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I am a mathematics student doing a course in thermodynamics and am working on the basics now. The question I am asked is:

Suppose two states differ in energy by one electronvolt. What can be said about the ratio of the population at $T = \pu{300 K}$?

Now it is clear that I will use the formula:

$$\frac{N_i}{N_j} = \exp\left(-\frac{E_i-E_j}{kT}\right)$$

where $k = \pu{1.381 \times 10^−23 J K-1}$. Now if let $E_i$ be the higher energy state, then am I right in saying that $E_i-E_j = \pu{1 eV}$, or should I convert this to some other unit?

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Any quantity that is used as a power has to be dimensionless (that is, a pure number with no units). This means that $kT$ has to have the same units as $E_i-E_j$, so the terms in the exponent cancel. If you're measuring temperature in $\pu{K}$ and Boltzmann's constant in $\pu{J/K}$, then the units of energy you have to use are joules, $\pu{J}$.

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  • $\begingroup$ Alternatuvely, you can convert $k_B$ to using $\mathrm{eV \cdot K^{-1}}$. $\endgroup$ – The_Sympathizer May 9 at 17:39

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