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# Statistical Thermo Gas Equation

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Derive the Ideal gas equation using statistical thermodynamics (ie partition function ect)

https://brainmass.com/physics/heat-thermodynamics/statistical-thermo-gas-equation-305879

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Hi C
Here it is.
I used the classical approach which is the simplest and most straightforward (no need to deal with quantum density of states and so forth and so on).
First let's examine the partition function of a single gas molecule with momentum p and mass m moving freely in a container of volume V (the collisions with the walls and other molecules are completely elastic, so there is no loss of energy).
The molecule's energy is given by:
(1.1)
The partition function is defined as:
(1.2)
Where and the summation is over all possible values of E.
We can write the sum as over all values of p:
(1.3)
Since is a continuous variable we can exchange the sum for an integral (recalling that we integrate over phase space, so we must integrate over the volume):
(1.4)
However, since the integrand is spatially independent, the spatial integral simply yields the volume of the box and we get:
(1.5)
In addition the integration over p is over all directions as well, so we get:
(1.6)
Since the momentums are independent we can write (1.6) as a product of single integrals:

(1.7)
Simple substitution:

(1.8)
The integral can be evaluated by first observing that:

Therefore:

Since the integration variables on the right hand side are completely independent we can write:

Using polar transform:

The integral becomes:

So finally:

And the single molecule's partition function is:
(1.9)
For N identical particles the partition function is simply:
(1.10)

The free energy is given as:
(1.11)

(1.12)

Using Maxwell relations we get the pressure:
(1.13)
Which in conjunction with (1.10) gives:

(1.14)
Hence we arrived at the ideal gas equation of state.

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