Kinetic Molecular Theory: Molecular Velocities, Temperature, and Kinetic Energy

All gas particles have kinetic energy, which is a function of the particle’s mass, in kilograms, and speed, or the magnitude of its velocity, in meters per second.

With every collision, the velocities of individual gas particles change. Therefore, a collection of gas particles actually has a distribution, or range, of velocities and kinetic energies.

This means that at any instant, some molecules are moving slower than others; however, the average kinetic energy remains the same.

The average kinetic energy is related to the average of the squares of the speeds, or mean-square speed — both of which remain constant at a given temperature for a given gas.

Now, the average kinetic energy of one mole of a gas is expressed by introducing Avogadro’s constant, N_A. The product of the mass per particle and Avogadro’s constant of particles per mole equals the molar mass of the gas in kilograms per mole.

Recall from the kinetic molecular theory that the average kinetic energy of a mole of gas is directly proportional to temperature. Through complex derivations, the proportionality constant is found to be 3/2 R.

Combining the two equations, rearranging the terms, and taking the square root on both sides relates the square root of the mean-square speed — also called the root-mean-square, or RMS, speed — the molar mass, and the absolute temperature of a gas.

The RMS speed is inversely proportional to molar mass and directly proportional to temperature.

Suppose two gases — helium and argon — are at the same temperature. As helium has the lower molar mass, the equation indicates that helium must have a higher RMS speed than argon.

A similar observation is made in a plot of the distribution of molecular speeds for three gases — helium, argon, and chlorine — at the same temperature.

Notice that even though all gases have the same average kinetic energy, the lightest gas, helium, has both the highest RMS speed and the broadest speed distribution, corresponding to the widest range of molecular velocities.

A plot of the speed distribution for any gas — say, argon — at different temperatures displays an increase in the RMS speed and a broadening of the speed distribution at higher temperatures.

In short, gases move faster at higher temperatures. For example, the aroma-causing gas particles from hot food move faster than particles from cold food. Thus, hot food is detected faster than cold food.