Thermal Physics 2

I have concluded that all of man's troubles have one cause, that he cannot sit still in a room by himself.

— Pascal

Average energy of a molecule

Toward the end of the 19th century great insight was gained into the meaning of thermodynamics by application of statistical methods. A leading ﬁgure in this work was

Ludwig Boltzmann. We quote here without proof one his most important ﬁndings:

In a system at equilibrium at temperature T, the probability that a particular particle will have energy E is proportional to e −E/kT .

In this, k is Boltzmann’s constant, and T is the Kelvin temperature.

From this statement about the probabilities, one can derive an important theorem about the average energy of a molecule:

Equipartition of energy

For a system in thermal equilibrium at temperature T, each degree of freedom contributes energy 1 kT to the average

2

energy of a molecule.

Here a degree of freedom is an independent part of the energy of a molecule. For example, the motion of the CM of the molecule is described by

2

2

2

KCM = 1 m(vx + v y + vz ) ,

2

which has three independent parts for the motion in the x, y, or z directions. These represent three degrees of freedom.

To be able to count the degrees of freedom we must make a model of the molecule. We will treat the atoms as point masses, and imagine that the bonds between atoms are like stiff springs connecting these masses.

PHY 53

1

Thermal Physics 2

A monatomic gas (such as helium) will consist simply of point masses, each with three degrees of freedom from the three terms in KCM as above.

A diatomic molecule will have four additional degrees of freedom besides these three:

•

The molecule can rotate about two independent axes passing through the CM and perpendicular to the line between the atoms. This gives two degrees of freedom.

Rotation about the line between point-like atoms gives no degree of freedom because the moment of inertia about that axis is zero and hence there is no rotational energy.

•

The atoms can vibrate back and forth along the line between them. The kinetic energy of this vibration gives one degree of freedom.

•

The potential energy of the “spring” in this vibration gives another degree of freedom. Since it has three degrees of freedom, a monatomic gas molecule should, if equipartition of energy holds true, have average energy 3 kT , while a diatomic molecule which has

2

seven degrees of freedom should have average energy

7 kT 2

. We will see later that these

predictions can be tested by experimental measurements of speciﬁc heats.

Thermal expansion

The increase in average energy of the molecules with increasing temperature is responsible for many familiar phenomena in our everyday experience. One of these is the fact that most solid or liquid substances expand when heated.

U

r1

r2 r E2

E1

PHY 53

2

Thermal Physics 2

The reason for this lies in the detailed shape of the potential energy curve for the interaction of nearest neighbor particles. A typical curve is shown above.

At small separations, when the molecules are close together, they experience a strong repulsion, represented by the rapidly rising potential energy curve.

This repulsion arises largely from the effects of the Pauli exclusion principle which comes into play when the electron systems of the molecules begin to overlap signiﬁcantly.

For larger separations there is a weaker attractive electrical force, which is what holds the molecules close to each other in the solid or liquid phase. The minimum in the curve represents a distance at which the molecules could remain at rest in stable equilibrium if they had the lowest possible energy. For higher energies the molecules oscillate between their turning points (the distances at which the total energy line crosses the potential energy curve).

Shown in the ﬁgure are two average