a momentum vector):

V = 4*pi*p^3/3
dV = 4*pi*p^2*dp dV = 4*pi*(2*m*E)*sqrt(m/(2*E))*dE (since p = sqrt(2*m*E) and dp = sqrt(m/(2*E))*dE) dV = 2*pi*(2*m)^(3/2)*sqrt(E)*dE How do I get rid of the m and how do I get in kB and T?

On the same theme:

Is there a DOS-expression D for the ideal gas which will both fit into MB_PDF(E, T) = D * e^(-E/(kB*T)) / Z (where Z normalizes the distribution) as well as giving an extensive entropy S = kB*ln(D) ? It should be the same quantity, right?

rabbed said:
How do I arrive at the density of states which hides inside the expression
You can't. The best you can do is arrive at the density of states divided by the partition function.

rabbed said:
I've only seen DOS that have "h" in them.. I want it to contain only E, pi, kB and T..
That doesn't make sense. The density of states depends on Planck's constant, so there is no way to write the DOS without it. Even Gibbs, way before Planck, figured out that there was a parameter ##h## that needed to be included in the partition function. It cancels out if you take the DOS divided by Z, see my comment above.

dextercioby
Thanks, looks like an interesting discussion that I will read.

So the way I've understood this is: Boltzmann's counting of states didn't give the same (extensive) entropy as the entropy of Clausius. Then Gibbs fixed this (Gibbs paradox) by considering indistinguishable particles?

This document seem to say that Boltzmann counting does give extensive entropy after all: https://www.hindawi.com/journals/jther/2016/9137926/

Does Gibbs give a DOS-expression D for the ideal gas which will both fit into MB_PDF(E, T) = D * e^(-E/(kB*T)) / Z (where Z normalizes the distribution) as well as giving an extensive entropy S = kB*ln(D) ?
Should it be the same quantity in both these places and in that case, does any h-parameters cancel out in the entropy expression also?

Just trying to sort things out.. :)

1. What is the concept of density of states in an ideal gas?

The density of states in an ideal gas is a measure of the number of energy states that are available for particles in the gas at a given energy level. It helps to understand the distribution of particles and their corresponding energies in an ideal gas.

2. How is density of states related to the thermodynamic properties of an ideal gas?

The density of states is directly related to the thermodynamic properties of an ideal gas, such as temperature, pressure, and volume. It determines the number of particles that can occupy a particular energy level and thus affects the overall behavior of the gas.

3. How does the density of states change with temperature and pressure in an ideal gas?

As temperature and pressure increase, the density of states also increases. This is because the higher energy states become more accessible to the particles, leading to a larger number of available states at a given energy level.

4. What is the significance of the density of states in understanding energy distribution in an ideal gas?

The density of states allows us to understand the distribution of particles and their corresponding energies in an ideal gas. It provides insight into the probability of a particle occupying a particular energy level and helps to determine the overall behavior of the gas.

5. How is the density of states calculated in an ideal gas?

The density of states in an ideal gas can be calculated using the Maxwell-Boltzmann distribution, which describes the distribution of particles at different energy levels. It can also be calculated by integrating the partition function, which is a measure of the number of states available to the particles.