Take one mole of helium and one mole of nitrogen in two rigid steel bottles. Warm each by one kelvin. The helium needs \(12.5\,\text{J}\) of heat and the nitrogen needs \(20.8\,\text{J}\). Yet the last lesson showed that at any temperature both kinds of molecule carry the same translational energy, \(\tfrac32 kT\). The nitrogen takes the extra heat somewhere else.
Ways to hold energy
A helium atom is a tiny ball. It can hold kinetic energy in only one way: by moving from place to place. Its energy is \(\tfrac12 mv_x^2 + \tfrac12 mv_y^2 + \tfrac12 mv_z^2\), three independent terms, one for each direction. A nitrogen molecule is two atoms joined by a bond, like a small dumbbell. It can move from place to place, and it can also tumble end over end. Tumbling stores energy that helium cannot store.
The degrees of freedom \(f\) of a molecule are the number of independent ways in which it can hold energy. Each is a separate term, quadratic in a velocity or a displacement, in the energy of the molecule: \(\tfrac12 mv_x^2\), \(\tfrac12 I\omega^2\) or \(\tfrac12 \kappa x^2\).
Count the motions of the dumbbell in Fig 4.1. Moving from place to place gives three terms, as for helium. Tumbling can happen about two axes, both perpendicular to the bond, and each gives a term \(\tfrac12 I\omega^2\). Spinning about the bond itself gives nothing. The atoms are almost points on that axis, so the moment of inertia \(I\) about it is almost zero, and the spin holds no energy. A rigid diatomic molecule therefore has \(f = 3 + 2 = 5\).
A molecule whose atoms do not lie on one line, such as water or methane, can turn about all three axes. It has \(f = 3 + 3 = 6\). A straight triatomic molecule such as carbon dioxide turns about only two axes, like a diatomic one, so its translation and rotation give \(f = 5\).
The bond itself behaves like a stiff spring. When the atoms vibrate along it, they carry kinetic energy, and the stretched bond carries potential energy. So each mode of vibration adds two degrees of freedom. At room temperature the vibrations of nitrogen and oxygen are not excited at all. They start only in a hot gas. So \(f = 5\) for air near room temperature, and \(f = 7\) for a diatomic gas hot enough to vibrate.
How many degrees of freedom does a molecule of oxygen have at room temperature?
Why is the rotation of a diatomic molecule about its own bond not counted as a degree of freedom?
Both atoms lie on the bond, and an atom's mass sits almost entirely in its tiny nucleus. So the moment of inertia about the bond is almost zero. A rotational energy \(\tfrac12 I\omega^2\) with almost no \(I\) cannot hold a share of the energy. Quantum theory makes this sharp: such a rotation needs far more energy to start than a collision at ordinary temperatures can give, so it never starts.
Equipartition: the energy is shared equally
Collisions move energy from molecule to molecule. A glancing blow can make a molecule tumble, and a head-on blow can slow it. After many collisions the energy is spread evenly over every way of holding it. The last lesson found \(\tfrac32 kT\) for the three directions of motion, which is \(\tfrac12 kT\) for each. Equipartition says that every other degree of freedom receives the same share.
In thermal equilibrium at absolute temperature \(T\), each degree of freedom of a molecule holds, on average, an energy \(\tfrac12 kT\). A molecule with \(f\) degrees of freedom therefore holds \(\tfrac{f}{2}kT\) on average.
A nitrogen molecule at \(300\,\text{K}\) holds \(\tfrac52 kT\). That is \(2.5 \times 1.38 \times 10^{-23} \times 300\), or \(1.04 \times 10^{-20}\,\text{J}\). Of this, \(\tfrac32 kT\) is in its motion from place to place and \(kT\) is in its tumbling. A helium atom at the same temperature holds only \(\tfrac32 kT\). This is the answer to the steel bottles: nitrogen stores extra energy in its tumbling.
Equipartition has a surprise, and Fig 4.2 shows it. A degree of freedom is not always switched on. Cold hydrogen behaves as if it had only three degrees of freedom. Rotation starts near \(100\,\text{K}\), and vibration only above a couple of thousand kelvin. Each motion needs a certain smallest energy before it can start, and at low temperatures the collisions are too gentle to supply it.
