Temperature and Mean Kinetic Energy

Temperature measures an average energy, not the speed of one chosen molecule. At one temperature, light molecules move faster than heavy molecules on average.

The physical pattern above can be proved by comparing two expressions for the same quantity, $pV$.

1. Write the kinetic-model equation and the molecular equation of state.
   $$pV=\tfrac{1}{3}Nm\langle c^2\rangle$$
   $$pV=NkT$$
2. Set the right-hand sides equal and cancel $N$.
   $$\tfrac{1}{3}m\langle c^2\rangle=kT$$
3. Multiply both sides by $\tfrac{3}{2}$, so that the left side becomes a kinetic energy.

• The left side is the mean translational kinetic energy of one molecule. Call it $E_K$.
• The right side is a constant times $T$. The thermodynamic temperature of an ideal gas is a direct measure of the mean translational kinetic energy of its molecules.
  $$E_K\propto T$$
• Translational means the energy of moving from place to place. A molecule made of two or more atoms can also spin, but that spinning energy is not included here.
• Two different gases at the same temperature have the same mean translational kinetic energy (9702/42/M/J/24 Q2(b)(iii)). The heavier molecules must move more slowly.
• Papers also ask you to do the three steps above as a derivation (9702/42/M/J/25 Q4(b)(ii)).

• One assumption of the kinetic model is that there are no forces between the molecules. So the molecules of an ideal gas have no potential energy (9702/41/M/J/24 Q3(a)(ii)).
• Topic 15 stops at the mean translational kinetic energy of one molecule. Do not add rotational or vibrational energy to this result.
• Internal energy, monatomic gases and the first law are taught in the Thermodynamics chapter. That is where the total energy of the gas is built carefully.