91Ó°ÊÓ

To understand average energy at absolute zero, we first need to know about Fermi energy, which is the maximum energy of electrons in a metal at absolute zero. The average energy, denoted as \(E_{\text{av}}\), describes the mean energy that electrons possess when they are cooled to absolute zero. At this temperature, all electrons occupy the lowest energy states available to them. For sodium, this energy is calculated using the Fermi energy formula:
- \(E_{\text{av}} = \frac{3}{5} E_{\text{F}}\)
- Where \(E_{\text{F}}\) is given as \(3.23 \text{ eV}\) for sodium.
Plugging in the values, we find that \(E_{\text{av}} = \frac{3}{5} \times 3.23 \approx 1.938 \text{ eV}\). This is the energy an electron typically exhibits in this state.

Kinetic energy refers to the energy possessed by an object due to its motion. In physics, it can be expressed with the formula:
- \(\frac{1}{2} mv^2 = E_{\text{kinetic}}\)
Where \(m\) represents the mass and \(v\) is the velocity of the object, in this case, an electron. To find the speed of an electron possessing the average energy calculated earlier, we rearrange the formula to solve for \(v\) (speed):

• \(v = \sqrt{\frac{2E_{\text{av}}}{m}}\)

Given:

• Electron mass \(m = 9.1 \times 10^{-31} \text{ kg}\)
• Average energy \(E_{\text{av}} = 1.938 \text{ eV}\)

We must convert \( ext{eV}\) into \( ext{Joules}\) for calculation purposes (1 eV = 1.602 x \(10^{-19} \text{J}\)):
\(E_{\text{av}} \approx 1.938 \times 1.602 \times 10^{-19} \text{ J}\).
This results in the velocity \(v \approx 1.37 \times 10^6 \text{ m/s}\). This speed shows how electrons zip around within atoms even at absolute zero!

The Fermi temperature is a concept that compels us to equate the energy of the fastest moving electron to the kinetic energy of a classical gas molecule. It is conceptualized by relating the Fermi energy to temperature using Boltzmann's constant \(k\). The formula is straightforward:

• \(kT = E_{\text{F}}\)

Thus, temperature:

• \(T = \frac{E_{\text{F}}}{k}\)

Where Boltzmann's constant \(k = 1.38064852 \times 10^{-23} \text{ J/K}\).
Substituting values and converting \( ext{eV}\) to \( ext{J}\), we have:

• \(T = \frac{3.23 \times 1.602 \times 10^{-19}}{1.38064852 \times 10^{-23}} \approx 3.78 \times 10^4 \text{ K}\)

This high temperature indicates the energy levels at which quantum effects predominate over classical statistics in metals, underpinning many electronic properties.