# Kinetic molecular theory

The kinetic molecular theory models the behaviour of gas on a microscopic scale. The theory is based on the following 5 postulates:

1. A gas is made up of a vast number of particles, and these particles are in constant random motion.
2. Particles in a gas are infinitely small; they occupy no volume.
3. Particles in a gas move in straight lines except when they collide with other molecules or with the walls of the container. Collisions with each other and with the walls of the container are elastic, so that the kinetic energy of the particles is conserved.
4. Particles in a gas interact with each other only when collisions occur.
5. The average kinetic energy of the particles in a gas is proportional to the absolute temperature of the gas and does not depend on the identity of the gas.

From these assumptions, the distribution of speed of a molecule can derived. This distribution is call the Maxwell-Boltzmann distribution and is given by the following equation.

$\frac{N(v)}{N_{total}} = \left(\frac{2}{\Pi}\right)^\frac{1}{2}\left(\frac{M}{2RT}\right)^\frac{3}{2}v^2e^{-\frac{Mv^2}{2RT}}$

Where N(v)/Ntotal is the fraction of molecules with speed v,

M is the molar mass of the gas,
R is the ideal gas constant
T is the temperature in K

Using this equation the effect of temperature on the speed on the molecules can be examined. The following graph shows how the speed distribution changes with temperature for a molecule, in this case the O2 molecule.

As the temperature increases, the fraction of the molecules moving at higher speed increases. Faster moving molecules collide more often and with greater force.

The following graph shows how the speed distribution changes with molar mass at a fixed temperature (300 K). Distributions of molecular speeds for He Ne and Ar atoms at the same temperature (300 K) are shown below.

If we hold the temperature at 300 K, the speed distribution for various gases will depend on the molar mass of the gas. As the molar mass decreases, the fraction of the molecules moving at higher speed increases. Therefore, lighter molecules travel faster than heavier molecules.

The Maxwell-Boltzmann distribution can also be expressed in terms of the translational energy of the molecule. When expressed in terms of energy the distribution goes as the following equation.

$\frac{N(v)}{N_{total}} = 2\left(\frac{E}{\Pi}\right)^\frac{1}{2}\left(\frac{1}{RT}\right)^\frac{3}{2}e^{-\frac{E}{RT}}$

Where N(E)/Ntotal is the fraction of molecules with speed E,

R is the ideal gas constant
T is the temperature in K

Using this equation the effect of temperature on the energy distribution of molecules can be examined. The following graph shows how the energy distribution changes with temperature for a molecule, in this case the O2 molecule.

Note that the energy distribution does not depend on the mass of the particle. The distribution only depends on the sample’s temperature. The following graphs shows that at 300 K, He Ne and Ar atoms have identical energy distributions.

The Kinetic-molecular theory says that:

1. Molecules interact with one another only through collisions (Postulate 4). The greater the number of collisions occurring per second, the greater the reaction rate.
2. Increase of collisions can be achieved by increasing the temperature, which results in increasing molecular speeds (Postulate 5). This can be also be achieved by increasing the concentration of the reactant molecules