15.4 Energy and Temperature of an Ideal Gas

Average Translational Kinetic Energy

Deducing Average Translational Kinetic Energy

We equate the macroscopic ideal gas equation with our newly derived microscopic kinetic theory equation.

1. $pV=NkT$
2. $pV=\frac{1}{3}Nm⟨c^2⟩$

Equating the two: $\frac{1}{3}Nm⟨c^2⟩=NkT$

Cancel $N$ from both sides: $\frac{1}{3}m⟨c^2⟩=kT$

We want the formula to look like kinetic energy ($E_k=\frac{1}{2}m{v}^2$). Multiply both sides by $\frac{3}{2}$: $\frac{1}{2}m⟨c^2⟩=\frac{3}{2}kT$

Therefore, the average translational kinetic energy of a single molecule is: $E_k=\frac{3}{2}kT$

Important

This equation proves that the average kinetic energy of an ideal gas molecule is directly proportional to the thermodynamic temperature: $E_k\propto T$

Internal Energy of an Ideal Gas

Internal Energy

Internal energy is defined as the sum of the random distribution of kinetic and potential energies of the particles in a system.

Internal Energy of an Ideal Gas

For an ideal gas:

• We assumed that there are no intermolecular forces.
• Therefore, there is no potential energy.
• Thus, the internal energy is solely the kinetic energy of the particles.