For Na metal there are approximately 2.6 ×10^22 conduction electrons/ cm^3, which behave approxi

step 1 For this problem, we have a sodium which has a Fermi energy of 3 .23EV. Now, the question wants

For Na metal there are approximately 2.6 ×10^22 conduction...

For Na metal there are approximately $2.6 \times 10^{22}$ conduction electrons/ $\mathrm{cm}^3$, which behave approximately as a free electron gas. From these facts, (a) give an approximate value (in eV ) of the Fermi energy in Na , (b) give an approximate value for the electronic specific heat of Na at room temperature.

For Na metal there are approximately $2.6 \times 10^{22}$ conduction electrons/ $\mathrm{cm}^3$, which behave approximately as a free electron gas. From these facts,
(a) give an approximate value (in eV ) of the Fermi energy in Na ,
(b) give an approximate value for the electronic specific heat of Na at room temperature.

Key Concepts

Fermi Energy

The Fermi energy is defined as the highest occupied energy level of a system of fermions at absolute zero temperature. In the context of the free electron model, it represents the energy level up to which electron states are filled, and it can be determined from the electron density. The Fermi energy is a crucial parameter because it sets the energy scale of the electronic properties of a metal, influencing factors such as electrical conductivity and heat capacity.

Electronic Specific Heat

The electronic specific heat of a metal describes the contribution of conduction electrons to the overall heat capacity. At low temperatures, only electrons within an energy kT of the Fermi level can be thermally excited, causing the electronic specific heat to be much smaller than what classical equipartition theory would predict. This dependence on the density of states at the Fermi energy, coupled with the quantum statistical behavior of electrons, results in a specific heat that varies linearly with temperature for a free electron gas.

Free Electron Model

The free electron model treats conduction electrons in a metal as free particles that occupy a ‘gas’ of energy states, neglecting electron-electron interactions and the periodic potential of the ion cores. This simplification is particularly useful for explaining many of the electronic properties of simple metals, such as electrical conductivity and thermal properties, by allowing the use of quantum statistics to predict the behavior of electrons under various conditions.

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