Question

In a perfect gas of electrons, the mean number of particles occupying a single-particle quantum state of energy $E_i$ is: $$ N_i=\frac{1}{\exp \left[\left(E_i-\mu\right) / k T\right]+1} . $$ (a) Obtain a formula which could be used to determine $\mu$ in terms of the particle density $n$ and various constants. (b) Show that the expression above reduces to the Maxwell-Boltzmann distribution in the limit $n \lambda^3 \ll 1$, where $\lambda$ is the thermal de Broglie wavelength. (c) Sketch $N_i$ versus $E_i$ for $T=0 \mathrm{~K}$ and for $T=\mu / 5 \mathrm{~K}$. Label significant points along both axes.

   In a perfect gas of electrons, the mean number of particles occupying a single-particle quantum state of energy $E_i$ is:
$$
N_i=\frac{1}{\exp \left[\left(E_i-\mu\right) / k T\right]+1} .
$$
(a) Obtain a formula which could be used to determine $\mu$ in terms of the particle density $n$ and various constants.
(b) Show that the expression above reduces to the Maxwell-Boltzmann distribution in the limit $n \lambda^3 \ll 1$, where $\lambda$ is the thermal de Broglie wavelength.
(c) Sketch $N_i$ versus $E_i$ for $T=0 \mathrm{~K}$ and for $T=\mu / 5 \mathrm{~K}$. Label significant points along both axes.

Problems and Solutions on Thermodynamics and Statistical Mechanics

U.S.T. of China… 1st Edition

Chapter 2, Problem 86

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Step 1

\] We need to find the total number of particles $N$ in the system, which can be expressed as the sum over all single-particle states: \[ N = \sum_i N_i = \sum_i \frac{1}{\exp \left[\left(E_i - \mu\right) / k T\right] + 1}. \] In the thermodynamic limit, we can Show more…

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In a perfect gas of electrons, the mean number of particles occupying a single-particle quantum state of energy $E_i$ is: $$ N_i=\frac{1}{\exp \left[\left(E_i-\mu\right) / k T\right]+1} . $$ (a) Obtain a formula which could be used to determine $\mu$ in terms of the particle density $n$ and various constants. (b) Show that the expression above reduces to the Maxwell-Boltzmann distribution in the limit $n \lambda^3 \ll 1$, where $\lambda$ is the thermal de Broglie wavelength. (c) Sketch $N_i$ versus $E_i$ for $T=0 \mathrm{~K}$ and for $T=\mu / 5 \mathrm{~K}$. Label significant points along both axes.

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