For a system composed of N identical molecules, the...

For a system composed of $N$ identical molecules, the Boltzmann distribution $$ \frac{n_{i}}{N}=e^{-\varepsilon_{i} k T} $$ gives the average fraction of molecules in the molecular state $i$ with energy $\varepsilon_{r}$ (i) Show that the ratio $n_{i} / n_{j}$ of the populations of states $i$ and $j$ depends only on the difference in energy of the two states. (ii) What is the ratio for two states with the same energy (degenerate states)?

For a system composed of $N$ identical molecules, the Boltzmann distribution
$$
\frac{n_{i}}{N}=e^{-\varepsilon_{i} k T}
$$
gives the average fraction of molecules in the molecular state $i$ with energy $\varepsilon_{r}$
(i) Show that the ratio $n_{i} / n_{j}$ of the populations of states $i$ and $j$ depends only on the difference in energy of the two states. (ii) What is the ratio for two states with the same energy (degenerate states)?

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Key Concepts

Degenerate States

Degenerate states are those energy states that have identical energy values. In the context of the Boltzmann distribution, since degenerate states have the same exponential factor, their population ratio is unity. This means that, in thermal equilibrium, particles are equally likely to occupy any of the degenerate states.

Population Ratio

The concept of population ratio refers to comparing the number of particles in two different energy states. By taking the ratio of the Boltzmann factors for these states, it becomes apparent that the ratio depends only on the difference in energy between the states, not on their absolute energy values. This highlights how thermodynamic equilibrium makes the relative occupancy of states a function solely of the energy difference and the temperature.

Boltzmann Distribution

The Boltzmann distribution is a fundamental concept in statistical mechanics that provides the probability of a system being in a particular energy state at thermal equilibrium. It shows that the probability is proportional to the exponential of the negative energy of the state divided by the product of Boltzmann's constant and the temperature, indicating how lower energy states are more likely to be populated than higher energy ones.

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