A system with nondegenerate energy levels has three energy...

A system with nondegenerate energy levels has three energy states: a ground state at $E=0$ and excited states at energies of $0.045 \mathrm{eV}$ and $0.135 \mathrm{eV}$. At a temperature of $650 \mathrm{~K},$ find the relative numbers of particles in the three states.

Key Concepts

Boltzmann Distribution

The Boltzmann distribution explains how particles are distributed among various energy states at thermal equilibrium. It states that the probability of finding a particle in a state with energy E is proportional to exp(–E/(kT)), where k is Boltzmann's constant and T is the temperature. This concept is central to determining relative populations in different energy levels.

Partition Function

The partition function is a normalization factor obtained by summing the Boltzmann factors of all possible states in the system. It ensures that the probabilities for all energy states add up to one. The partition function is crucial because it allows one to calculate the absolute numerical values for the populations of each energy state.

Energy Levels

Energy levels refer to the specific, discrete energies that a quantum system can have. In statistical mechanics, each energy level contributes differently to the overall behavior of the system, and the relative probability of each level depends on its energy as determined by the Boltzmann factor.

Temperature and Thermal Energy

Temperature is a measure of the average kinetic energy of the particles in a system and is directly related to the thermal energy available, given by kT. The ratio of the energy levels to kT determines how significantly different energy states are populated; at higher temperatures, higher energy states are more likely to be occupied.

Nondegeneracy

Nondegeneracy indicates that each energy level has a unique state (degeneracy of 1), simplifying calculations of statistical populations. In a nondegenerate system, the probability factors are solely given by the Boltzmann factors without any additional degeneracy factors.

Relative Population of States

The relative number of particles in each energy state is determined by the connection between the energy levels and temperature through the Boltzmann distribution. By comparing the Boltzmann factors for each state and normalizing with the partition function, one can compute the proportion of particles occupying each energy level.

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