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|3-2: Molecular Motions and Heat Capacities|
of heat required
to raise the temperature
of a substance
especially for 1 g
of the substance,
it is called
specific heat capacity
"per unit mass".)
since the heat
necessary to warm up
1 g of water by
is 1 cal,
the specific heat
of water is
If such a physical quantity in the macroscopic world as specific heat can be derived from the molecular or atomic degrees of freedom in the microscopic world, how wonderful it would be! To explain this, let us study the law of equipartition of energy.
[The Law of Equipartition of Energy]
When we learned about the motion of molecules on the page, 1-4: The Motion of Molecules, we estimated the speed of the molecules in a gas. There, putting a gas in a cubic container as shown in Fig. (A) and assuming the x -component of the average velocity of the molecule to be vx , we calculated the pressure applied onto the wall A of the container. Consequently, we obtained a relation
where P, V and M are the pressure, the volume and the total mass of the gas, respectively.
Now, consider 1 mole of gas. Let m be the mass of a single molecule and T the absolute temperature of the gas. The total mass of the gas is M = m NA, where NA is the Avogadro constant. Combining the above relation and famous Boyle-Charles's law
Here k is a fundamental constant called the Boltzmann constant.
The quantity on the left-hand side of the above equation is the kinetic energy of a molecule. This implies that the mean value of the kinetic energy of one molecule is kT /2.
In this discussion, we consider that the molecular motion is only in the x direction. In a realistic case, a molecule has also the y- and z-components of the velocity. Let the x- , y- and z-components of the average molecular velocity be vx , vy , and vz respectively. Then the average kinetic energy of the molecule, , is written
This result tells us that the mean value of the kinetic energy of each degree of freedom in the x, y and z directions is kT /2. In other words, the equal amount of energy kT /2 is distributed to each of all the degrees of freedom of the molecules. This is called the law of equipartition of energy.
According to the theories of statistical mechanics, this equipartition law can be generalized to be said that the average kinetic energy of each dynamical degree of freedom is equal to kT /2. The proof of this theorem is given on the following page
3-2-A: Generalization of the Law of Equipartition of Energy .
But you may skip this page, if you feel it difficult or tedious because of mathematical formulae and complicated expressions.
[Heat Capacities of Gases]
Consider an ideal (or perfect) gas. It means a gas in which the atomic or molecular interactions can be neglected. In 1 mole of the gas, there are as many molecules as Avogadro's constant NA.
In the case of a monoatomic molecular gas like a noble gas (inert gas), the number of the degrees of freedom of one molecule is 3, which means the directions of the x-, y-, and z- axes. Hence the total number of degrees of freedom in 1 mole of the gas is 3NA.
An ordinary gas like hydrogen or oxygen is in general diatomic molecular gas. In this case, as seen in the following figure, there are 5 degrees of freedom; 3 of them are those of the center-of-mass motion in the directions of the x-, y-, and z- axes, and the rest 2 are those of the rotational motion around the center-of-mass, i.e. the degrees of freedom of and . Therefore the total number of degrees of freedom in 1 mole of a diatomic molecular gas is 5NA.
Degrees of freedom of a diatomic molecule
Since there are 2 degrees of freedom of the rotational motion, and , of the rotational motion, in addition to 3 degrees of freedom of the translational center-of-mass motion, the total number of degrees of freedom is 5.
If we consider
that the same amount
of kinetic energy
degrees of freedom
according to the law
of equipartition of energy,
the total energy E
of 1 mole of the gas
Accordingly, the molar heat capacities (heat capacities per mole) of gases are
These theoretical results are compared with the experimental data in the following Table 1.
We can see from Table 1 that the law of equipartition of energy is well satisfied. The molecular structure of carbon monoxide resembles to that of oxygen, so that those molar heat capacities are rather similar to each other. However the molecular structures of carbon dioxide, water and methane are more complicated than such a diatomic molecule as oxygen, and their heat capacities appear to differ a little.
[Heat Capacities of Solids]
In a solid, the inter-molecular interaction is so strong that the molecules cannot freely move. Therefore we cannot count the number of degrees of freedom as done in a gas or liquid.
A solid is considered to be a collection of oscillators. This figure is drawn on a two-dimensional plane, but it should be noted that a realistic solid is three-dimensional.
It is considered
that each molecule
in a solid
connects with adjacent ones
via springs as shown
in the above figure.
are infinitesimally oscillating
around their equilibrium
we can consider
that there are
as many harmonic oscillators
as the number of molecules
in the solid.
One molecule has
three directions of motion,
i.e. the directions of the
so that we can consider
that there are
in 1 mole of solid.
on the page,
Generalization of the Law
of Equipartition of Energy,
the constant amount
of energy kT
to every harmonic oscillator.
Hence the total energy
of a solid is
and the molar heat capacity of a solid is
This result is nothing other than Dulong and Petit's law discovered experimentally in 1819. Some experimental data of molar heat capacities of solids are shown in Table 2.
[Problems on Heat Capacities]
As discussed above, the molar heat capacities appear to be explained well by using the law of equipartition of energy. Certainly, there is no problem when the temperature of the system is high, whereas problems might appear when the temperature becomes low. For example, as seen in the following figure, the molar heat capacity of zinc becomes smaller and smaller and approaches 0 when the temperature comes close to 0 K. Every solid shows a similar property. This is a serious problem that can never be explained with the classical theory.
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