Derivation of Kinetic Gas Equation

Kinetic Gas Equation 

 Let us consider a cubical container has only one type of gas molecules.

Cubical vessel have side length=l.  

So the total volume (V) of the vessel is= l³

Total inside surface area of the vessel is = 6l²

The mass of each gas molecules is= m

The total number of gas molecules inside the vessel is= n

So total mas of gas is= mn

Now according to kinetic theory of gases, these molecules are moving randomly in different directions with different velocities= v₁, v₂, v₃, v₄, v₅………..v­_n. Now from physics we know that a vector can be resolve into three component, for any velocity say v₁ can be resolve into three component v_x, v_y, v_z. along the three axis X, Y, Z respectively. now we know that:

v₁²=v_x²+v_y²+v_z²

consider the v_x velocity component along the X-direction, we have two opposite faces A & B. The momentum of the molecule before colliding the face B is mv_x. since the collisions are perfectly elastic so the molecules will return with the same velocity but in opposite to the initial direction so new momentum of the molecule is -mv_x.

Therefore the change in momentum is = mv_x-(-mv_x) = 2mv_x

Now for the next collision, the molecule has to travel from the wall B to wall A, collide there and return back from wall A to wall b, so for next collision the molecule should have to travel twice the AB=2l distance. But the molecule has v_x velocity along the X-axis i.e. the molecule travels v_x distance in 1 sec time.

Number of collisions for 2l distance is= 1

Therefore the number of collisions in 1 sec for v_x distance travel is    

         

So change in momentum in 1 sec on the B wall is                         

But in X-direction there is an another face A, so the total change in momentum along the X-direction

due to two faces A & B is

Similarly for both Y & Z-direction the change in momentum in 1 sec are

       

 respectively.                                

                                

Therefore for 1 molecule the total change in momentum in 1 sec is

But from newton’s 2^nd law of motion we know that change of momentum in 1 sec is the force, so the force (F) by the molecule 1 is

Similarly the 2^nd molecule gives force 

…………………………………………………………………………………

Similarly the n^th molecule gives force 

Therefore the total force (F) is

                             

      

Where u is called the root mean square velocity or rms velocity  

Now, we know tha the pressure (P) is

But we know that the total inside surface area of the vessel is = 6l²

So in this case the pressure exerted on the six wall’s of the container is