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

The kinetic molecular theory qualitatively explains the behaviors described by the various gas laws. The postulates of this theory may be applied in a more quantitative fashion to derive these individual laws.

Collectively, the molecules in a sample of gas have average kinetic energy and average speed; but individually, they move at different speeds. Molecules frequently undergo elastic collisions in which the momentum is conserved. Since the colliding molecules are deflected off at different speeds, individual molecules have widely varying speeds. However, because of the vast number of molecules and collisions involved, the molecular speed distribution and average speed are constant. This molecular speed distribution is known as a Maxwell-Boltzmann distribution, and it depicts the relative numbers of molecules in a bulk sample of gas that possesses a given speed.

The kinetic energy (KE) of a particle of mass (m) and speed (u) is given by:

Expressing mass in kilograms and speed in meters per second will yield energy values in units of joules (J = kg·m²/s²). To deal with a large number of gas molecules, we use averages for both speed and kinetic energy. In the KMT, the root mean square velocity of a particle, u_rms, is defined as the square root of the average of the squares of the velocities with n = the number of particles:

The average kinetic energy for a mole of particles, KE_avg, is then equal to:

where M is the molar mass expressed in units of kg/mol. The KE_avg of a mole of gas molecules is also directly proportional to the temperature of the gas and may be described by the equation:

where R is the gas constant and T is the kelvin temperature. When used in this equation, the appropriate form of the gas constant is 8.314 J/mol⋅K (8.314 kg·m²/s²·mol·K). These two separate equations for KE_avg may be combined and rearranged to yield a relationship between molecular speed and temperature:

If the temperature of a gas increases, its KE_avg increases, more molecules have higher speeds and fewer molecules have lower speeds, and the distribution shifts toward higher speeds overall, that is, to the right. If temperature decreases, KE_avg decreases, more molecules have lower speeds and fewer molecules have higher speeds, and the distribution shifts toward lower speeds overall, that is, to the left.

At a given temperature, all gases have the same KE_avg for their molecules. The molecular velocity of a gas is directly related to molecular mass. Gases composed of lighter molecules have more high-speed particles and a higher u_rms, with a speed distribution that peaks at relatively higher velocities. Gases consisting of heavier molecules have more low-speed particles, a lower u_rms, and a speed distribution that peaks at relatively lower velocities.

This text is adapted from Openstax, Chemistry 2e, Section 9.5: Kinetic-Molecular Theory.

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.