Confusion about the canonical partition function and probabilities

Question

I'm in a first course on statistical mechanics at the moment and I'm having trouble wrapping my head around an example problem involving the canonical partition function. The question setup has a three particle system with three different energy levels corresponding to $0$ J, $1000$ J, and $2000$ J. The microstates that arise from this have energies of $0$, $1000$, $\dots$, $6000$ J. We were then asked to graph the probabilities of being in a specific microstate using the formula:

$$p_i = \frac{e^{-\beta E_i}}{Q} = \frac{e^{-\beta E_i}}{\sum_{k=1}^{m} e^{-\beta E_k}}$$

In theory, this formula makes perfect sense to me. However, if we plug in $\beta = \frac{1}{k_BT}$ and take $T$ to be around $300$ K, we get that $\beta \approx 2.4 \times 10^{24}$ J$^{-1}$. This results in the Boltzmann factor term, $e^{-\beta E_i}$, to be extremely close to $0$ for each microstate except in the case where $E_i = 0$ J where the Boltzmann factor would be $1$. This results in a probability distribution of being almost entirely in the $0$ J microstate but this isn't the answer my professor gave so I'm wondering where exactly my logic goes awry? Thank you so much for your help!

Answer 1

As Vadim says, your beta is off by a few orders of magnitude. Nevertheless your reasoning is correct.

Think about how much energy 1kJ is! That's (very) roughly the energy in a whole mole (not a molecule, but $10^{23}$ of them) of gas at room temperature.

A temperature of $300\mathrm{K}$ corresponds to $0.025\mathrm{eV}$. That's the scale on which thermal excitations happen.