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 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”.
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 = rate of change of momentum
Putting the values of DM and t
rate of change of momentum =putting values of dm
rate of change of momentum = expression 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 = gas pressure(v1x2 + v2x2 + v3x2 + —————— + vnx)/N
   Here
square of mean velocities = (v1x2 + v2x2 + v3x2 + —————— + vnx)/N
square mean of velocity(v1x2 + v2x2 + v3x2 + —————— + vnx)/N
therefore
F = (mN/L) square mean of velocity
F =( mN/L )square mean of velocity —————– (1)
   Since resultant velocity is given by:
resultant velocity—- (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.
components of velocities
   Therefore, in equation (2) replacing Vy and Vz by Vx
value of velocity
   OR
velocity v2
   OR
value of square of velocity
Putting the value of square of velocity in equation (1)

P = r square of velocity
P =r 1/3 square of velocity
pressure

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