Question

A system consists of 110 particlesEach particle can have only two non-degenerate energy states $\epsilon_1$ and $\epsilon_2$ with $\epsilon_2 > \epsilon_1$The energy difference between these two states is 0.10 eVThe temperature at which only 10 particles of the system will be present in the energy state $\epsilon_2$ is _ _ _ K. (rounded off to two decimal places)

Hint:

For non-degenerate two-level systems, use $\frac{N_2}{N_1}=e^{-\Delta E/kT}$ directly

Answer 1

Correct Answer: 503.86

Step 1: Boltzmann ratio.
\[ \frac{N_2}{N_1} = e^{-\Delta E / kT} \] with 10 particles up and 100 down, so the ratio is 0.1.

Step 2: Rearrange for temperature.
\[ T = \frac{\Delta E}{k \ln 10} \]

Step 3: Put in values.
With $\Delta E = 1.6\times10^{-20}$ J, this gives about 503.86 K.

Step 4: Answer.
\[ \boxed{503.86 \text{ K}} \]