Question

At what temperature is the ratio of the number of hydrogen atoms in the \(n=3\) state to the number of hydrogen atoms in the \(n=8\) state equal to \(5.1383 \cdot 10^{5} ?\)

Short answer

Answer: The temperature at which the ratio of hydrogen atoms in the states n=3 to n=8 is equal to \(5.1383 \cdot 10^{5}\) is approximately \(1.42 \cdot 10^{4}\) K.

Step-by-step solution

Write down the Boltzmann distribution formula

The Boltzmann distribution formula states that the ratio of the populations of two energy levels is proportional to the exponential of the negative difference in energy between the levels divided by the Boltzmann constant \(k_B\) and the temperature \(T\): $$\frac{N_2}{N_1} = \frac{g_2}{g_1} e^{-\frac{\Delta E}{k_B T}}$$ Where \(N_1\) and \(N_2\) are the populations of the two levels, \(g_1\) and \(g_2\) are the degeneracies of the levels, and \(\Delta E\) is the difference in energy between the levels.

Calculate the energy difference between the levels

We are given the principal quantum numbers of n=3 and n=8. We can use the formula for the energy of a hydrogen atom to find the energy difference between the two levels: $$\Delta E = E_3 - E_8 = -\frac{13.6 \,\text{eV}}{3^2} + \frac{13.6\, \text{eV}}{8^2} = -1.51\, \text{eV} $$

Calculate the degeneracy ratio

Since hydrogen has only one electron, the degeneracy of each level is given by the formula \(g_n = 2n^2\). We can calculate the degeneracy ratio for the \(n=3\) and \(n=8\) levels as follows: $$\frac{g_8}{g_3} = \frac{2(8^2)}{2(3^2)} = \frac{128}{18}$$

Substitute the given ratio and the calculated values into the Boltzmann distribution formula

We are given the ratio of the number of hydrogen atoms in the \(n=3\) state to the number of hydrogen atoms in the \(n=8\) state as \(5.1383 \cdot 10^{5}\). We can substitute this value along with the calculated values for \(\Delta E\) and the degeneracy ratio into the Boltzmann distribution formula: $$5.1383 \cdot 10^{5} = \frac{128}{18} e^{-\frac{-1.51\, \text{eV}}{k_B T}}$$

Solve for the temperature T

Now we can solve the equation for \(T\), remembering to convert the energy difference from electron volts to joules and using the Boltzmann constant in units of \(\text{J/K}\): $$ T = \frac{-1.51 \cdot 1.6 \cdot 10^{-19}\, \text{J}}{k_B \cdot \ln{\left(\frac{18}{128} \cdot 5.1383 \cdot 10^{5}\right)}} = \frac{-1.51 \cdot 1.6 \cdot 10^{-19}\, \text{J}}{1.38 \cdot 10^{-23} \,\text{J/K} \cdot \ln{\left(\frac{18}{128} \cdot 5.1383 \cdot 10^{5}\right)}} $$ After evaluating this expression, we obtain the temperature as: $$T \approx 1.42 \cdot 10^{4}\, \text{K}$$ Thus, the temperature at which the ratio of hydrogen atoms in the states \(n=3\) to \(n=8\) is equal to \(5.1383 \cdot 10^{5}\) is approximately \(1.42 \cdot 10^{4}\) K.