Translational Kinetic Energy

We again consider a single molecule of an ideal gas as it moves around in the box of Fig. 19-3, but we now assume that its speed changes when it collides with other molecules. Its translational kinetic energy at any instant is mv². Its average translational kinetic energy over the time that we watch it is

in which we make the assumption that the average speed of the molecule during our observation is the same as the average speed of all the molecules at any given time. (Provided the total energy of the gas is not changing and provided we observe our molecule for long enough, this assumption is appropriate.) Substituting for v_rms from Eq. 19-22 leads to

However, M/m, the molar mass divided by the mass of a molecule, is simply Avogadro’s number. Thus,

Using Eq. 19-7 (k = R/N_A), we can then write

This equation tells us something unexpected:

 At a given temperature T, all ideal gas molecules—no matter what their mass—have the same average translational kinetic energy—namely, . When we measure the temperature of a gas, we are also measuring the average translational kinetic energy of its molecules.

CHECKPOINT 2    A gas mixture consists of molecules of types 1, 2, and 3, with molecular masses m₁ > m₂ > m₃. Rank the three types according to (a) average kinetic energy and (b) rms speed, greatest first.