1. Kinetic Theory of Gases & the Equipartition Theorem

2. Results of Kinetic Theory
The Kinetic Energy of individual particles in an ideal monatomic gas is related to the Temperature of the gas as:
v is the thermal average velocity.
(½)mv2 = (3/2)kBT

3. Maxwell-Boltzmann Energy Distribution
This distribution shows that there is a wide range of molecular speeds that varies with temperature.

4. Equipartition of Energy
The Kinetic Theory result tells us that the average translational KE is proportional to the temperature.
Types of allowed motion are referred to as Degrees of Freedom (DOF)
Linear motion in three directions
(x,y,z)  3 DOF.

5. This is true for single point masses that possess no structure.
Each degree of translational freedom contributes (½)kBT to the thermal energy:
KEx + KEy + KEz =
(½)kBT + (½)kBT + (½) kBT
KEtotal = (3/2)kBT
This is true for single point masses that possess no structure.

6. For molecules (multi-atom particles) there are added degrees of freedom due to rotations & vibrations.

7. The Equipartition Theorem.
Each additional DOF requires (½)kBT of energy.
So, each new DOF contributes (½)kBT to the total internal energy of the gas.
This is
The Equipartition Theorem.

8. Sum of all the kinetic & potential energies.
Internal Energy, U
The Internal Energy U of a gas is the
Sum of all the kinetic & potential energies.
U = KE + PE
For an ideal gas there is no PE of interaction, so U is comprised solely of the various types of kinetic energies, i.e. translational, rotational and vibrational.

9. Internal Energy of an Ideal Diatomic Molecular Gas
There are
3 Translational DOF + 2 Rotational DOF
= 5 DOF Total.
Each DOF contributes (½)kBT, so, for an ideal gas of N diatomic molecules, the internal energy is,
U = (5/2)NkBT