Internal Energy of an Ideal Gas
For a monatomic ideal gas (such as helium, neon, or argon), the only contribution to the energy comes from translational kinetic energy. The average translational kinetic energy of a single atom depends only on the gas temperature and is given by equation
Kavg = 3/2 kT.
The internal energy of n moles of an ideal monatomic (one atom per molecule) gas is equal to the average kinetic energy per molecule times the total number of molecules, N:
Eint = 3/2 NkT = 3/2 nRT
where n is the number of moles. Each direction (x, y, and z) contributes (1/2)nRT to the internal energy. This is where the equipartition of energy idea comes in – any other contribution to the energy must also contribute (1/2)nRT. As can be seen, the internal energy of an ideal gas depends only on temperature and the number of moles of gas.
If the gas molecules contain more than one atom, there are three translation directions, and rotational kinetic energy also contributes, but only for rotations about two of the three perpendicular axes. The five contributions to the energy (five degrees of freedom) give:
Diatomic ideal gas:
Eint = (5/2)NkT = (5/2)nRT
This is only an approximation and applies at intermediate temperatures. At low temperatures only the translational kinetic energy contributes, and at higher temperatures two additional contributions (kinetic and potential energy) come from vibration.
The internal energy will be greater at a given temperature than for a monatomic gas, but it will still be a function only of temperature for an ideal gas.
The internal energy of real gases also depends mainly on temperature, but similarly as the Ideal Gas Law, the internal energy of real gases depends also somewhat on pressure and volume. All real gases approach the ideal state at low pressures (densities). At low pressures molecules are far enough apart that they do not interact with one another. The internal energy of liquids and solids is quite complicated, for it includes electrical potential energy associated with the forces (or “chemical” bonds) between atoms and molecules.
Joule’s Second Law
For any gas whose equation of state is given exactly by pV = nRT (or pv = RT), the specific internal energy depends on temperature only. This rule was originally found in 1843 by Joule experimentally for real gases and is known as Joule’s second law:
The internal energy of a fixed mass of an ideal gas depends only on its temperature (not pressure or volume).
The specific enthalpy of a gas described by pV = nRT also depends on temperature only. Note that the enthalpy is the thermodynamic quantity equivalent to the total heat content of a system. It is equal to the internal energy of the system plus the product of pressure and volume. In intensive variables the Joule’s second law is therefore given by h = h(T) = u(T) + pv = u(T) + RT.
These three equations constitute the ideal gas model, summarized as follows:
pv = RT
u = u(T)
h = h(T) = u(T) + RT
Internal energy involves energy on the microscopic scale. It may be divided into microscopic potential energy, Upot, and microscopic kinetic energy, Ukin, components:
U = Upot + Ukin
where the microscopic kinetic energy, Ukin, involves the motions of all the system’s particles with respect to the center-of-mass frame. For an ideal monatomic gas, this is just the translational kinetic energy of the linear motion of the atoms. Monoatomic particles do not rotate or vibrate. The behavior of the system is well described by kinetic theory of gases. Kinetic theory is based on the fact that during an elastic collision between a molecule with high kinetic energy and one with low kinetic energy, part of energy will transfer to the molecule of lower kinetic energy. However, for polyatomic gases there is rotational and vibrational kinetic energy as well.
The microscopic potential energy, Upot, involves the chemical bonds between the atoms that make up the molecules, binding forces in the nucleus and also the physical force fields within the system (e.g. electric or magnetic fields).
In liquids and solids there is significant component of potential energy associated with the intermolecular attractive forces.