GAS PRESSURE



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 container, they transfer an amount of their momentum to       the walls. Since a number of molecules collide the walls of 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”.

kinetic theory of gas
kinetic theory of gas
  
 Area of each wall = A = L2
   Volume of container = V = L3
   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 x-axis is equal to    vx
Initial momentum of the molecule = m x – v= -mvx
Final momentum of the molecule = mvx
Change in momentum = mvx – (-mvx)
Change in momentum = mvx + mvx

D M = 2mvx…………(a)
time taken for one collision
s = v t 

t = s/v…………(b)
in one collision distance covered is ,
s = 2L
v = vx
Putting the values of v and s in equation (b)
t = 2L/vx
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 = mvx2/L
   Thus the total force on the wall “a”
F = F1 + F2 + F3 + —————— + Fn
                                    F = mv1x2/L + mv2x2/L + mv3x2/L + —————— + mvnx2/L
                     F = m/L(v1x2 + v2x2 + v3x2 + —————— + vnx)
Multiply and dividing by N on R.H.S.
                                                               F = (v1x2 + v2x2 + v3x2 + —————— + vnx)/N
   Here
square of mean velocities = (v1x2 + v2x2 + v3x2 + —————— + vnx)/N
(v1x2 + v2x2 + v3x2 + —————— + vnx)/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 same in all three dimensions.
   Therefore, in equation (2) replacing Vy and Vz by Vx
   OR
   OR

Putting the value of  in equation (1)

    P = r 
    P =r 1/3 

Where r = density of gas
 = root mean square velocity of gas molecules.