Order the following transitions in the hydrogen atom from smallest to largest frequency of light absorbed: \(n=3\) to \(n=6, n=4\) to \(n=9, n=2\) to \(n=3,\) and \(n=1\) to \(n=2 .\)

For a hydrogen atom, the Rydberg formula relates the frequency of the absorbed light, \(v\), to the energy levels involved in the transition: \[v=R_H\left(\frac{1}{n_i^2}-\frac{1}{n_f^2}\right)\] where \(R_H\) is the Rydberg constant for hydrogen (\(R_H\approx 3.29\times10^{15}s^{-1}\)), \(n_i\) is the initial energy level ("lower" energy state), and \(n_f\) is the final energy level ("higher" energy state).

We will now apply the Rydberg formula to determine the frequency of the light absorbed for each of the given transitions: 1. \(n=3\) to \(n=6\): \(v_1=R_H\left(\frac{1}{3^2}-\frac{1}{6^2}\right)\) 2. \(n=4\) to \(n=9\): \(v_2=R_H\left(\frac{1}{4^2}-\frac{1}{9^2}\right)\) 3. \(n=2\) to \(n=3\): \(v_3=R_H\left(\frac{1}{2^2}-\frac{1}{3^2}\right)\) 4. \(n=1\) to \(n=2\): \(v_4=R_H\left(\frac{1}{1^2}-\frac{1}{2^2}\right)\) Now, we will compute the frequencies for each transition: 1. \(v_1=3.29\times10^{15}\left(\frac{1}{9}-\frac{1}{36}\right)\approx 2.47\times10^{15}s^{-1}\) 2. \(v_2=3.29\times10^{15}\left(\frac{1}{16}-\frac{1}{81}\right)\approx 1.91\times10^{15}s^{-1}\) 3. \(v_3=3.29\times10^{15}\left(\frac{1}{4}-\frac{1}{9}\right)\approx 4.95\times10^{15}s^{-1}\) 4. \(v_4=3.29\times10^{15}\left(\frac{1}{1}-\frac{1}{4}\right)\approx 8.23\times10^{15}s^{-1}\)

Finally, we can order the transitions from smallest to the largest based on the frequency of light absorbed: - Smallest frequency: \(n=4\) to \(n=9\), with \(v_2\approx 1.91\times10^{15}s^{-1}\) - Next smallest frequency: \(n=3\) to \(n=6\), with \(v_1\approx 2.47\times10^{15}s^{-1}\) - Next largest frequency: \(n=2\) to \(n=3\), with \(v_3\approx 4.95\times10^{15}s^{-1}\) - Largest frequency: \(n=1\) to \(n=2\), with \(v_4\approx 8.23\times10^{15}s^{-1}\) So, the order of transitions from smallest to largest frequency of light absorbed is: \[n=4 \to n=9, n=3 \to n=6, n=2 \to n=3, \text{ and } n=1 \to n=2 \]