Regarding quantum physics(fermi energy)

[lotrsimp12345]

The average electron energy in an electron gas is known to be <e>=3E(f)/5. The energy for N such electrons would be 3NE(f)/5. Find the pressure in terms of U and V, where V is the volume of the electron gas.

My solution.

Fermi energy
E(f)=(h^2/2m)*((3N/8*pie*V)^(2/3))
therefore E(F) is proportional to V^(-2/3)
since it is proportional i just have to plug into 3NE(f)/5 or the other one correct?

Thanks.




so which formula do i plug V^(-2/3) into E(F)? The 3NE(f)/5 or 3E(f)/5.

 

[nickjer]

Hah, you have me confused. Let's write down what you have:

U=(3/5)NE_F
and
E_F = C V^{-2/3}

where I stuck all those nasty constants that don't depend on V in the term C.

So, you also know P=-dU/dV.

You have all the pieces, just put them together now.

 

[lotrsimp12345]

but the pressure needs to be in terms of U and V.

 

[nickjer]

First use substitution to remove the Fermi energy from the first equation. Then take the derivative with respect to V. And substitute back in U to get the pressure in terms of U and V.

 

[lotrsimp12345]

thanks.

 

[lotrsimp12345]

after i take derivative i have p=-dU/dV=(6/15)NC*(V^(-5/3))
you can't substitue this into U because it isn't=U to derivative of du/dv.

 

[nickjer]

First, 6/15 can be simplified. Second, you can pull a U out of that final equation of yours. Just remember what U is equal to, and factor that out of the equation above, and then make it U.

 

[lotrsimp12345]

Can i do this U=3/5*N*C*(V^(-2/3)). so solve for N and plug into p=-dU/dV=(6/15)NC*(V^(-5/3))? don't really understand what you mean by pull out U from -du/dv.

 

[nickjer]

You can do that as well, or even solve for C and plug it in. All the constants will cancel out in the end and you will be left with U and V for your pressure equation.

 

[lotrsimp12345]

thank you so much.

 

[lotrsimp12345]

now he asks to find bulk modulus. B=-V*(dp/dv) but he wants it in terms on U only and if i take derivative and simplify i am left with 2U/3V again but am unsure of how to get further in terms of plugging in for V.

 

[nickjer]

You need to show some work. And remember that U also depends on V.

 

[lotrsimp12345]

so use U=(3/5*N*C*V^(-2/3)) solve for V and substitute?

 

[lotrsimp12345]

here's what i did B=-V(dp/dv)=-V((d/dV(2U/3V))=2U/3V. But i need my answer in terms of of only U.

 

[nickjer]

You didn't take the derivative of U with respect to V in that formula. But that formula may not be the best to take the derivative of. Try a formula you quoted earlier...

p=-dU/dV=(6/15)NC*(V^(-5/3))?

Take the derivative of that pressure with respect to volume, then substitute the U back in.

Edit: Nevermind, I see your problem. What does the problem ask for exactly?

 

[lotrsimp12345]

Problem:
The bulk modulus, B, is defined to be B=-V(dp/dv). Find expression of B in terms of U.

 

[nickjer]

Well, I get a similar answer as with the U/V, except I get a different coefficient than you. But I still also get the volume in my bulk modulus. So not sure how to express it only as U.

 

[lotrsimp12345]

thanks for help i will just ask teacher tom.