Consider a perfect gas in a cube of side 'l'. Let total
number of molecules of gas = N
Mass of each molecule = m_{0}
Area of each face of cube A = l^{2}
Volume of the cube V = l^{3}
Total mass of the gas m = Nm_{0}
Density of the gas r =


Pressure exerted by gas is same along X, Y, Z directions.
\ Number of
molecules travelling parallel to each axis =
\ Velocities of
the


molecules travelling parallel to Xaxis.

C_{1}, C_{2}, …, C_{ N/3}.
Consider the motion of first typical molecule along xdirection.
Momentum of the molecule (before colliding with face ABCD) = m_{0}C_{1}
Momentum after collision = m_{0}C_{1}
\ Change in
momentum of molecule =  2m_{0}C_{1}
\ According to
the principle of conservation of linear momentum
Change in momentum of face ABCD = 2m_{0}C_{1}
Between two successive collisions with the same face ABCD, the molecule
travels distance 2l with velocity C_{1}.
Time interval between the two collisions =


\ Number of collisions per unit time =


\ Change in
momentum of surface per second = Change in momentum per collision ×
Number of collisions per second
\ Total rate of
change of momentum of the surface due to N/3 molecules =
According to Newton's 2nd law, rate of change of momentum
= force creating on the surface (F)
\ Pressure
exerted by the gas molecules on the surface
Kinetic Energy per Unit Volume
Since, m_{0}C^{2}
= K.E. of gas molecule
\ N. m_{0}C^{2}
= Total K.E. of gas
To show that Average K.E. per Molecule is kT
Total K.E. of gas =


PV (From (ii))

But, PV = RT (n = 1) for an ideal gas
But gas constant R = N_{0}k
Where k = Boltzmann constant
N_{0} = Avogadro's number
Þ K.E. per
molecule is directly proportional to its absolute temperature.
If number of molecules in 1 mole of a gas = N_{0}
If M is mass of 1 mole of gas, then K.E. per unit mass =


RMS Velocity
^{}
\ PV =


MC^{2} (for one mole gas, mass of the gas (m)
= molecular weight (M))

But, PV = RT (n = 1)
\


MC^{2 }= RT Þ \ C^{2 =}


R
and m are constant, C µ or
C^{2} µT
P =


rC^{2 }^{\}^{ C2= }


,C=


Since, Cµ
C_{0} = RMS velocity at 0° C.
C = RMS velocity at t° C.
T = t + 273
