Let's break down and solve the problem step by step: --- ### **Problem Restatement** - **Surface temperature of star:** \( T = 925\, \text{K} \) - **Hydrogen atoms:** Find the ratio of the number in the \( n = 3 \) excited state to the number in the \( n = 2 \) excited state. - **Assume Maxwell-Boltzmann statistics.** --- ## **Step 1: Maxwell-Boltzmann Population Ratio** The ratio of the number of atoms in two energy states \( i \) and \( j \) is given by: \[ \frac{N_i}{N_j} = \frac{g_i}{g_j} \exp\left(-\frac{E_i - E_j}{k_B T}\right) \] Where: - \( g_n = 2n^2 \) is the degeneracy of level \( n \) for hydrogen. - \( E_n = -13.6\,\text{eV}/n^2 \) is the energy of level \( n \). - \( k_B = 8.617 \times 10^{-5} \, \text{eV/K} \) is the Boltzmann constant. --- ## **Step 2: Calculate Degeneracies** - For \( n = 3 \): \( g_3 = 2 \times 3^2 = 18 \) - For \( n = 2 \): \( g_2 = 2 \times 2^2 = 8 \) --- ## **Step 3: Calculate Energy Levels** - \( E_3 = -13.6\,\text{eV}/9 = -1.511\,\text{eV} \) - \( E_2 = -13.6\,\text{eV}/4 = -3.400\,\text{eV} \) - \( E_3 - E_2 = (-1.511) - (-3.400) = 1.889\,\text{eV} \) --- ## **Step 4: Calculate Boltzmann Factor** \[ k_B T = (8.617 \times 10^{-5}\, \text{eV/K}) \times 925\,\text{K} = .797\,\text{eV} \] \[ \exp\left(-\frac{E_3 - E_2}{k_B T}\right) = \exp\left(-\frac{1.889}{.797}\right) = \exp(-2.37) \approx .093 \] --- ## **Step 5: Put It All Together** \[ \frac{N_3}{N_2} = \frac{g_3}{g_2} \exp\left(-\frac{E_3 - E_2}{k_B T}\right) \] \[ = \frac{18}{8} \times .093 = 2.25 \times .093 \approx .21 \] --- ## **Final Answer** \[ \boxed{.21} \] **So, the ratio of hydrogen atoms in the \( n = 3 \) excited state to those in the \( n = 2 \) excited state at a temperature of 925 K is approximately \( .21 \).**