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

In summary, the conversation discusses the bulk modulus equation, 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. The equations used include p = (2U)/(3V), <E> = (3Ef)/5, U = N<E>, and B = -V(dp)/(dV). The conversation also mentions solving for V in the equation for Ef, V = (3*∏^2*N)*(2*Ef*m/(hbar)^2)^(3/2
  • #1
Jellybabe
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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.
 
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  • #2
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.
 

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

What is the bulk modulus of a Fermi gas?

The bulk modulus of a Fermi gas is a measure of its resistance to compression. It is a physical property that describes how much pressure is needed to decrease the volume of the gas by a certain amount.

How is the bulk modulus of a Fermi gas calculated?

The bulk modulus of a Fermi gas can be calculated using the equation K = n2EF/3V, where n is the number density of the gas, EF is the Fermi energy, and V is the volume of the gas.

What factors affect the bulk modulus of a Fermi gas?

The bulk modulus of a Fermi gas is affected by the number density of particles, the Fermi energy, and the volume of the gas. It also depends on the temperature and the mass of the particles in the gas.

Why is the bulk modulus of a Fermi gas important?

The bulk modulus of a Fermi gas is an important parameter in the study of condensed matter physics. It helps to understand the behavior of materials under pressure and is used to calculate other thermodynamic properties of the gas, such as its compressibility and thermal expansion coefficient.

Can the bulk modulus of a Fermi gas be changed?

Yes, the bulk modulus of a Fermi gas can be changed by altering the external pressure or temperature of the gas. It can also be affected by the addition of impurities or changing the number density of particles in the gas.

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