Welcome to Thermodynamics 6.
In this video we explore the relation between heat capacity and the third law of thermodynamics.
As we will see, the third law is intimately connected with quantum theory.
Indeed, it was a primary motivation for the development of a quantum theory of matter.
Let’s think about what happens to the entropy of a system as its temperature approaches
absolute zero.
For an ideal monatomic gas, the Sackur-Tetrode equation gives the entropy as a function of
temperature and volume.
In the limit T goes to zero, the argument of the logarithm goes to zero, and the expression
goes to minus infinity.
A very strange result.
We can argue that this result is not valid because gases cannot exist at very low temperatures,
so the expression is not applicable in this limit.
As we saw in the previous video, the Sackur-Tetrode equation accurately predicts the entropy of
a monatomic gas.
However, in general, when cooled, real gases will condense to a liquid, and then freeze
to a solid.
The equation is not applicable to these states.
Let’s consider the problem from a more abstract thermodynamic point of view.
Suppose we have a system at temperature T .
We extract heat delta-Q . This changes the system’s entropy by dS
equals minus delta-Q over T . As heat is removed, and the system temperature
approaches absolute zero, it seems like the magnitude of these entropy increments should
grow without bound, since the denominator approaches zero.
So, the system entropy would tend toward minus infinity.
Entropy equals Boltzmann’s constant times the logarithm of Gamma, the number of ways
to arrange the components of the system.
If S approaches minus infinity, then Gamma would have to approach zero.
But, that makes no sense.
The minimum possible value of Gamma should be one.
The system is in some arrangement.
So, there has to always be, at least, one way to arrange the components of a system.
Let’s look at this from a slightly different point of view.
Recall the definition of heat capacity “C”.
To change an object’s temperature by d-T, we have to add heat delta-Q equal to C, d-T.
In the present case we are extracting heat, which is adding negative heat, so we have
an additional minus sign.
Then, d-S equals C-over-T d-T.
We see that the entropy increments will not blow up if the ratio C-over-T remains finite
as T goes to zero.
That is, if the heat capacity goes to zero at least as fast as the temperature does.
Let’s now give one statement of the third law of thermodynamics.
The entropy of a system approaches a constant value (which is zero in certain cases) as
its temperature approaches absolute zero.
From our present discussion, we see that a corollary of this is that the heat capacity
of any system must approach zero as the temperature goes to zero.
There is an intimate connection between heat capacity and the third law.
Now, let’s consider the heat capacity of solids, since that is the state we generally
deal with at very low temperatures.
In 18 19 the Dulong-Petit (Petee) law was proposed.
Dulong and Petit noticed that the experimental heat capacity of many solids is near 25 joules
per mole, kelvin.
For example, in the table at left are values for several metals at room temperature, ranging
from 24.3 for aluminum to 26.4 for lead.
Recall that one mole is Avogadro’s number of molecules, about 6 times 10 to the 23.
The plot at right shows molar heat capacity vs. atomic number Z, for most of the elements
which are solids at room temperature.
The great majority of these values were unknown at the time of Dulong and Petit (Petee).
Still, we see that, roughly speaking, the values do tend to cluster about the Dulong-Petit
value, indicated by the horizontal red line.
Previously, we found that our model of an ideal monatomic gas as N tiny, colliding,
billiard balls predicts a heat capacity, specifically heat capacity at constant volume, of 3/2 N-k,
with k Boltzmann’s constant.
If N is Avogadro’s number, then N-k is the gas constant R, 8.314 joules per mole, kelvin.
And, the molar heat capacity is 12.47.
The values for Helium, Neon, and Argon gas agree with this prediction to four digits.
The Dulong-Petit (Petee) law predicts that many solids have a molar heat capacity about
twice this, 3-R equals 24.94 joules per mole, kelvin.
This leads us to ask, why do some solids have twice the molar heat capacity of some gases?
In place of our simple model of a monatomic gas as a collection of freely moving, tiny
billiard balls.
We might model a solid as a collection of tiny billiard balls held together by little
springs, representing chemical bonds.
The balls can move, but the springs will tend to return them to their equilibrium positions
in the crystal lattice.
In addition to the kinetic energy due to the motion of the balls, the solid will also contain
potential energy due to the compression and stretching of the springs.
In the third video of the mechanics series, we saw that when a mass on a spring oscillates,
the kinetic and potential energies of the system fluctuate sinusoidally, such that the
average kinetic and potential energies are equal.
So, an explanation for why some solids have twice the molar heat capacity of some gases,
is that in a solid atoms store the same kinetic energy, as they would in the gas state at
the same temperature, but in addition the chemical bonds store an equal amount of potential
energy.
However, this higher heat capacity for a solid, like that of a monatomic gas, is independent
of temperature.
We still have the issue, that the heat capacity should go to zero as the temperature does,
so that the entropy at zero temperature remains finite.
During the 19th century, the development of cryogenic technology enabled the measurement
of heat capacity at very low temperatures.
Researchers found that, as shown here for platinum, heat capacity does indeed approach
zero, as temperature does.
Good news for the third law.
And, notice that as temperature increases, the value approaches the Dulong-Petit value
of 25 joules per mole, kelvin.
But how do we explain this phenomenon?
In 1907, the Einstein model provided an answer.
Albert Einstein is, of course, famous for his theory of relativity.
But, his reputation as one of the greatest, arguably the greatest physicist is due as
much to his contributions to many other fields of physics, including thermodynamics.
Starting with our balls connected by springs model of a solid.
We treat each atom as a three-dimensional harmonic oscillator.
Able to oscillate independently in any of the three dimensions of space.
In general, this is a difficult problem to analyze.
Because the force on each atom depends on the positions of its neighbors.