Classical physics cannot explain the steps in Fig 4.2. Quantum theory can: the energy of rotation and of vibration comes only in fixed amounts. If \(kT\) is much smaller than the first amount, collisions cannot excite that motion, and it is "frozen". Kelvin called this failure of equipartition a dark cloud over physics in 1900, and it was one of the first clues to quantum theory.
What is the average energy of one molecule of methane, a non-linear molecule, at temperature \(T\), with no vibration excited?
The internal energy of an ideal gas
The internal energy \(U\) of a gas is the total energy of the random motion of all its molecules. In an ideal gas the molecules exert no forces between collisions, so there is no potential energy between them. Each of the \(N\) molecules holds \(\tfrac f2 kT\), so \(U = N\tfrac f2 kT\). With \(Nk = nR\) this becomes \(U = \tfrac f2 nRT\).
Now the steel bottles can be settled. Warming one mole by one kelvin raises its internal energy by \(\tfrac f2 R\). For helium, \(f = 3\), that is \(1.5 \times 8.31 = 12.5\,\text{J}\). For nitrogen, \(f = 5\), it is \(2.5 \times 8.31 = 20.8\,\text{J}\). A rigid bottle does no work, so all the heat goes into internal energy. Counting motions predicts both measured numbers.
The formula holds a deep fact: the internal energy of an ideal gas depends on its temperature alone. It does not depend on the pressure or the volume, or on how the gas reached its state. Two moles of nitrogen at \(300\,\text{K}\) hold \(12\,465\,\text{J}\), whether squeezed into a cylinder or spread through a room. The next chapter builds the first law of thermodynamics on this fact.
Internal energy counts only random motion, measured from the centre of mass of the gas. A cylinder of nitrogen on a Rajdhani at \(36\,\text{m/s}\) is no hotter than one on the platform. The ordered motion it shares with the train is kinetic energy of the whole cylinder, not internal energy. Suppose the train stopped suddenly and all that ordered energy turned into random motion. The rise would be \(\Delta T = Mv^2/fR = 0.028 \times 36^2/(5 \times 8.31)\), which is \(0.87\,\text{K}\), less than a degree.
A vessel at \(300\,\text{K}\) holds \(1\,\text{mol}\) of helium and \(2\,\text{mol}\) of oxygen. Find the internal energy of the mixture. Take \(R = 8.31\,\text{J/(mol K)}\).
Internal energies add. Helium, \(f = 3\): \(U_1 = \tfrac32 \times 1 \times RT = 1.5\,RT\).
Oxygen, \(f = 5\): \(U_2 = \tfrac52 \times 2 \times RT = 5\,RT\).
Total: \(U = 6.5\,RT = 6.5 \times 8.31 \times 300 = 16\,205\,\text{J}\), about \(16.2\,\text{kJ}\).
Find the internal energy of \(3\,\text{mol}\) of oxygen at \(400\,\text{K}\), treating oxygen as a rigid diatomic gas. Take \(R = 8.31\,\text{J/(mol K)}\). Give the answer in joules.
A sealed cylinder of gas sits in a Metro coach that speeds up from rest to \(20\,\text{m/s}\). What happens to the temperature of the gas while the coach runs at that speed?
On 27 April 1900, Lord Kelvin spoke in London on "Nineteenth-century clouds over the dynamical theory of heat and light". Physics seemed almost complete, he said, but two clouds hid the view. One was the failed search for the ether. The other was equipartition: measured heat capacities of gases did not match the counts of degrees of freedom. Within thirty years the first cloud had become relativity and the second quantum theory.
- Counting the spin about the bond of a diatomic molecule.The moment of inertia about the bond is almost zero, so that spin holds no energy. A rigid diatomic molecule has \(f = 5\).
- Including vibration at room temperature.The vibrations of nitrogen and oxygen are frozen near room temperature. Add two per mode only for a hot gas, and only when the problem says so.
- Adding the kinetic energy of a moving container to the internal energy.Internal energy is the random motion relative to the centre of mass. The container's motion is ordinary kinetic energy.