Page 16 - Hsslive-Plus-One-Physics-Class-Notes-Part-III (1)

Heat, Thermodynamics & Kinetic Theory {15} Vinodkumar M, St. Aloysius H.S.S, Elthuruth, Thrissur.
Expression for pressure exerted by a gas. Y
Consider a gas in a cubical vessel of unit side. Let m be the mass
of one molecule and n be the total number of molecules. Let W and W
1 2
be two opposite walls perpendicular to X axis.
m
Consider a molecule moving towards the wall W with a velocity v. W
1 2 v W
This molecule will collide with wall W and bounce back with same velocity. 1 X
1
Hence momentum of the molecule before collision = m v.
Momentum of the molecule after collision = – m v.
z
change in momentum of the molecule due to a single collision of the wall
W = – m v – m v. = – 2m v.
1
change in momentum of the wall W due to a single collision of the molecule = 2m v.
1
Now as the distance between the walls W and W is unity, the molecule will travel v times between
1 2
the walls in one second.
Number of collisions of the molecule with wall W per second = v/2.
1
Hence rate of change of momentum of wall W ie the force exerted by molecule on wall
1
v
W = change in momentum x number of collisions = 2m v . = m v 2
1 2
1
Now since the vessel contains n molecules, we can assume that at least n molecules will be moving in X direction.
3
1 2
Force exerted by the gas on wall W = m n v .
1 3

1 2
Since area of wall is unity, pressure exerted on wall W , P = m n v . This is true for all sides of the vessel.
1 3
1 2
Hence pressure exerted by the gas on the walls of the vessel, P = m n v .
3
Now, since different molecules are moving with different velocities, it will be convenient to consider rms
velocity.

1 2 1 2
P = m n v rms . ie P = m n c .; where c 2 is mean square velocity..
3 3
Now mn = total mass of the gas. Since volume is unity, mn = density of the gas, .
1 2
Pressure, P =  c
3
Kinetic Energy of gases.
Consider one mole of a gas of mass m and volume V.
1 2 1 M 2
Then, pressure P =  c = c .
3 3 V

1 2 2 1 2
ie PV = M c ie PV = M c
3 3 2

1 2
But PV = RT and M c is the average KE of one mole of gas.
2
2
RT = average KE of one mole of gas.
3

3
 KE of one mole of gas = RT; R = universal gas constant.
2