Q. 7.20 - An Introduction to Thermal Physics

Question

At the center of the sun, the temperature is approximately ${10}^7 \mathrm{K}$ and the concentration of electrons is approximately ${10}^{32}$ per cubic meter. Would it be (approximately) valid to treat these electrons as a "classical" ideal gas (using Boltzmann statistics), or as a degenerate Fermi gas (with $T\approx 0$ ), or neither?

Step-by-Step Solution

Answer

The gas cannot be treated as degenerate at $T=0$, and ordinary classical ideal gas at $T\gg T_F$ because the temperature T is not much less than or greater than $T_F$.

Step 1. Given Information

We are given that the temperature at the centre of the sun is approximately ${10}^7 \mathrm{K}$ and the concentration of electrons is approximately ${10}^{32}$ per cubic meter.

Step 2. Finding fermi temperature

Calculating the fermi temperature for the electron gas at the center of the Sun to check if the given statement is correct or not.

The Fermi temperature for the electron gas at the center of the Sun is given by,

$T_F=\frac{E_F}{k}$

The fermi energy of electrons can be expressed in terms of free electron density as follows,

${\epsilon }_F=\frac{h^2}{8m_e}{\left(\frac{3N}{\pi V}\right)}^{\frac{2}{3}}$

Step 3. Finding fermi temperature

Putting the values, we get

$$\begin{gathered} T_F=\frac{1}{k}\frac{h^2}{8m_e}{\left(\frac{3N}{\pi V}\right)}^{\frac{2}{3}} \\ {T}_F=\frac{{\left(6.63\times {10}^{-34} \mathrm{J}·\mathrm{s}\right)}^2}{\left(1.38\times {10}^{-23} \mathrm{J}/\mathrm{K}\right)\left(8\right)\left(9.1\times {10}^{-31} \mathrm{kg}\right)}{\left(\frac{3\left({10}^{32}{ \mathrm{m}}^{-3}\right)}{\pi }\right)}^{\frac{2}{3}} \\ =9.1\times {10}^6 \mathrm{K} \end{gathered}$$

The Fermi temperature is of the same order of magnitude as the temperature of the Sun $\left({10}^7 \mathrm{K}\right)$.

Step 4. About the Statement

Here, the temperature T is not much less than or greater than $T_F$. Hence, the approximation is not very accurate.

Due to this reason, the gas cannot be treated as degenerate at $T=0$, and ordinary classical ideal gas at $T\gg T_F$.