GAS PRESSURE



Gaseous molecules are in continuous motion. They collide with each other and with the walls of the container. When they collide with the walls of the container, they transfer an amount of their momentum to the walls. Since a number of molecules collide the walls of the container, therefore the walls of the container are constantly under the influence of the force. This force expressed per unit area is called “GAS PRESSURE”. Mathematically 
P = F/A

EXPRESSION FOR GAS PRESSURE


Consider “N” molecules of a gas enclosed in a cubical container of each side equal to “L”.
mass of each molecule is “m”. 



Area of each wall = A = L^{2}
Volume of container = V = L^{3} 
Consider the motion of those molecules moving along x axis towards the wall marked “a”.
Taking the example of a molecule moving from right to left . Velocity of molecule along xaxis is equal to v_{x} 
Initial momentum of the molecule = m x – v_{x }= mv_{x}
Final momentum of the molecule = mv_{x}
Change in momentum = mv_{x} – (mv_{x})
Change in momentum = mv_{x} + mv_{x}
D M = 2mv_{x}…………(a)
time taken for one collision
s = v t
t = s/v…………(b)
in one collision distance covered is ,
s = 2L
v = v_{x}
Putting the values of v and s in equation (b)
t = 2L/v_{x}

rate of change of momentum =

Putting the values of DM and t

rate of change of momentum =

rate of change of momentum =

rate of change of momentum =

But rate of change of momentum is equal to the applied force.

F = mv_{x}^{2}/L

Thus the total force on the wall “a” 
F = F_{1} + F_{2} + F_{3} + —————— + F_{n}

F = mv_{1x}^{2}/L + mv_{2x}^{2}/L + mv_{3x}^{2}/L + —————— + mv_{nx}^{2}/L

F = m/L(v_{1x}^{2} + v_{2x}^{2} + v_{3x}^{2} + —————— + v_{nx}^{2 })

Multiply and dividing by N on R.H.S.

F = (v_{1x}^{2} + v_{2x}^{2} + v_{3x}^{2} + —————— + v_{nx}^{2 })/N

Here

square of mean velocities = (v_{1x}^{2} + v_{2x}^{2} + v_{3x}^{2} + —————— + v_{nx}^{2 })/N

= (v_{1x}^{2} + v_{2x}^{2} + v_{3x}^{2} + —————— + v_{nx}^{2 })/N

therefore
F = (mN/L)
F =( mN/L ) —————– (1)

Since resultant velocity is given by: 
— (2)

Velocity of gas molecules in different directions may be different but on the average and randomness of
the molecular motion we can assume that the components of velocities are the same in all three dimensions. 

Therefore, in equation (2) replacing V_{y} and V_{z} by V_{x} 

OR 

OR 


Putting the value of in equation (1)
Where r = density of gas
= root mean square velocity of gas molecules.
