One mole of a dilute gas of \(\mathrm{He}^{3}\) atoms (which are spin- \(1 / 2\) fermions) at temperature \(T\) \(=140 \mathrm{~K}\) is contained in a box of volume \(\mathrm{V}=1.0 \mathrm{~cm}^{3}\). (a) Compute the lowest order (in density) correction to the classical ideal gas pressure. (b) What fraction of the total pressure is due to the Fermi statistics of the atoms?

Fermi-Dirac statistics help us understand the behavior of particles that follow the Pauli exclusion principle. Fermions, such as electrons, protons, and neutrons (including our helium-3 atoms), cannot occupy the same quantum state simultaneously. This principle significantly affects their distribution and properties, especially at low temperatures. The occupation probability of fermions at a given energy level is determined by the Fermi-Dirac distribution function: \( f(E) = \frac{1}{e^{(E-\text{{\mu}})/k_B T} + 1} \). Here, \(E\) is the energy, \(\mu\) is the chemical potential, \(k_B\) is Boltzmann's constant, and \(T\) is the temperature. These statistics are essential in calculating corrections to classical models, as they account for quantum effects that become significant in densely packed or low-temperature conditions.

When dealing with fermions, classical mechanics alone isn't sufficient. Quantum corrections become necessary to account for the inherent properties of particles on a quantum scale. In this particular exercise, we see that for a fermion gas ---- like helium-3 --- quantum corrections lead to a modification in pressure calculations. The quantum correction to the pressure formula, which appears as: \( P_{\text{quantum}} = - \frac{3}{4} \frac{n^2 \bar{h}^2}{m T} \), reflects the significant role of reduced distances and quantum effects. Here, \(\bar{h} = \frac{h}{2\pi}\) is the reduced Planck constant, \(n\) is the density, \(m\) is the particle mass, and \(T\) is the temperature. These corrections demonstrate the deviation from classical ideal gas behavior and decrease the overall pressure because of the exclusion principle.

The ideal gas law is a fundamental equation relating pressure, volume, number of moles, and temperature of an ideal gas. The law is expressed as \( PV = nRT \), where \(P\) is pressure, \(V\) is volume, \(n\) is the number of moles, \(R\) is the gas constant, and \(T\) is the temperature. For a specific number of particles, like in our example with helium-3, it can also be written as \( P = \frac{n k_B T}{N_A} \). This equation assumes that particles do not interact with one another and that they occupy a negligible volume. The law is derived under classical assumptions, making it an estimator rather than an absolute measure for real gases, especially at low temperatures or high densities, necessitating quantum corrections for more accurate results.

Density is a crucial factor in determining the properties of any gas. In this exercise, the density \(n\) of the helium-3 gas is calculated using Avogadro's number and the given volume: \( n = \frac{N_A}{V} \). Given the values \(N_A = 6.022 \times 10^{23}\) mol\(^{-1}\) and \(V = 1.0 \) cm\(^3\), converted to \(1 \times 10^{-6}\) m\(^3\), the density calculation becomes straightforward: \( n = \frac{6.022 \times 10^{23}}{1 \times 10^{-6}} = 6.022 \times 10^{29} \) m\(^{-3}\). Understanding how to calculate density is fundamental, as it feeds into further calculations involving pressure, energy levels, and various corrections. It becomes even more significant in scenarios involving quantum effects where particle density plays a pivotal role in determining the properties of the system.