Consider a two-level system in which the excited state, separated from the ground state by energy ε, is doubly degenerate. The fraction of the molecules in the excited state, as T → ∞, is

Concept:

→ The partition function of a two-level system with doubly degenerate excited state is given by:

where k is the Boltzmann constant and T is the temperature.

The fraction of molecules in the excited state is given by:

Explanation:

→ At T → ∞, we have , which means that the denominator becomes much larger than the numerator. Therefore,

, due ton which exponential term will become 0.

When T → ∞

→ g_excited = 2. The ground state is non-degenerate, so g_total = 2 (the degeneracy of the excited state) + 1 (the degeneracy of the ground state) = 3.

Conclusion:
Therefore, as T → ∞, the fraction of molecules in the excited state approaches 2/3.