So we've already obtained the translational and electronic
partition functions for a monatomic gas and they're going to be exactly the same
for a diatomic gas. So what we need to obtain now, determine, are the molecular
partition functions for rotation and vibration. So let's start with the rotation. Recalling that the energy and degeneracy for
rotation are given here. The energy is k times
characteristic temperature. It's the way it's written. In this case, J times J plus 1 and the
degeneracy is 2J plus 1. So the rotational partition
functions equal to the sum over J of 2J plus 1 e to the minus
Epsilon J over kT. Or it can be written in terms of a characteristic
temperature as well. So it turns out if you
expand the exponential, you get this series, and in the approximation of large temperature compared to the characteristic
temperature, this reduces to just
T over Theta r. So the partition function
for rotation takes on this particularly simple form. I should have said right from
the beginning that this is of course the rigid
rotator approximation. Now the rotational temperature is typically below 100 Kelvin. So this is usually a
pretty good approximation most of the time at least
for first cut purposes. There is a complication
for homonuclear molecules, that's molecules such as H2, O2, and N2 that are made
of the same atoms. Symmetry requirements
place restrictions on the possible values of the rotational quantum
number, and in fact, only even or odd ones are
allowed so that generally, the rotational partition
function is written as 1 over Sigma T over Theta
r where Sigma is this symmetry factor
which is one for a heteronuclear molecule and two for a homonuclear molecule. So the fraction and it is interesting to look at
the fraction of molecules, given rotational state that's given by the Boltzmann
ratio there. So it's N_J over
N is g sub J e to the minus Epsilon over kT or in terms of
characteristic temperature, it's minus J times
J plus 1 Theta r over T. So if you
just plot that out, you get a peaked distribution, and the diagram here is for CO, it shows the distribution at a temperature of 300 Kelvin and a temperature 2,000 Kelvin, and you can see that the peak of the rotational
energy distribution does shift quite a
bit with temperature. So that's rotation for
the rigid rotator. For vibration, considering
the harmonic oscillator, we know that the
energy is equal to h Nu times V plus one-half. However, if we wish to
deal with mixtures, there's the issue of referencing. So the potential
energy function for a real diatomic molecule will look something similar
to that in the graph, and so it depends on
where you set zero. But in that coordinate system, the absolute energy of vibration is set to
zero when there's complete separation of the atoms that make up the molecule. D_e is the energy
difference between complete separation
and the bottom of the attractive well,
a potential well. It's called the electronic
binding energy, and it makes sense because the potential energy
function is equal to the negative of the integral of the force function
between the two atoms, integrate it from r to infinity, and so as the atoms
approach from a distance, there's initially no force. The potential energy is zero, and it decreases
because of attraction, and then eventually
because of repulsion between the two nuclei, it turns the corner as
it were and increases. So because of this and because our referencing is important when working with
reacting mixtures, usually we set the
same zero reference for each molecule when calculating the
partition function. So this is one way to do it. So we can write the
vibrational energy as our h Nu times V plus one-half, and then subtract off the
electronic binding energy. So that means that essentially, we've caused a similar reference zero for each molecule
we would put in. So then the partition function can be calculated as below. Adding an e to the D_e over
kT out of the exponential, and since it's a constant, not a function of
the quantum number, you can pull it out of a sum, and you can also
write it in terms of characteristic vibrational
temperature as shown there. Well, the sum, e to the
minus Theta over Theta, the characteristic
temperature over T times v plus one-half can be expanded into an
infinite series, and the term, the
bracketed term on the right is a
binomial expansion. So it turns out
that you can write the vibrational
partition function for the harmonic oscillator either in the form of
exponents as shown first on the bottom equation
or in terms of hyperbolic functions and particularly this sine
h or cinch function. So that's a simplification. So if we put all
that together into the full partition function, we end up with the
fundamental relation for when t is much
greater than Theta r, the characteristic rotational
temperature and the electronic partition function is just equal to the
ground-state degeneracy. This would be the
expression you would get, and then from that you can calculate the
equations of states. So you can find that the
energy is three-halves kT plus kT plus the bracketed term. So the three-halves kT is the translational contribution
to the internal energy; kT is the rotational contribution and the bracketed term is the
vibrational contribution. We'll say a little bit
more about that in a bit. Then the expression for
the entropy is given there and the expression
for the specific heat. So the temperature dependence of the specific heat is shown here, and this is for some
particular molecule, I can't remember what it is, and it is C_v over the
Boltzmann constant over k, and you can see it goes
from two and a half up to seven-halves as you
increase the temperature. So the two and a half is the three-halves translation plus the two-halves
rotational contribution, and of course if we blew up the portion of the graph
close to zero temperature, you'd see translation
kicking, rotation kick in. But normally, those are not important
because we typically deal with temperatures of
300 Kelvin or so higher, and then as you increase the temperature at
a certain point, this specific heat
asymptotes out at seven-halves k. All that is for rigid rotator
harmonic oscillator. We know that real molecules are not rigid rotators and
harmonic oscillators, and in fact, we have presented an expression like the
one shown on this. The two motions are not
completely independent. In other words, as you increase
the vibrational energy, it changes the average spacing of the atoms and therefore the
moment of inertia changes. So the rotational
energy levels change with vibrational
levels and so on, and in that case when you're
talking about real effects, then you just have to calculate the partition
function numerically, and that is what people that calculate highly
accurate properties do. For instance, the folks at the National Institute of
Standards and Technology, they calculate the
partition function numerically in order to take these real
effects into account. So thanks very much
and have a great day.