10. Specific Heat and Equipartition Principle.

One of the great successes of the kinetic theory of gases is the prediction of the specific heat,i.e., the energy that must be supplied to increase the temperature by one degree. For a monoatomic ideal gas the average kinetic energy of an atom is 3/2 kT (*). The kinetic energy is the only energy in a monoatomic gas, for which the atoms have three degrees of freedom corresponding to the three space dimensions. Therefore, an energy 1/2 kT is associated to each degree of freedom. This result indicates that

The molecular heat capacity at constant volume is 3/2 times the Boltzmann constant, in excellent agreement with the experimental results.
For polyatomic gases the specific heat is higher, indicating the existence of additional forms in which the molecules can absorb energy. As a consequence, it is necessary to supply more energy to increase the temperature. Let us consider an experience with a diatomic gas.

By clicking the button we run a simulation (**) of a gas made of diatomic molecules. The molecules possess translation and rotation kinetic energy. As the simulation is two dimensional, there are two degrees of freedom associated to translation and only one to rotation. In the actual three dimensional case there are three translational and two rotational degrees of freedom, so that there are five degrees of freedom able to absorb the energy supplied to the gas. Then, the specific heat at constant volume will be 5/2 kT.

The simulation shows ten diatomic molecules in a plane, with an initial energy around 9000 units distributed in such a way that there is much more energy associated to rotation than to translation. Due to the constraint of being in a plane

Equipartition of Energy Theorem
If the molecules are in thermal equilibrium with their neighbors they will absorb, on the average, equal amount of energy in each of their degrees of freedom.
From the results obtained in the previous experience, and assuming that the gas is three dimensional, select one of the following options: 

1. A diatomic gas absorbs more energy due to rotational than to translational motion.
1. The energy associated to each degree of freedom does not depend on the temperature.
1. The energy is equally distributed, on the average, among all the degrees of freedom of the system.

Index 1.Introduction 2.Pressure 3.Temperature 4.Internal Energy
5.Heat 6.Work 7.First Law 8.Entropy 9.Velocity Distribution
(*) TIPLER. PHYSICS. 15.5 Molecular interpretation of temperature. Kinetic Theory of Gases.

(**) Simulation developed by Ernesto Martin & Rafael Chicon