Show that, when \(N\) ideal-gas atoms come to equilibrium, and $$ \begin{aligned} &\frac{g_{i}}{N_{i}}=\frac{Z}{N} e^{c_{1} / k T} \\ &\frac{Z}{N}=\frac{(k T)^{5 / 2}}{P}\left(\frac{2 \pi m}{h^{2}}\right)^{3 / 2} \end{aligned} $$ Taking \(\epsilon_{i}=\frac{3}{2} k T, T=300 K, P=10^{3} \mathrm{~Pa}\), and \(m=10^{-26} \mathrm{~kg}_{1}\) calculate \(\boldsymbol{g}_{i} / N_{i} .\)

The partition function is a crucial concept in statistical mechanics and thermodynamics. It encodes all the possible states a system can occupy, allowing us to calculate various thermodynamic properties. For an ideal gas at equilibrium, the partition function per particle, denoted as \(\frac{Z}{N}\), is given by: \[ \frac{Z}{N} = \frac{(k T)^{5 / 2}}{P}\bigg(\frac{2 \pi m}{h^{2}}\bigg)^{3 / 2} \] Here, \(k\) is the Boltzmann constant, \(T\) is the temperature, \(P\) is the pressure, \(m\) is the molecular mass, and \(h\) is Planck's constant. This function helps in determining how particles distribute themselves in different energy states within the gas. As temperature and pressure vary, the partition function adjusts, reflecting changes in the distribution and energy of the particles.

The Boltzmann constant, symbolized as \(k\), is fundamental in the field of thermodynamics and statistical mechanics. It bridges the macroscopic and microscopic worlds by linking the average kinetic energy of particles in a gas to the temperature of the gas. Numerically, \(k = 1.38 \times 10^{-23} \text{J/K}\). This constant appears in the equation for the partition function: \[ \frac{Z}{N} = \frac{(k T)^{5 / 2}}{P}\bigg(\frac{2 \pi m}{h^{2}}\bigg)^{3 / 2} \] It ensures that the temperature, measured macroscopically, can be related to the energy and distribution of individual microscopic particles. For example, in our problem, \(k\) allows us to compute how energy states are populated at a given temperature.

Thermodynamic equilibrium is a state where all macroscopic properties of a system remain constant over time. For an ideal gas in equilibrium, the energy distribution among atoms doesn't change. Various properties like pressure and temperature are uniform throughout the system. Mathematically, we use equilibrium conditions to express ratios and relations of states. For instance, the partition function per particle helps calculate the fractional populations of energy states using the equation: \[ \frac{g_{i}}{N_{i}} = \frac{Z}{N} e^{c_{1} / k T} \] Achieving equilibrium means the probability of finding a system in any given state remains constant.

Pressure is the force exerted by particles colliding with the walls of their container, and it is measured in Pascals (Pa). In an ideal gas, pressure is related to temperature, volume, and the number of particles. For our specific problem, the pressure \(P\) is given as \(10^3 \text{Pa}\). Within the formula for the partition function per particle: \[ \frac{Z}{N} = \frac{(k T)^{5 / 2}}{P}\bigg(\frac{2 \pi m}{h^{2}}\bigg)^{3 / 2} \] Pressure inversely affects the partition function, meaning as pressure increases, the available energy states per particle decrease, holding temperature and mass constant. This relationship is key to understanding the behavior of a gas under different conditions.

Temperature measures the average kinetic energy of particles in a system and is fundamental to thermodynamic calculations. For an ideal gas, temperature directly influences the energy distribution among particles. Our example utilizes \(T = 300 \text{K}\). The partition function per particle is heavily temperature-dependent: \[ \frac{Z}{N} = \frac{(k T)^{5 / 2}}{P}\bigg(\frac{2 \pi m}{h^{2}}\bigg)^{3 / 2} \] Higher temperatures increase \( \frac{Z}{N} \), leading to more energy states being populated. This information is crucial for predicting and understanding how gas particles behave under various thermal conditions.

Molecular mass refers to the mass of a single molecule, typically measured in kilograms (kg). In our problem, it is given as \(10^{-26} \text{kg}\). The mass of gas particles directly influences how they spread across different energy levels. The partition function per particle includes the molecular mass: \[ \frac{Z}{N} = \frac{(k T)^{5 / 2}}{P}\bigg(\frac{2 \pi m}{h^{2}}\bigg)^{3 / 2} \] Lighter molecules have higher velocities at a given temperature compared to heavier ones, impacting their energy distribution. Understanding molecular mass helps explain the behavior and characteristics of different gases.