# Total energy of free electron gas

• RingNebula57
In summary, the problem says that experimentally it has been shown that the specific heat of the conduction electrons at constant volume in metals depends on temperature, and the experimental value at room temperature is about two orders of magnitude lower than its classical counterpart.
RingNebula57
Hello everyone! I encountered a problem about the specific heat of electronic gas and I do not understand a formula... so the problem says that experimentally it has been shown that the specific heat of the conduction electrons at constant volume in metals depends on temperature, and the experimental
value at room temperature is about two orders of magnitude lower than its classical
counterpart. This is because the electrons obey the quantum statistics rather than
classical statistics. According to the quantum theory, for a metallic material the
density of states of conduction electrons (the number of electronic states per unit
volume and per unit energy) is proportional to the square root of electron energy ,
then the number of states within energy range for a metal of volume V can be
written as:
dS = E^(1/2)* C* V *dE
where C is the normalization constant, determined by the total number of electrons of
the system.
The probability that the state of energy E is occupied by electron is f(E) (it is an exponential formula, not really relevant for what I am about to ask), f(E)is called Fermi distribution function.

Now, my question is the following:

Why is the total energy of the system:
U = ∫ E * f(E) * dS ?

I am new into probability , but I have studied a little and didn't find exactly how to derive the formula. I thought at first that the total energy of the electrons is U = ∫ E * dN ( where dN is the number of electrons with energy E). After that I tried to equal dN with f(E)*dS, so I said that the probaility to find an electron within the energetic range E and E+dE is dP = f(E)*dE and this led to dN = dS/dE * dP. So the number of electrons is the probability times the density of states?
I am not convinced that my answer is corect because after the energy formula arrives another formula which says that the total number of electrons is N = C *V * ∫ E^(1/2) * dE = ∫ dS.
So what is the logic behind these formulas?

RingNebula57 said:
Why is the total energy of the system:
U = ∫ E * f(E) * dS ?

It is just adding up the energy that each particle contributes.
This may make more sense to you:
http://www.qudev.ethz.ch/phys4/PHYS4_lecture14v1_2page.pdf

Or you can try:
http://folk.ntnu.no/ioverbo/TFY4250/til8eng.pdf
... p10 onwards.

RingNebula57

## What is the total energy of a free electron gas?

The total energy of a free electron gas is the sum of the kinetic energy and potential energy of all the electrons in the system. It is a measure of the total energy that is available for electrons to move and interact within the gas.

## How is the total energy of a free electron gas calculated?

The total energy of a free electron gas can be calculated using the Fermi-Dirac distribution, which takes into account the number of electrons, temperature, and energy states available to the electrons. This distribution is based on quantum mechanics and provides an accurate description of the behavior of electrons in a gas.

## What factors affect the total energy of a free electron gas?

The total energy of a free electron gas is affected by the number of electrons in the system, the temperature, and the energy states available to the electrons. It may also be influenced by external factors such as electric and magnetic fields.

## What is the significance of the total energy of a free electron gas?

The total energy of a free electron gas is an important quantity in understanding the behavior of electrons in materials. It can provide insight into the electrical and thermal properties of a material, as well as its ability to conduct electricity.

## How does the total energy of a free electron gas relate to the electronic structure of a material?

The total energy of a free electron gas is closely related to the electronic structure of a material. The distribution of electrons in different energy states contributes to the total energy and can affect the material's properties such as conductivity and magnetism.

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