Hello. In this video, we're going to talk
about monatomic gases. Monatomic gases are the easiest of all the
gases to talk about because a monatomic atom only stores energy as translation
and electronic motion. So that makes developing the fundamental relationship
a little bit easier. For monatomic gases,
we'll only need the translational and
electronic partition functions, as those are the two
modes of energy storage. To evaluate the translational
partition function, we need the energy, and we've already developed
that expression in our course on quantum mechanics. If the motion in the three coordinate
directions are independent, then we can write the translational
partition function as the product of the
partition function at each of the three independent
coordinate directions, and that is shown at the
bottom of the slide. Here, we come to
taking advantage of the fact that the energy spacing, the translational quantum
state energy spacing, at any kind of
realistic temperature is extraordinarily small. Remember that your typical
translational quantum number is 10 to the eighth, or something like that.
Enormous numbers. So making the approximation
that we can replace the sum in the partition function with an integral is
perfectly reasonable. So that's the first
equation on this slide, where we've just replaced the sum with an integral plus a dnx. We're integrating over dnx. This is for the x-direction, and we do exactly the same thing for the y and z directions. If we let a new independent
variable x be equal to the coefficient of
n in the exponent, then we get the
expression down below, and the integral
in that expression is a well-known
definite integral, it has the value of the
square root of pi over 2. The result is quite simple. It is the translational
partition function, and multiplying the three
together as we discussed, you end up with an
expression 2pi m, the mass of the atom, KT over H squared, all to the three-halves
power times the volume. It turns out that this is the only place where
the volume appears, and it's due to the dependency of translational quantum
states on the volume. We do need to deal
with degeneracy, recall that translational
states are degenerate. SO if we're interested in
energy rather than state, we need to account
for the degeneracy. We can define f of ede as the fraction of
particles in an interval Epsilon to Epsilon
plus d Epsilon, f of Epsilon, which I keep
calling E, it's energy, is a distribution
function for the energy, and that can be written using the Boltzmann ratios e to
the minus Beta Epsilon k over the sum of e to the
minus Beta over Epsilon k, the sum over all quantum numbers, quantum indices, times dg. To relate d Epsilon into dg, consider that we can
write the energy as h bar squared over 8mV to the
two-thirds times n squared, where n is the total
translational quantum number, and it's equal to the
sum of the squares of the three coordinate
direction quantum numbers. It turns out that
n is the radius of a one-eighth sphere in the positive octant of translational quantum
number space. In other words, if you take an xyz Cartesian
coordinate system, and then make a point where an x and y and
a z are positive, n is the distance to that point from the origin of the
coordinate system. Therefore, each state
with energy between Epsilon and d-Epsilon occupies a spherical shell of
that one-eighth sphere, where n is between
n and n plus d, and in the volume of that shell, dg is four pi n
squared dn over 8. If you differentiate
Epsilon with respect to n, then you get this expression, and that is an energy
distribution function for translational energy. Now, the translational energy
is one-half mc squared, or v squared, depending
on your notation. Typically, when talking about
molecular or atomic speeds, individual speeds at
the microscopic level, we use the term c for speed. So Epsilon is equal
to mc squared over 2, therefore, d-Epsilon is mcdc. If you want the
speed distribution, which is where I'm heading here, you have to have that f of Epsilon d-Epsilon is
equal to f of cdc. If you do that, you get the so-called Maxwellian
speed distribution. This is a Gaussian
function, anything, either the independent variables
squared over or under, or over some other number, is known as a Gaussian function. So the Maxwellian
speed distribution is a so-called
Maxwellian function. We will run into this
later, this idea. Here's a plot of the
speed distribution. The average speed is just
the integral of cf of cdc, and that turns out to 8KT over Pi m to the one-half power. That is translation. Then the other partition
function we need is electronic. This is both simpler
and more complex. There is no simple expression for the electronic states of an atom, as we have learned, and she electronic
motion is degenerate. So basically, you just
have to do the sum. The partition function is
equal to the sum over k. The quantum energy state index of the degeneracy times e
to the minus Beta Epsilon. In this case, we're summing over energy states rather
than quantum states. Recall that Nk is equal to e
to the minus Beta Epsilon k, and that N in the Maxwell-Boltzmann
limit is e to the minus Gamma sum over k, e to the minus Beta Epsilon
k. The probability of finding a particle in a given
quantum state is N_k over n, which is equal to e to
the minus Beta Epsilon k over the sum over
all the k's of e to the minus Beta Epsilon over k. That's called the
Boltzmann ratio, and that comes in very handy in a variety of applications. Just a number just to
drive home the fact that electronic energies are quite large compared to vibrational and
rotational energies. For nitrogen, the first
excited electronic state has an energy of 19,227
inverse centimeters, and the ground state and
excited state degeneracies are 4 and 10. So if the temperature is 300 K, then the Boltzmann
ratio is equal to 1.67 times 10 to the minus 40. That's a pretty small number. For most atoms, you can just
make the assumption that the electronic partition function is just equal to the
ground-state degeneracy. If you have some low
lying electronic states, then you just have
to perform the sum. But typically, for normal
thermodynamic system analysis, you might only have to add in one or two terms to the sum. If you put all of this together, you can form the
fundamental relationship, and here it is. This is the mass L
function in which we've replaced U with one over t, and N with minus Mu over
t. There is the function. This is the fundamental
relationship for a monatomic ideal gas. The differential form of
that relation is shown, and we know the Euler
relation is shown. So we can get the equations
of state for the ideal gas, and it is interesting for the first equation
is the internal energy, and that's equal to
three-halves NKT. That's the so-called
Caloric equation of state. So the specific heat at constant volume
is three-halves NK. P over T, if you crank through, we've got P over T. But
if you calculate N, you get PV over KT, and that is the Ideal gas law. Some other things you
could get from this, the average energy per
atom is three-halves KT. Easy number to calculate. The entropy is k times
five-halves minus Mu over kT. It gets to the specific heat, which is the partial of u with T at constant V is three-halves k. Just some useful energy units, we've thrown around
some of these before. So it is often the case that an electron volt is a
useful unit of energy. It's the energy to
move an electron through a potential of one volt, or 1.602 times 10
to minus 19 Joules. We talked about
inverse centimeters. Energy of a photon
is equal to h Nu, which is equal to hc over Lambda. So Lambda is inversely
proportional to the energy, and 1 over Lambda is often
called lambda prime. That's equal to the
energy over hc, and 1EV in units of inverse centimeter
comes out to 8,066. Going back and recalling the expression for the
Maxwell-Boltzmann limit, if we evaluate this for molecular hydrogen at 300
Kelvin and one atmosphere, the left-hand side has
a value of 40,747, and that clearly
satisfies the inequality. That's it for this video. Thanks, and have a great day.