So far, we’ve seen that the heat capacity of a solid approaches zero at very low temperatures.
Classical physics cannot explain this. But Einstein was able to model this behavior qualitatively
by treating each atom of the solid as a quantum harmonic oscillator. The predicted heat capacity
of the Einstein model, however, approaches zero more rapidly than is experimentally observed.
The quantitative short comings of the Einstein model were rectified in 19 12 by the Debye
model. Developed by Peter Debye.
We start as in the Einstein model, with the solid treated as a collection of balls connected
by springs. But, we do not make the simplifying mean-field
assumption. Instead, we accept that the atoms are coupled, with the force on each dependent
on the motion of its neighbors. Let’s see what kinds of oscillations are
possible by numerically solving for the oscillations of a one-dimensional array of 8 balls connected
by springs. We set up several copies, with the initial
ball velocities corresponding to one of the possible modes of sinusoidal oscillation.
We find a range of possible frequencies. In the lower-frequency modes, near the top, many
of the balls tend to move in the same direction as one or both of their neighbors, resulting
in overall less compression and stretching of the springs. In the high-frequency modes,
near the bottom, neighboring balls tend to move in opposite directions, resulting in
overall more compression and stretching of the springs.
With coupling, we have a spectrum of oscillation frequencies, instead of a single frequency
as in the Einstein model. The additional low frequency components of
the Debye model, correspond to smaller energy quanta. These should increase heat capacity
at low temperatures, which is where the values of the Einstein model were too small.
Calculation of the heat capacity is lengthy, and we simply present the result. C equals
3-N-k, times 3, quantity T-over-T-D cubed, times the integral from x equals 0 to T-D
over-T, of x-to-the-fourth, e-to-the-x, over quantity e-to-the-x minus one, squared. Here
T-D is the Debye temperature, which is on the order of the Einstein temperature. It
is often determined experimentally. In the high-temperature limit, C approaches
the Dulong-Petit value 3-N-k, as does the Einstein model.
In the low-temperature limit, C behaves as 3-N-k times 4-pi-to-the-fourth over-5, times
quantity T-over T-D, cubed. So, heat capacity varies as T-cubed near absolute zero.
Here are both the Debye and Einstein models compared to the observed heat capacity of
platinum. They agree with each other and experiment at high temperatures. But only the Debye model
accurately tracks the variation of heat capacity at low temperatures.
Here is a plot of the room-temperature, molar heat capacities of most solid elements versus
atomic number Z. The horizontal red line shows the classical prediction of the Dulong-Petit
law, 25 joules per mol, kelvin. The actual values fluctuate about this line. The Debye
model should explain values below the line. On the right, these heat capacities are plotted
versus Debye temperature. There is a very strong correlation between increasing Debye
temperature and decreasing heat capacity. Room temperature is about 300 kelvin. We see
that all elements with heat capacities less than 24 joules per mole, kelvin, have Debye
temperatures greater than 300. Let’s look at three examples. Carbon, in
the form of graphite, has a Debye temperature of 15 50 kelvin. At room temperature, its
observed molar heat capacity is only 8.5 joules per mol, kelvin. Far below the prediction
of the Dulong-Petit law. But we can understand this using the Debye model, which predicts
a value of 8.6.
Similarly for Silicon, with a Debye temperature of 692, observed heat capacity of 19.8, and
a Debye model value of 19.3. And, for Germanium, with a Debye temperature of 403, observed
heat capacity of 23.3, and a Debye model value of 22.8.
Something the Debye model does not do, is to explain heat capacities above the Dulong-Petite
value. Although, notice on the right plot, that these are correlated with lower Debye
temperatures. We will come back to this issue. Now that we have an explanation of how heat
capacity due to atomic vibrations varies with temperature. Let’s see if we can figure
out heat capacities of multi-atomic gases. We’ve seen that the molar heat capacity
of monatomic gases is accurately described by the simple model of billiard balls bouncing
around inside a box. The value is 3/2 R, which equals 12.47 joules per mole, kelvin.
However, this result does not apply to two-atom, diatomic molecules, such as nitrogen and oxygen.
Those gases have molar heat capacities near 5/2 R, but the oxygen value is measurably
larger than the nitrogen value. First, let’s tackle an important issue for
describing the heat capacity of a gas. The value is highly dependent on how we confine
the gas. So far we’ve primarily considered heat added to gases confined to a constant
volume. But we can also confine a gas under constant pressure.
Suppose we place a gas sample at temperature T-1, in a cylinder with cross-sectional area
‘A’. The gas presses against a moveable piston, creating a closed volume with height
x-1. One side of the piston is exposed to the atmosphere at pressure P-0. The gas will
expand or contract until the internal pressure is also P-0.
The volume ‘V’ is the area ‘A’ times the height ‘x’. From the ideal gas law,
P-0 V-1 equals N-k T-1. Suppose we add heat to the gas to increase
its temperature to T-2. If the volume were held constant, the gas pressure would increase.
However, with the gas pressure held constant, the volume will increase. The piston will
move to height x-2. Now the ideal gas law reads, P-0 V-2 equals
N-k T-2. The piston has moved a distance x-2 minus
x-1. This requires work equal to force times displacement.
