# Confusion about the canonical partition function and probabilities

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!

• I get a different value for $\beta$ - still big, but not as big as yours. Nov 17, 2020 at 8:24

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.