Ideal Diatomic Gas

In addition to translation, diatomic molecules possess vibrational and rotational degrees of freedom. Since the quantum mechanical problem for most diatomics cannot be solved exactly, we use approximate energy levels to obtain the partition function. There are several useful approximations. The Born-Oppenheimer approximation is used to separate electronic and nuclear motion in the quantum mechanical wave function. Since the electrons are 2000 times less massive than even the smallest nuclei their motion is rapid compared to that of the nuclei. The B-O approximation states that the electrons respond rapidly to any nuclear motion and therefore represent a potential U(Q) in the field of which the nuclei move. We are concerned here with nuclear motion in a diatomic molecule. First we must transform to center-of-mass coordinates to separate the nuclear motion into translational and internal motions. We then consider the internal motions as composed of rotational and vibrational motion. The approximation that separates rotational and vibrational motion is the rigid rotator approximation. Vibrational motion is discussed in the context of the harmonic approximation.

Center of mass coordinates

The coordinates x_{1}, y_{1}, z_{1} of mass 1 and x_{2}, y_{2}, z_{2} of mass 2 can be separated rigorously into center-of-mass coordinates X, Y, Z, and relative coordinates x_{12}, y_{12}, z_{12}. In the center-of-mass coordinate system the diatomic molecule behaves as if it were a single point with mass M = m_{1} + m_{2} (see Problem Set 4). Effectively we can separate the hamiltonian into translation of the entire molecule and internal contributions due to rotation and vibration.

H = H_{trans} + H_{int}

The energies are

e = e_{trans} + e_{int}

The partition function is

q = q_{trans}q_{int}

Notice that as before the energies are additive and the partition function due to each type of energy enters the problem in a multiplicative fashion since

The translational problem has already been solved for a monatomic ideal gas. The translation partition function is

The rigid-rotator approximation

The relative motion of the two nuclei in a diatomic consists of two degrees of freedom for rotation and one for vibration. The reduced mass, m figures in both of these where

The separation of variables to obtain the reduced mass is demonstrated in problem set 4.

We consider the intermolecular potential due to the bond between the nuclei.

In this expansion R_{0} represents the equilibrium bond distance in the diatomic molecule. The linear term vanishes because ¶U/¶R is zero at the minimum of the potential surface. The second derivative with respect to the intramolecular potential is also known as the force constant, k. Thus k = (¶^{2}U/¶R^{2}). The potential can be written

neglecting higher order terms.

There is no potential for rotation. If we assume that rotation occurs for a fixed value of the interatomic distance, i.e. R_{0} then we can make the rigid-rotator approximation. This approximation allows the hamiltonian to be written as

H_{int} = H_{vib} + H_{rot}

The energies are

e_{int} = e_{vib} + e_{rot}

The partition function

q_{rot,vib }= q_{rot}q_{,vib }

Rotational partition function for a heteronuclear diatomic molecule

The solution of the quantum mechanical rigid-rotator results in energy levels

where I is the moment of inertia. For a diatomic molecule I = mR_{0}^{2}. The degeneracy is W_{J} = 2J + 1. The rotational partition function can be calculated using these quantities.

Note that the spacing of the energy levels is quite small relative to thermal energy at ambient temperature. This means that we can approximate the sum with an integral.

It turns out that d[J(J+1)] = d[J^{2}+J] = (2J+1)dJ. So we can express the integral as

This is an exponential integral with a solution

Provided the energy level spacing 8p^{2}I/h^{2} is small compared to kT. McQuarrie defines a rotational temperature Q_{r} = 8p^{2}I/h^{2}k. The rotational temperature is largest for HD where Q_{r }= 42.7 K and so the temperature should be of the order of 40 K or above in order to make the above approximation for HD. For example for CO, Q_{r }= 2.8 K and so the above approximation is good above 3 K. The treatment for homonuclear diatomics is similar to this but must include nuclear spin states. We will not discuss this further in the course (you may skip section 6.5 in McQuarrie).

The vibrational partition function

The solutions of the Schrodinger equation for vibrational motion provide a quantized set of vibrational states. Within the harmonic approximation the problem to be solved is

The solutions are Gaussian functions multiplied by polynomials shown below.

For our purposes the most important point is that the energies form a set of equally spaced levels given by e_{v} = (v + 1/2)hn. The relationship between the energies and the reduced mass m and the force constant is obtained by solving the harmonic oscillator problem. Classically, the problem can be expressed as the hamiltonian

H = T + U = 0

If we let x = e^{i}wt then p_{x} = m¶x/¶t = iwme^{i}wt. Substituting this into the above expression we have

-w^{2}m + k = 0.

This equation is satisfied provided that

The quantum mechanical analog of this solution is a Gaussian solution to the Schrodinger equation above.

The vibrational energy levels are evenly spaced with a separation of hn or áw. If we take the zero-point level as our "zero of energy" then e_{v} = váw.

The partition function becomes

Note that we have used the result for a geometric progression that

This the way the partition function is written in many physical chemistry texts. Note that the zero-point-energy has been neglected. McQuarrie includes the zero-point energy term in the partition function. This is more rigorously correct thught as we shall see when calculating the vibrational entropy, it is the relative population of the higher levels that matters and so the zero point energy term drops out in that expression. Including the zero-point energy the vibrational partition function is

We can define a vibrational temperature just as we did a rotational temperature above, Q_{v} = hn/k_{B}. The vibrational temperature is typically hundreds or thousands of Kelvin indicating that vibrational levels are not highly populated at ambient temperatures. Spectroscopists usually report vibrational energy levels in wavenumbers. The most common unit is the reciprocal centimeter, cm^{-1}. Boltzmann's constant is k_{B} = 0.697 cm^{-1}/K, thus thermal energies or k_{B}T are of the order 200 cm^{-1} at 300 K. This conversion factor allows one to easily calculate the vibrational temperature for a particular vibration using the above formula.

If the temperature is sufficiently high the excited vibrational levels are highly populated. The sum can be replaced by an integral.

This is classical limit as well and this comparison is important for calculation of the classical partition function.

Calculation of the average energy

Just as we have seen for the translation partition function we can calculate thermodynamic averages for the vibrational and rotational states of molecules. The average energy is obtained from

Note that for energy calculations it does not matter whether the molecules are distinguishable or indistinguishable. This is because if we write

then

and clearly the terms depending only on N will vanish when we take the derivative with respect to b. Thus, we have

and in the present case we can calculate the average vibrational from the vibrational partition function.

Thus,

The derivative of this quantity with respect to b gives

Using the vibrational temperature, Q_{v} = hn/k_{ }this can be rewritten as

Note that in the high temperature limit this expression becomes

The rotational energy can be calculated in the high temperature limit since the rotational temperature is typically quite low. Thus,

We can write ln(q_{rot}) as a sum of two terms.

Only the second term depends on b so the average rotational energy becomes

Calculation of the heat capacity

The heat capacity can be determined from