prove that the kinetic energy per molecule of an ideal gas is 3/2 kbt, where kb is boltzmann constantprove that the kinetic energy per molecule of an ideal gas is 3/2 kbt, where kb is boltzmann constant
Answers
Explanation:
Pressure is explained by kinetic theory as arising from the force exerted by molecules or atoms impacting on the walls of a container, as illustrated in the figure below. Consider a gas of N molecules, each of mass m, enclosed in a cubical container of volume V=L3. When a gas molecule collides with the wall of the container perpendicular to the x coordinate axis and bounces off in the opposite direction with the same speed (an elastic collision), then the momentum lost by the particle and gained by the wall (
Δ
p
) is:where vx is the x-component of the initial velocity of the particle.
The particle impacts one specific side wall once every
Δ
t
=
2
L
v
x
,
(where L is the distance between opposite walls). The force due to this particle is:
F
=
Δ
p
Δ
t
=
mv
2
x
L
.
The total force on the wall, therefore, is:
F
=
Nm
¯¯¯¯¯
v
2
x
L
,
where the bar denotes an average over the N particles. Since the assumption is that the particles move in random directions, if we divide the velocity vectors of all particles in three mutually perpendicular directions, the average value of the squared velocity along each direction must be same. (This does not mean that each particle always travel in 45 degrees to the coordinate axes. )This gives
¯¯¯¯¯
v
2
x
=
¯¯¯¯¯
v
2
/
3
. We can rewrite the force as
F
=
Nm
¯¯¯¯¯
v
2
3
L
.
This force is exerted on an area L2. Therefore the pressure of the gas is:
P
=
F
L
2
=
Nm
¯¯¯¯¯
v
2
3
V
=
nm
¯¯¯¯¯
v
2
3
,
where V=L3 is the volume of the box. The fraction n=N/V is the number density of the gas. This is a first non-trivial result of the kinetic theory because it relates pressure (a macroscopic property) to the average (translational) kinetic energy per molecule which is a microscopic property.