Thermodynamics problem - Find the kinetic energy of ideal gas, given P and V

[arddi2007]

Homework Statement

Find the Kinetic Energy of the ideal gas $$\overline{Ek}$$=? if its volume is 10 liters (10^-2 m³) and it is under the pressure of p=5·10⁵.

Homework Equations

p·V = N·k·T (where p-pressure, V-volume, N-number of particles (molecules), k- Boltzmann constant, T-absolute temperature (in Kelvins))
Ek=Mm·$$\overline{v}$$²/2 (where Mm-the mass of the molecule and $$\overline{v}$$-the quadratic velocity of the molecule)
Ek=3/2 k·T (where k-the Boltzmann constant and Ek - kinetic energy (of the ideal gas in this case))

The Attempt at a Solution

With the help of Wikipedia, I was really close to solving this one. But at the end I got a little confused and now I need help. This is my progress:

p·V=N·k·T
p= N·k·T/V
p=N/V · 2/3 Ek
p·V = 2/3 N · Ek

Knowing that Ek=Mm·$$\overline{v}$$²/2 :

p·V= 2/3 · N · Mm·$$\overline{v}$$²/2

Since p·V=N·k·T:

N·k·T=N·Mm·$$\overline{v}$$²/3

then: T= Mm·$$\overline{v}$$²/3·k

This is the part when it gets tricky. I assumed that since Ek=2/3 k·T, it would be that:

T=2/3 Ek/k

and then we would substitute T in p·V=N·k·T. But it Wikipedia it says that Ek=2/3 k·T·N. Can anyone confirm that and tell me where that equation was derived from?

I really need to solve this equation. As you can see, I nearly solved it but I just need a small push. I really need to get this done by tonight so any help at all would be greatly appreciated. Thank you very much for your time!

 

[Doc Al]

With the help of Wikipedia, I was really close to solving this one. But at the end I got a little confused and now I need help. This is my progress:

p·V=N·k·T
p= N·k·T/V
p=N/V · 2/3 Ek
p·V = 2/3 N · Ek

OK. Looks to me like you're just about done. Note that Ek is the average KE per molecule so N · Ek would be the total KE of the gas.

 

[tiny-tim]

hi arddi2007!

Find the Kinetic Energy of the ideal gas $$\overline{Ek}$$=? if its volume is 10 liters (10^-2 m³) and it is under the pressure of p=5·10⁵.
…
p·V = N·k·T (where p-pressure, V-volume, N-number of particles (molecules), k- Boltzmann constant, T-absolute temperature (in Kelvins))
…
… Wikipedia it says that Ek=2/3 k·T·N. Can anyone confirm that and tell me where that equation was derived from?

i don't remember this stuff well,

but if p·V = N·k·T and Ek=2/3 k·T·N, doesn't that immediately give you Ek = 2/3 p·V ?

 

[arddi2007]

OK. Looks to me like you're just about done. Note that Ek is the average KE per molecule so N · Ek would be the total KE of the gas.

hi arddi2007!


i don't remember this stuff well,

but if p·V = N·k·T and Ek=2/3 k·T·N, doesn't that immediately give you Ek = 2/3 p·V ?

Thank you, I just needed the confirmation. So, all that is left to do is to substitute and from there we get:

Ek = 2/3 p·V

The part that I did not get was that Ek is the average KE per molecule, while the total KE of the gas is N·Ek. Thank you both very much!

 

[Doc Al]

Just an addendum to my last post:

This is the part when it gets tricky. I assumed that since Ek=2/3 k·T,

Must be a typo. Ek = 3/2 kT.

But it Wikipedia it says that Ek=2/3 k·T·N.

Really?

 

[Doc Al]

Thank you, I just needed the confirmation. So, all that is left to do is to substitute and from there we get:

Ek = 2/3 p·V

Careful with that.

The part that I did not get was that Ek is the average KE per molecule, while the total KE of the gas is N·Ek.

Right!

 

[arddi2007]

Thank you very much for the heads up. I normally wouldn't make that mistake on my notebook but it's sometimes hard to focus when typing it.

By the way, I checked with my professor and he confirmed the solution is correct. Thank you all so much!