Average kinetic energy of molecules

[shounakbhatta]

Hello,

The formula for average kinetic energy of molecule is:

k=3/2KT.

Can anyone please explain the derivation without using calculus?

Thanks,

-- Shounak

 

[Andy Resnick]

Each degree of freedom contains 1/2 kT of energy; for point objects there are only 3 degrees of freedom. Diatomic molecules have 5 degrees of freedom

 

[jtbell]

Try here:

http://hyperphysics.phy-astr.gsu.edu/hbase/kinetic/kinthe.html

followed by

http://hyperphysics.phy-astr.gsu.edu/hbase/kinetic/kintem.html

 

[technician]

To start right at the beginning the problem is to find an expression for the pressure (force) due to molecules colliding with a wall.
Consider a molecule of mass m moving with speed (velocity) c inside a container of dimensions l x a x b (volume = lab)
The molecule strikes face ab and rebounds perfectly elastically so change in
momentum = 2mc
Force on face ab = change in momentum x Number of collisions per second
so force due to 1 molecule = 2mc/(2l/c) = mc²/l
So pressure on ab due to 1 molecule = mc²/lab = mc²/V
If there are N molecules in the container then, on average, N/3 move in the x, y and z directions.
Therefore pressure on ab due to N molecules = Nmc²/3V
or PV = Nmc²/3
Now KE of molecules = mc²/2 so PV = 2N(KE)/3
If there is 1 mole of molecules then N = N_a
Therefore PV = 2N_a x KE/3 for 1 mole
You also know that PV = RT for 1mole
so 2N_a x KE/3 = RT
This give KE = 3/2 RT/N_a
R/N_a is a ratio of 2 constants and is known as Boltzmann's constant, k.

I hope that this is MORE than what you needed.

 

[shounakbhatta]

Thank you very much for the derivation. The N/3 factor comes into play due to the x,y,z three directions, right?

 

[technician]

correct.
In addition the equation should have an 'average' velocity known as the root mean square velocity.
I did not include that to keep the typing as simple as possible.
Also the KE is called TRANSLATIONAL KINETIC ENERGY. There are other kinetic energies (rotational and vibrational) but it is translational KE that 'shows' as Temperature

 

[shounakbhatta]

Hello Technician,

Thank you very much for all the reply.

 

[shounakbhatta]

Hello jtbell,

I have just one small thing to ask you. The link that you have provided tells, the ideal gas law PV=nRT which is again:PV=2/3N(1/2mv^2).

Can you please show me how it came?

Thanks.