What is the equation for the bulk modulus of a Fermi gas?

[Jellybabe]

Homework Statement

It is just a line of equation from my Stat Mech textbook, that says

B = -V(dp)/(dV) = (10U)/(9V) = (2nEf)/3

where B is the bulk modulus, V is the volume, p is the pressure, U is the energy, n is the number per unit volume and Ef is the fermi energy.

Homework Equations

p = (2U)/(3V)
<E> = (3Ef)/5
U = N<E>
B = -V(dp)/(dV)

The Attempt at a Solution

I have found that p = (2nEf)/5 = (2NEf)/(5V) and verified that this is correct, so I get that the (partial) derivative wrt V is: -(2NEf)/(5V^2), then multiplying this by -V to get B = (2nEf)/5, which is a factor of 5/3 out.

Alternatively going from p = (2U)/(3V) and taking the derivative wrt V gives: -(2U)/(3V^2), multiplying this by -V then gives B = p = (2U)/(3V).

I don't know what I'm missing or whether it's a typo in the book.

Also apologies for the lack of LaTex, I haven't used it before and wasn't sure exactly how the equations were going to turn out.

 

[Stormarov.45]

The energy at 0 K is U=3/5*N*Ef

Ef = ((hbar)^2/(2*m))*(3*∏^2*N/V)^(2/3), (hbar) is Planks constant in k space (h/2∏)
N=the number of orbitals electrons can occupy
m= mass of electron at rest
solve the equation Ef for V, (volume)

V=(3*∏^2*N)*(2*Ef*m/(hbar)^2)^(3/2)

plug U and V into the equation for the bulk modulous B=10U/9V and N, the term you don't know, drops out.