Number of states of 3D ideal gas ↩

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aliases: [Number of states of 3 dimensional ideal gas, Number of states of ideal gas in 3D, Number of states of ideal gas in 3 dimensions]
shorthands: {}
created: 2022-06-01 15:09:05
modified: 2022-06-01 15:33:16

Let's consider an ideal gas in 3 dimensions with its usual Hamiltonian function (see Hamiltonian of ideal gas), with

particles. Let's calculate its number of states:

Where the orange is the combinatorial part, comes in because the the particles of the ideal gas are indistinguishable. The green integral is then can be solved by dividing it into two parts the spatial and the momentum part.

The spatial part is just the volume of the space where the gas in confined to the power, since there are particles:
The momentum part depends on the energy:

It is the volume of the section of momentum space where . Expanding the expression:

Which is the equation of a dimensional sphere with a radius of , since it is equivalent to:

The volume of this sphere is then:

From these, the number of states becomes: