# Derivation of Kinetic Gas Equation

## Derivation of Kinetic Gas Equation

Let us consider a given mass of a gas to be confined in a cubical vessel. Let,

l = length of the edge of the cube

n = total number of molecules enclosed

m = mass of each molecules

c = root mean square velocity of the molecules

These molecules are moving in all possible directions and the velocity in any direction may be resolved into three components along the three axes X, Y , Z at right angles to one another. The three components x, y and z are related to c as-

c

^{2}= x

^{2}+ y

^{2}+ z

^{2}

Let us now consider only one molecule moving between two opposite faces and striking the face A again and again.

The velocity of the molecule before striking the face A is x and since it perfectly elastic, it rebounds with the same velocity(−x).

Momentum of the molecule before it strikes = mx

Momentum of the molecule after impact = −mx

Change in momentum after each impact = mx − (−mx) = 2mx

Now the molecules strikes the same face after travelling a distance 2l cms.(to the opposite face and back) with a velocity x cms. per second.

∴ Number of impacts per second on the same face = (x/2l)

∴ Number of impacts per second on the two opposite faces along the X-axis = 2 × (x/2l) = x/l

Hence, the total change of momentum per second due to the impact of one molecule on two opposite walls of the cube along X-axis = 2mx x (x/l) = (2mx

^{2}/l).

Similarly, total change of momentum on the two opposite faces along Y-axis is 2my

^{2}/l and along Z-axis is 2mz

^{2}/l

Hence, the total change of momentum on all the six faces of the cube per second per molecule is

= (2mx

^{2}/l) + (2my

^{2}/l) + (2mz

^{2}/l)

= 2m/l(x

^{2}+ y

^{2}+ z

^{2})

= 2mc

^{2}/l

where, c

^{2}is root mean square velocity as mentioned before.

∴ Total change of momentum due to n molecules-

= 2mnc

^{2}/l

But the change of momentum per second is force and force per unit area is pressure-

∴ P = Force/Area

or, P = (2mnc

^{2}/l) / l x 6l

^{2}

(Area of one face is l

^{2}and cube has six faces having area = 6l

^{2})

or, P = 1/3(mnc

^{2}/l

^{3})

or, P = 1/3(mnc

^{2}/V)

(As Volume 'V' = l

^{3})

or, PV = (1/3)mnc

^{2}

This equation is called kinetic equation of gas.

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