Derivation of the Kinetic Equation - QS Study
QS Study

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

c2 = u2 + v2 + w2

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 2l to the face B and back.

  • The time required to travel from A to B and back to A = 2l/u
  • Hence the number of impacts on the wall A by one molecule in one second will be u/21

The total change in momentum per second for one molecule due to impacts with wall A = 2mu.u/2l = mu2/l

As the number of molecules is n the total change of momentum in one second for all the molecules = mnu2/l

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

f = mnu2/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.

u2 + v2 + w2 = c2 = 3u2

or, u2 = c2/3

therefore, force = f = mnc2/3l

Dividing both sides by l2 one obtains,

f/l2 = mnc2/3l3

But, f/l2 is force per area, which is equal to the pressure, P, and l3 is the volume, V, of the container cube,

Hence P = mnc2/3V

PV = 1/3 mnc2

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.