Ideal Monatomic Gas

Ideal gas: dilute, noninteracting monatomic species that can be represented well by pV = NkT (or pV = nRT), under 1 atm. and T > ambient temperature.

Since we are dealing with atoms (indistinguishable particles), we can use the results obtained in the previous lecture, i.e.
There are only two degrees of freedom; one in translation, and one in electronic in the dilute monatomic gas. We can use the separable approximation to the molecular partition function such that
There is another degree of freedom; nuclear degree of freedom. Since we are only dealing here with chemical changes (not nuclear reaction!), the nuclear energy state of the system should not change (unless you are doing cold fusion!). Therefore, we neglect its contribution from the partition functions. Below we examine the partition functions in each degree of freedom.

Translational Partition Function
Consider a 3-D box with length l. The energy of the state given by quantum mechnics is 

                            (1)
And, the molecular partition function for translation has a form, 
                                      (2).
By substituting Eqn. (1) into Eqn. (2), one can obtain 
           ,
and 
                            (3).
The infintie sum in Eqn. (3) is problematic (no closed form). However, if we approximate the summation by integral we have, 
           .
Then, we have
translational partition function (4)

This replacement of summation to integral can be justified by considering the changes in the exponents in one state to the next. 

           ,
and at ambient temperature, m = 10-22 g and l = 10 cm, D is 
           .
Since at ambient temperature, n is typically in order of 1010, D is a very small number, and summation can be approximated by integral.

The average translational energy is given as 

           .
Therefore, the average translational energy is 
           .
Since and since therefore
L = Thermal De Broglie Wavelength
The de Broglie wavelength should be small in comparison to the size of the container, then it is said to satisfy Boltzmann statistics.

L3/V << 1condition for Boltzmann statistics to be valid

Electronic Partition Function
We write partition function for electronic part as 

           .
It is customarily define the ground electronic state as 0 of energy, e1 = 0. Then, the above equation becomes
Electronic Partition Function (5)
We can show, however, for the most purpose, we can perhaps neglect terms from the second term. Since the fraction of molecules, say He atoms, in the lowest triplet state, 3S1 state, is written as 
           ,
which tells us that the fraction is a very very small number at room temperature!

Thermodynamic Functions

Helmholtz Free Energy
Internal Energy
pressure
Entropy
Chemical Potential