Partition Function for Equispaced Energy Levels

The molecular partition function ($Z$) is defined as the sum over all states ($i$) of the Boltzmann factor, $e^{-\beta E_i}$, where $\beta = 1/(k_B T)$ and $E_i$ is the energy of the state $i$.

For a system with equispaced energy levels, let the energy levels be $E_i = i\varepsilon$, starting from the ground state energy $E_0 = 0$. Here, $\varepsilon$ is the energy spacing between adjacent levels ($i = 0, 1, 2, \dots$).

The partition function is formulated as:

$Z = \sum_{i=0}^{\infty} e^{-\beta E_i}$

Substituting the energy levels:

$Z = \sum_{i=0}^{\infty} e^{-\beta (i\varepsilon)}$

This can be written as:

$Z = \sum_{i=0}^{\infty} (e^{-\beta \varepsilon})^i$

This is an infinite geometric series of the form $\sum_{i=0}^{\infty} r^i$, where the first term is $a = (e^{-\beta \varepsilon})^0 = 1$ and the common ratio is $r = e^{-\beta \varepsilon}$.

The sum of an infinite geometric series is given by $\frac{a}{1-r}$, provided $|r| < 1$. In this case, $a=1$ and $r = e^{-\beta \varepsilon}$. Since $\beta > 0$ and $\varepsilon > 0$, we have $0 < e^{-\beta \varepsilon} < 1$, satisfying the condition $|r| < 1$.

Therefore, the partition function is:

$Z = \frac{1}{1 - e^{-\beta \varepsilon}}$

This result matches the expression in Option 4.