The force is pressure times area, P-0 A. So the work W is P-0 A times x-2 minus x-1.
Which is P-0 times the volume change, V-2 minus V-1.
Using the gas-law expressions, this is N-k times T-2 minus T-1.
The heat capacity at constant pressure, C-P, is defined by added heat Q equals C-P times
temperature change T-2 minus T-1. This does not equal the heat we would need
to add to the gas at constant volume to get the same temperature increase, C-V times temperature
change, where C-V is the heat capacity at constant volume. For the constant volume case,
all added heat goes into increasing the internal energy, hence the temperature, of the gas.
For the constant-pressure case we need to add the additional heat that gets converted
into work, N-k times the temperature change. Canceling the common factor T-2 minus T-1,
we have C-P equals C-V plus N-k. If N is Avogadro’s number, then N-k is the
gas constant R, and the molar heat capacity at constant pressure equals the molar heat
capacity at constant volume, plus R. For an ideal monatomic gas, the molar heat
capacity at constant volume is 3/2 R. So the molar heat capacity at constant pressure is
5/2 R. For a gas we need to be clear which form of heat capacity we are dealing with.
For liquids and solids, expansion with heating is a much smaller effect, and we will assume
the heat converted to work in expansion is negligible. So, for liquids and solids, we
will take the heat capacity at constant volume or constant pressure to be effectively the
same.
Now, let’s look at a fundamental principle in the theory of heat capacity. The concept
of equipartition of energy. We considered a form of this in Video 2 of
this series. There we simulated a gas in which ½ of the atoms have mass m-1 and the other
½ mass m-2, equals 4 times m-1. Setting the heavier atoms in motion.
The lighter atoms eventually end up moving faster, on average, than the heavy atoms.
This difference in speed is just that required to give the two groups of atoms the same,
on average, kinetic energy. We say that the total energy of the gas has been equally partitioned
among the atoms. So, one aspect of equipartition of energy
is that at the same temperature, atoms of different masses will have the same average
kinetic energy. Also in Video 2, we saw that the microscopic
significance of temperature is that ½ k-T equals the average kinetic energy of an atom’s
motion in a single spatial dimension, ½ m mean of v-x squared, or ½ m mean of v-y squared,
or ½ m mean of v-z squared. The total energy of an atom is therefore 3 times ½ k-T.
So, another aspect of equipartition of energy is that each translational degree of freedom
will, on average, contain energy ½ k-T. In Video 3 of the mechanics series, we saw
that for a mass oscillating on a spring, a harmonic oscillator, energy continuously transitions
between potential and kinetic forms, with the average kinetic energy equal to the average
potential energy. Therefore, another aspect of equipartition
of energy is that harmonic oscillators will have the same average potential and kinetic
energies, each ½ k-T.
Now, let’s see what happens when we mix single-atom, the red and green balls, and
two-atom, diatomic molecules, the blue balls. All balls represent point masses. The blue
balls are rigidly attached so that their centers always maintain the same distance.
We give the red ball an initial velocity, and allow the system to evolve.
All three molecules undergo translational motion in two spatial dimensions. In addition,
the diatomic molecule undergoes rotational motion about an axis perpendicular to the
screen. Each molecule has two degrees of translational freedom. The diatomic molecule also has one
degree of rotational freedom. Here the top two plots show the total kinetic
energy of the red and green balls over time. The third plot shows the translational kinetic
energy of blue balls, and the fourth plot shows their rotational kinetic energy. There
are 7 degrees of freedom in this system. Two of translation for each of the 3 molecules,
and 1 of rotation for the diatomic molecule. The labels at left show the average of each
plot as a fraction of the total energy of the system. The red ball’s 2 degrees of
freedom contain, on average, 2.09 sevenths of the total energy. For the green ball it’s
1.87 sevenths. For the blue molecule’s translational energy, it’s 1.96 sevenths. Each of these
values is very close to 2, sevenths. The number of translational degrees of freedom of the
molecule over the total number of degrees of freedom.
The bottom plot shows the rotational kinetic energy of the blue molecule. The average value
is 1.07 sevenths. Very close to 1 seventh. The number of rotational degrees of freedom
of the molecule over the total number of degrees of freedom.
This demonstrates how yet another aspect of equipartition of energy is that each rotational
degree of freedom will, on average, contain the same energy as a translational degree
of freedom, namely, ½ k-T.
More formally, in general, if the total energy, the so-called Hamiltonian of the system, is
a quadratic function of m momenta, p, and n coordinates, q.
Of the form, a-1 p-1 squared, plus a-2 p-2 squared, and so on, plus b-1 q-1 squared,
plus b-2 q-2 squared, and so on. Here the q’s can be translational coordinates, distances,
or rotational coordinates, angles. The p’s are the corresponding linear or angular momenta.
Then the number of quadratic degrees of freedom in the system is m-plus-n. And, the molar
heat capacity at constant volume is the number of degrees of freedom over 2, times the gas
constant ‘R’. Note that “degrees of freedom” can be
defined in different ways. Our definition is that a degree of freedom corresponds to
a linear or angular coordinate, or the associated momentum, that appears in a quadratic term
in the expression for total energy. These are sometimes called quadratic degrees of
freedom. To emphasize this, we use the unusual notation D-O-F for degrees of freedom.