**Derivation of the Kinetic Equation**

The pressure exerted by a gas is the total effect of the impacts of the gas molecules on the walls of the container. The expression for the pressure of a gas in terms of the molecular velocity may be derived as follows:

Consider a cube with sides* l* cm each, which contains n molecules of a gas, the mass of each molecule being m.

Let the root-mean-square velocity of the molecules be c. This may he resolved into three mutually perpendicular components u, v and w along the x, y and z axes respectively, c and its components (which are vector quantities) are related by the expression

**c ^{2} = u^{2} + v^{2} + w^{2}**

The molecules are moving at random within the container and are colliding with each other and the walls of the container. Consider one molecule starting from the wall B and moving in a straight line perpendicularly to the wall A opposite to it, and rebounds:

- The momentum of the molecule before impact with the wall A = mu
- The momentum of the molecule after the impact with the wall A = -mu.
- Therefore, the change in momentum = mu-(-mu) = 2mu

This will be the momentum imported to the wall by each impact. Before the molecule can strike the wall A again it has to travel a distance 2*l *to the face B and back.

- The time required to travel from A to B and back to A = 2
*l*/*u* - Hence the number of impacts on the wall A by one molecule in one second will be u/2
*1*

The total change in momentum per second for one molecule due to impacts with wall A = 2mu.u/2*l* = mu^{2}/*l*

As the number of molecules is n the total change of momentum in one second for all the molecules = mnu^{2}/*l*

The rate of change of momentum is equal to the force on the wall A. Hence force is given by:

**f = mnu ^{2}/l**

Experiment shows that the force exerted on the walls is the same for all the walls. Hence the velocities resolved along three axes must be equal. i.e.

u^{2} + v^{2} + w^{2} = c^{2} = 3u^{2}

or, u^{2} = c^{2}/3

therefore, force = f = mnc^{2}/3*l*

Dividing both sides by *l ^{2}* one obtains,

**f/ l^{2} = mnc^{2}/3l^{3}**

But, f/*l ^{2}* is force per area, which is equal to the pressure, P, and

*l*is the volume, V, of the container cube,

^{3}Hence P = mnc^{2}/3V

**PV = 1/3 mnc ^{2}**

This is the kinetic equation for gases. Although the above deduction has been made for a cubical vessel the equation is valid for a vessel of any shape as the vessel can he divided into a large number of small tubes for each of which this equation is valid.