Ideal Diatomic 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.

Again we are dealing with indistinguishable particles, we can use the same results as we had in the previous lecture. We have 

                              (1)
The traslational partition function is similar to monatomic case, 
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Rigid Rotor-Harmonic Oscillator Approximation
Diatomic molecules have rotational as well as vibrational degrees of freedom. If we approximate rotation and vibration to be separable, i.e. 

           .
The partition function can, then, be written as Eqn. (1). Here we also have used Born-Oppenheimer approximation to separate out the vibration and electronic degrees of freedom.

Rigid Rotor Approximation
The separable approximation used for rotation is called rigid rotor approximation, where rotation does not affect changes might occur in the body. The change might be of elongation due to centrifugal force. Quantum mechanically, the rotational eigenvalues and their degeneracy are given by 

                         (2)
where I is the moment of inertia, mre2, of the diatomic molecule and J is the rotational quantum numbers (J = 0, 1, 2, ...). Eqn. (2) can be written in terms of rotational constant, B 
           .
Therefore, we can write rotational partition function as

High Temperature Limit
We can use similar summation --> integration transformation can be done if the energy levels are close to each other. If we define
which is called the characteristic temperature of roation. If the term, Qrot/T, is small enough we can approximate the summation by an integral. Therefore, we have
for Qrot << T.

Low Temperature Limit
The low temperature limit may be calculated explicitly including the summation, such that for HD molecule
which give us a good approximation as long as Q rot > 0.7T.

Intermediate Temperature
For intermediate region, we need other approximation.

We use Euler-MacLaurin summation formula to approximate a function. The Euler-MacLaurin formula is often used in a numerical integration of a function.
where Bj is the Bernoulli numbers (B1 = 1/6, B2 = 1/30, B3 = 1/42,...

For rotational partition function, one obtains
It is interesting to note that the distribution of the rotation at certain temperature has a maximum. If we write the fraction of molecules in jth rotational state
and differentiate the fraction with respect to J, we can arrive at
Case in point, the following figure shows the rotational population as a function of J.

Harmonic Oscillator Approximation
We approximate vibrational motion by harmonic oscillator. The approximation is good as long as the motion is stiff and deep enough such that anharmonic effects are minimal.

We first expand potential energy (interaction between atoms) function V(r) at equilibrium geometry in a Taylor series.
Since evaluation is done at the equilibrium geometry, the second term on the right-hand side is zero and the first term is chosen to be zero, and truncating the expansion at the second order. Then, the V(r) can be written as
where k is a force constatnt. This is called Harmonic approximation. When the potential, V(r), is used for vibrational Schrodinger equation, then one obtains 
             for n = 0, 1, 2, ...
where u is the frequency and n is a vibrational quantum number. The degeneracy is always one in the case of diatomic molecules. Note also that n starts at zero! The frequency is written as
and m is a reduced mass, 
           .
Using the above expressions the vibrational partition function is written as 
           ,          (3)
and we have used a series expansion in a geometric series. Contrary to the rotational partition function we have a closed expression for the partition function. In arriving at Eqn. (3), we have used no approximation (other than harmonic approximation).

The fraction of molecules in the vibrational state n can be obtained by 

           .
The fraction of molecules that in the excited states is 
           .
The fraction is exponentially decaying function! Therefore, the ground state is the only state that are populated in vibrations.
Here we used Br2 molecule, which has small force constant, therefore excited levels are easier than others to be occupied. Unlike rotation, we have the ground states to be the most populated state.

Electronic Partition Function
If we take the zero of energy in the electronic state to be the dissociated atom limit, we can write the partition function for electronic part as We can write 

           .
Generally, only the ground state is populated, and therefore, we can truncate the equation to have only the first term. Here, De denotes the dissociation energy of the diatomic molecule (not to be confused with the D0).

Thermodynamic Functions
The total molecular partition function found in Eqn. (1) can now be written as 

           .        (4)
We have used rigid-rotor harmonic oscillator approximation to derive Eqn. (4). In deriving, we have restrictions that Qrot << T and only the ground electronic state is populated. Using Eqn. (4), we can arrive at the following thermodynamic functions;

Energy
Heat capacity
Entropy
Pressure
Chemical potential