Maxwell Law of Distribution of Velocities for Gas Molecules

Gas molecules cannot maintain the same velocity for any length of time because they continuously collide with one another. At any instant, however, different molecules have different velocities. Maxwell and Boltzmann showed, using statistical and probability considerations, that the velocities of gas molecules follow a definite distribution.

The distribution of molecular velocities depends mainly on the temperature and molar mass of the gas. According to the Maxwell distribution law, the fraction of molecules having speeds between c and c + dc is given by:

^dn_c/n = 4π (M/2πRT)^3/2 c² e^−Mc2/(2RT) dc

where dn_c is the number of molecules having speeds between c and c + dc, n is the total number of molecules, T is the absolute temperature, M is the molar mass of the gas, and R is the gas constant.

The ratio dn_c/n represents the fraction of the total number of molecules having speeds between c and c + dc.

Dividing both sides of the above equation by dc, we obtain:

(1/n)(dn_c/dc) = 4π (M/2πRT)^3/2 c² e^−Mc2/(2RT)

The expression (1/n)(dn_c/dc) represents the distribution of molecules with respect to their speeds. If the fraction of molecules is plotted against speed, a characteristic Maxwell distribution curve is obtained.

The fraction of molecules having speeds between c and c + dc is represented by the area under the curve between these two speeds. Thus, the area under a small strip of width dc represents dn_c/n.

The total area under the normalized Maxwell distribution curve is 1, representing the complete population of gas molecules. The curve shows that only a small fraction of molecules have very low or very high speeds, while a relatively large fraction have speeds in an intermediate range.

The speed corresponding to the peak of the distribution curve is called the most probable speed. It is the speed possessed by the maximum fraction of molecules.

Effect of Temperature on the Distribution of Molecular Velocities

When the temperature of a gas is increased, the average kinetic energy of its molecules increases. Consequently, the molecular speeds increase and the most probable speed also increases.

As the temperature increases, the Maxwell distribution curve becomes broader and lower, and its peak shifts towards higher speeds.

If V₁ is the most probable speed at temperature T₁ and V₂ is the most probable speed at temperature T₂, then:

T₂ > T₁  ⇒  V₂ > V₁

Conversely, when the gas is cooled, the distribution curve becomes taller and narrower, and its peak shifts towards lower speeds. Thus, increasing temperature broadens the distribution and shifts the maximum towards higher speeds.

Key Point: At higher temperature, a larger range of molecular speeds is observed, the most probable speed increases, and the Maxwell distribution curve becomes broader and lower. The total area under the normalized curve, however, remains equal to 1.