Tuesday, March 26, 2019

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………..v­n. 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 vx, vy, vz. along the three axis X, Y, Z respectively. now we know that:
v12=vx2+vy2+vz2

consider the vx velocity component along the X-direction, we have two opposite faces A & B. The momentum of the molecule before colliding the face B is mvx. 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 -mvx.
Therefore the change in momentum is = mvx-(-mvx) = 2mvx
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 vx velocity along the X-axis i.e. the molecule travels vx distance in 1 sec time.
Number of collisions for 2l distance is= 1
Therefore the number of collisions in 1 sec for vx 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 2nd 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 2nd molecule gives force 

…………………………………………………………………………………
Similarly the nth 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


    
 
    
 

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