So, the oscillations are coupled.
Einstein made the simplifying assumption that these forces are due to the average, or mean
field, of the other atoms.
In this case the atoms can be treated as moving independently.
This is equivalent to having the springs connected to each atom, in turn connected to fixed supports,
instead of to other atoms.
This assumption avoids the problem of coupled oscillations.
Now the oscillations of each atom, in each dimension, are independent and of the same
frequency.
And, the thermal behavior of the entire solid is equivalent to 3-N independent, one-dimensional
oscillators.
Here’s an atom oscillating in the single ‘x’ dimension.
Classically this would be a sinusoidal function of time, with some amplitude ‘A’, and
frequency nu, determined by the atom’s mass and the stiffness of the springs, the chemical
bonds.
Einstein’s brilliant idea was to apply the, at that time, new quantum theory to this oscillator.
Previously, quantum theory had only been applied to electromagnetic oscillations.
In this theory, the energy of an electromagnetic field, oscillating at frequency nu, can only
have the discrete values n-h nu.
Where ‘n’ is a non-negative integer, and ‘h’ is Planck’s constant.
Technically there can also be an additive constant to the energy, but that won’t concern
us here.
Einstein guessed that quantum theory might be a general principle of nature, applying
to any oscillations, even the mechanical oscillations of atoms in a solid.
The constant h-nu, the quantum of energy, can also be written as k-T-E, where ‘k’
is Bolztmann’s constant, and T-E is called the Einstein temperature.
T-E equals h-nu over-k.
So the oscillator energy is assumed limited to the discrete values n-k T-E.
In the previous video on statistical mechanics, we developed the Boltzmann distribution.
This gives the occupation frequency, f-n, of the oscillator state with energy n-k T-E,
as f-n equals one over-Z, e-to-the minus n-k T-E over k-T . Canceling common k’s, we
are left with e-to-the minus n-T-E over T. The occupation frequency is the fraction of
time a system will spend in a state with the given energy, when the temperature is T. This
can also be interpreted as the probability that the system will be in that state.
For convenience, let’s set a-equal to e-to-the minus T-E-over-T. This is less than one.
Then, f-n equals, one-over-Z, a-to-the-n.
The occupation frequencies have to sum to one, because the probability that the oscillator
has any possible energy is 100 percent.
This requires the normalizing factor ‘Z’, the so-called partition function, to equal
the sum from n-equals zero to infinity, of a-to-the-n.
This sum evaluates to one over one-minus-a.
Now we can calculate the average energy of the one-dimensional quantum oscillator.
This is the sum from n-equals zero to infinity, that is, the sum over all possible energy
states, of f-n, the probability of that state, times E-n, the energy of that state.
This is k-T-E over-Z, times the sum of n-a-to-the-n.
This sum evaluates to a-over quantity one-minus-a squared.
Substituting for-a, and simplifying, we get k-T-E over, e-to-the T-E over-T minus one.
If we set k-T-E equal to one, and call T-over T-E ‘x’, this has the functional form
one over e-to-the one-over-x minus one, with ‘x’ proportional to temperature.
Here’s a plot of that function.
For large ‘x’, large temperature, the function becomes a line.
This is the classical result that the energy of a system is proportional to temperature.
The slope of that line is the heat capacity.
But for small ‘x’, low temperature, the curve flattens out.
So, the heat capacity goes to zero, as required by the third law, and observed experimentally.
For a solid of N atoms, each of which can oscillate
independently in three dimensions, the average energy is 3-N times our 1-D result.
So, U equals 3-N, k-T-E, over e-to-the T-E-over-T minus one.
Let’s look at the high-temperature limit, T-goes to infinity.
In this case T-E over-T goes to zero.
Since e-to a small exponent is well approximated by one plus the exponent, e-to-the T-E over-T
can be replaced by one plus T-E over-T. The result is the system energy ‘U’ reduces
to 3-N k-T.
The rate of change of U with respect to T is the heat capacity C-equals 3-N-k.
The prediction of the Dulong-Petit law.
Now, let’s look at the low temperature limit, T-goes to zero.
In this case T-E over-T goes to infinity.
Since e-to a very large exponent is a huge number, the denominator, e-to-the T-E over-T
minus one, is essentially just e-to-the T-E over-T.
Writing one over e-to-the T-E over-T, as e-to-the minus T-E over-T, we find U goes to 3-N k-T-E
e-to-the minus T-E over-T. The slope, or derivative, of this function
can be calculated with a computer algebra tool.
We find a heat capacity of 3-N-k, quantity T-E over-T, squared, e-to-the minus T-E over-T.
This goes to zero as T goes to zero.
For reference, the exact heat capacity of the Einstein model is given at top in blue.
Here’s the Einstein model compared to the observed heat capacity of platinum.
The agreement is good for higher temperatures.
At low temperatures the Einstein model has qualitatively correct behavior, with heat
capacity decreasing to zero.
But it decreases too fast and fails to give quantitatively correct results.
Still, it indicates that the reason heat capacity decreases to zero with temperature is the
quantum mechanical nature of matter.
The reason why quantum theory predicts zero heat capacity at zero temperature is as follows.
The entire solid is thermally equivalent to 3-N, one-dimensional, quantum harmonic oscillators.
The energy of a single oscillator can only be zero, k-T-E, 2-k-T-E, and so on.
At temperature T, the thermal energy available in a single interaction is on the order of
k-T.
If T is much less than the Einstein temperature T-E, then almost no interactions will have
enough energy to excite an oscillator from its ground state to its first excited state
with energy k-T-E.
So the oscillators will generally not be able to absorb energy.
This results in the entire solid generally not being able to absorb energy, which means
its heat capacity is zero.