11.2 Show that for the ideal Fermi gas the Helmholtz free energy per particle at low temperatures is given by A 572 2 kT | ... 12
11.2 Show that for the ideal Fermi gas the Helmholtz free energy per particle at low temperatures is given by A 572 2 kT | ... 12
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![**11.2** Show that for the ideal Fermi gas the Helmholtz free energy per particle at low temperatures is given by
\[
\frac{A}{N} = \frac{3}{5} \epsilon_F \left[ 1 - \frac{5 \pi^2}{12} \left( \frac{kT}{\epsilon_F} \right)^2 + \cdots \right]
\]
Where:
- \( A \) is the Helmholtz free energy.
- \( N \) is the number of particles.
- \( \epsilon_F \) is the Fermi energy.
- \( k \) is the Boltzmann constant.
- \( T \) is the temperature.
This formula expresses the Helmholtz free energy per particle for an ideal Fermi gas at low temperatures, using an expansion in terms of a small parameter \(\left(\frac{kT}{\epsilon_F}\right)\). The terms following the ellipsis (\(\cdots\)) represent higher-order corrections that become significant at higher temperatures.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fddaece62-c748-41a0-80d0-fafdb1836a11%2F6d645f17-84ee-4100-8bb1-ee1b7925444f%2F0k8u3xi_processed.png&w=3840&q=75)
Transcribed Image Text:**11.2** Show that for the ideal Fermi gas the Helmholtz free energy per particle at low temperatures is given by
\[
\frac{A}{N} = \frac{3}{5} \epsilon_F \left[ 1 - \frac{5 \pi^2}{12} \left( \frac{kT}{\epsilon_F} \right)^2 + \cdots \right]
\]
Where:
- \( A \) is the Helmholtz free energy.
- \( N \) is the number of particles.
- \( \epsilon_F \) is the Fermi energy.
- \( k \) is the Boltzmann constant.
- \( T \) is the temperature.
This formula expresses the Helmholtz free energy per particle for an ideal Fermi gas at low temperatures, using an expansion in terms of a small parameter \(\left(\frac{kT}{\epsilon_F}\right)\). The terms following the ellipsis (\(\cdots\)) represent higher-order corrections that become significant at higher temperatures.
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