Read the Three Molecular Speeds Before Comparing Gases
Kinetic theory links the pressure and temperature of a gas to the random motion of its molecules. Most exam questions come down to which of three characteristic speeds is being asked about, and to the fact that these speeds depend on temperature and molecular mass only.
Motion and force questions become simpler when direction, starting condition, time and the acting forces are marked first.
The three speeds and their order
The root-mean-square speed is v_rms = sqrt(3RT/M) = sqrt(3kT/m), the average speed is v_avg = sqrt(8RT/(πM)), and the most probable speed is v_mp = sqrt(2RT/M).
For speeds of 2u, 10u and 11u the mean is 23u/3, about 7.7u, and the rms speed is sqrt(75)u, about 8.7u, so the rms speed exceeds the mean by about u. For speeds 2, 3, 4, 5, 6 the rms speed is sqrt((4+9+16+25+36)/5) = sqrt(18), about 4.2. Averages of odd powers of the velocity components vanish for random motion, but even powers such as <v⁴> cannot be zero.
Dependence on temperature and mass
v_rms is proportional to sqrt(T/M), so two gases have equal rms speeds when T/M is equal.
The average translational kinetic energy per molecule is (3/2) kT and depends on temperature alone, so two gases at the same temperature have the same average kinetic energy per molecule whatever their pressures or masses, while their speeds differ because of mass. Heating to double the temperature while a diatomic gas dissociates into atoms changes both T and M, so apply both factors to v_rms.
Pressure and energy density
The pressure of an ideal gas is P = (1/3) ρ v_rms² = (2/3) E, where E is the kinetic energy per unit volume, so pressure is two-thirds of the energy density. Moving the container as a whole does not change the pressure, since pressure depends on random thermal speeds. If the number of molecules doubles while the total kinetic energy stays the same, the average kinetic energy per molecule halves, so the temperature halves.
Moon's missing atmosphere and mean free path
A planet or moon keeps an atmosphere only if the rms speed of the gas molecules stays well below the escape velocity; the moon has none because the rms speed of molecules there exceeds the moon's escape velocity. The mean free path is the average distance travelled between successive collisions and is inversely proportional to the number density and to the square of the molecular radius.
OlympIQ exam lens
- ✓v_mp < v_avg < v_rms with ratio 1 : 1.13 : 1.22; v_rms = sqrt(3RT/M)
- ✓Equal rms speeds need equal T/M; find the unknown temperature from that ratio
- ✓Average translational kinetic energy per molecule is (3/2) kT, independent of mass and pressure
- ✓P = (1/3) ρ v_rms² = (2/3) x energy density; bulk motion of the container does not change P
- ✓No atmosphere forms where molecular rms speeds exceed escape velocity; mean free path is inversely proportional to n and r²
Do not say that gases at the same temperature have the same molecular speeds. They have the same average kinetic energy per molecule, so the lighter gas moves faster because its speed scales as one over the square root of the mass.
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