Kinetic Theory of Gas problem

[sy7kenny]

Homework Statement

Helium gas with T1 = 500K and P1 = 0.02MPa in a rigid container with volume V = 1 cm^3.
Then Helium goes through a process where atoms with kinetic energies greater than kB*T1, where kB is Boltzmann constant, are instantaneously removed from the container.
Atoms remaining in the container attain a M-B velocity distribution with a final Temperature T2.
Calculate T2 and the final pressure P2.

Homework Equations

(1) PV = N * kB * T
M-B speed distribution:
(2) f(v) = 4 * pi * v^2 * (m / (2*pi*kB * T))^(3/2) * exp( - m * v^2 / (2 * kB * T))

The Attempt at a Solution

First, I find the number of molecules using (1) and the total number of atoms in state 1 is 2.987e18.
Then find the limiting velocity by setting E = kB* T = 1/2 * m * v^2 to solve for v,
integrate from 0 to v ( f(v) dv ) to find the fraction of the atoms up to this speed v = 0.463.
The remaining atoms will be this fraction multiply by the number of atoms = 1.34e18 atoms remaining.
Then I am lost on solving for T2 with this remaining atoms that attain M-B velocity distribution.

Any help will be appreciated. Thanks!

 

[DrClaude]

Then I am lost on solving for T2 with this remaining atoms that attain M-B velocity distribution.

I'll give you a hint: when you remove the atoms, you are also taking away their energy.

 

[sy7kenny]

That means if I can find the total energy in state 1, I can also find what is left in state 2?

Thanks for your fast reply DrClaude.

 

[DrClaude]

That means if I can find the total energy in state 1, I can also find what is left in state 2?

Since you have figured out the cutoff velocity, it should be easy to get the kinetic energy of the atoms that are left.

 

[sy7kenny]

Hmm sorry DrClaude, I am still a bit confused.
Does it mean sense that if I integrate from 0 to cutoff velocity ( f(v) dv) and set it = 1, I can find a temperature that satisfy this?

 

[DrClaude]

Does it mean sense that if I integrate from 0 to cutoff velocity ( f(v) dv) and set it = 1, I can find a temperature that satisfy this?

What you need to do is use the distribution function to find the energy of the atoms in the velocity range 0 to cutoff at T₁. Then you should find a T₂ that gives you the same energy.

Note that calculating the energy is not simply integrating f(v) dv. (This should be in your textbook.)

 

[sy7kenny]

My bad. From the text I found, the energy is:

E = integral ( 0 to cutoff velocity) ( 1/2 * m *v^2 * f(v) dv) , then uses the value to find T2, where T2 = E/kB.

And I have T2 and number of remaining atoms, getting P2 should be no problem.

Thanks for the help!