Question

Calculate the relative populations of two quantum states separated by an energy of $40\times {10}^{-21}J$if the temperature is .Calculate the relative populations of two quantum states separated by an energy of if the temperature is ${25}^{°C}$ .

Short answer

The relative population of two quantum states is$5.97\times {10}^{-5}$

Step-by-step solution

Given information

The energy difference of the populations of two quantum and temperature given as:

$$\begin{gathered} hv=40\times {10}^{-21}JT \\ ={25}^{°}C \\ =298K \end{gathered}$$

Concept of Maxwell–Boltzmann distribution

The Maxwell–Boltzmann distribution is concerned with the energy distribution between identical but distinct particles. It denotes the likelihood of the distribution of states in a system with varying energies.

The so-called Maxwell distribution law of molecular velocities is a specific instance.

Calculate the relative probability

The relative probability of finding molecules in the excited state $\eta$ and in the ground state$\eta =\circ$ is given as shown below.

$\frac{P\left(n\right)}{P\left(0\right)}=\exp \left(\frac{-\left[{\epsilon }_n-{\epsilon }_0\right]}{k_{B}T}\right)$

Or in other words, $\frac{P\left(n\right)}{P\left(0\right)}=\exp \left(\frac{-h\nu }{k_{B}T}\right)$.

Where $k_B$is the Boltzmann constant $=1.38\times {10}^{-23}{\text{J}}^{-1}{\text{K}}^{-1}$.

Use the equation in the previous step can get $N_{\text{high}}/N_{\text{low}}$.

Then, use $n=1$

localid="1663402453106" $$\begin{gathered} \frac{P\left(1\right)}{P\left(0\right)}=\exp \left(\frac{-40\times {10}^{-21}}{1.38\times {10}^{-23}\times 298}\right) \\ \frac{P\left(1\right)}{P\left(0\right)}=5.97\times {10}^{-5} \end{gathered}$$

So, conclude that the relative population of two energy levels is $5.97\times {10}^{-5}$.

and the result is$\frac{N_{\text{high}}}{N_{low}}=5.97\times {10}^{-5}$.