Which of the following electron transitions in hydrogen would absorb the largest amount of energy? A. \(n=2\) to \(n=3\) B. \(\quad n=2\) to \(n=4\) C. \(n=1\) to \(n=4\) D. \(n=3\) to \(n=1\) E. \(n=7\) to \(n=1\)

The formula for the energy (E) associated with a particular energy level (principal quantum number, n) in a hydrogen atom is given by: \( E = -\dfrac{13.6\, \mathrm{eV}}{n^2} \) When an electron in a hydrogen atom undergoes a transition between two energy levels (levels \(n_i\) and \(n_f\)), the energy absorbed or emitted is given by: \( \Delta E = E_{n_f} - E_{n_i} \) This formula can be applied to each of the transitions given in the exercise.

We will now calculate the energy absorbed for each of the given transitions using the formula mentioned earlier: A) \( \Delta E_A = E_3 - E_2 = -\dfrac{13.6\, \mathrm{eV}}{3^2} + \dfrac{13.6\, \mathrm{eV}}{2^2} \) B) \( \Delta E_B = E_4 - E_2 = -\dfrac{13.6\, \mathrm{eV}}{4^2} + \dfrac{13.6\, \mathrm{eV}}{2^2} \) C) \( \Delta E_C = E_4 - E_1 = -\dfrac{13.6\, \mathrm{eV}}{4^2} + \dfrac{13.6\, \mathrm{eV}}{1^2} \) D) \( \Delta E_D = E_1 - E_3 = -\dfrac{13.6\, \mathrm{eV}}{1^2} + \dfrac{13.6\, \mathrm{eV}}{3^2} \) E) \( \Delta E_E = E_1 - E_7 = -\dfrac{13.6\, \mathrm{eV}}{1^2} + \dfrac{13.6\, \mathrm{eV}}{7^2} \)

Now we will compare the values of the energy absorbed for each of the transitions: \( \Delta E_A \) = \( -\dfrac{13.6\, \mathrm{eV}}{9} + \dfrac{13.6\, \mathrm{eV}}{4} > 0 \) \( \Delta E_B \) = \( -\dfrac{13.6\, \mathrm{eV}}{16} + \dfrac{13.6\, \mathrm{eV}}{4} > 0 \) \( \Delta E_C \) = \( -\dfrac{13.6\, \mathrm{eV}}{16} + \dfrac{13.6\, \mathrm{eV}}{1} > 0 \) \( \Delta E_D \) = \( -\dfrac{13.6\, \mathrm{eV}}{1} + \dfrac{13.6\, \mathrm{eV}}{9} < 0 \) \\ \( \Delta E_E \) = \( -\dfrac{13.6\, \mathrm{eV}}{1} + \dfrac{13.6\, \mathrm{eV}}{49} < 0 \) The negative values for \(\Delta E_D\) and \(\Delta E_E\) indicate that the energy is getting emitted rather than absorbed from energy transitions D and E. Therefore, we need to consider only the energy absorbed values from transitions A, B, and C. After quick observation, we can see that the largest value is \(\Delta E_C\), as it has a larger denominator in the first term.

To identify the electron transition, we find which option corresponds to the largest energy calculation in the previous step. Hence, the electron transition in hydrogen that would absorb the largest amount of energy is C: \(n = 1\) to \(n = 4\).