Step 1: Conceptual Framework:
This problem integrates hydrogen atom energy level transitions with angular momentum quantization based on the Bohr model. The objective is to determine the electron's final energy level post-absorption and subsequently calculate the alteration in its angular momentum.
Step 2: Relevant Formulas/Methodology:
1. Hydrogen atom electron energy in the \(n\)-th orbit: \(E_n = -\frac{13.6}{n^2}\) eV.
2. Electron angular momentum in the \(n\)-th orbit: \(L_n = n \frac{h}{2\pi} = n\hbar\).
3. Energy change: \(\Delta E = E_{final} - E_{initial}\).
4. Angular momentum change: \(\Delta L = L_{final} - L_{initial}\).
Step 3: Detailed Analysis:
Part 1: Determine the final energy level (\(n_f\)).
The electron begins in the ground state, meaning its initial principal quantum number is \(n_i = 1\).
The initial energy is \(E_1 = -\frac{13.6}{1^2} = -13.6\) eV.
The electron absorbs 12.09 eV.
The final energy is \(E_f = E_i + \Delta E = -13.6 \, \text{eV} + 12.09 \, \text{eV} = -1.51 \, \text{eV}\).
Using the energy formula, the final quantum number \(n_f\) is calculated:
\[ E_f = -\frac{13.6}{n_f^2} \]
\[ -1.51 = -\frac{13.6}{n_f^2} \]
\[ n_f^2 = \frac{13.6}{1.51} \approx 9 \]
\[ n_f = 3 \]
Consequently, the electron transitions from the n=1 state to the n=3 state.
Part 2: Compute the change in angular momentum (\(\Delta L\)).
Initial angular momentum (\(n_i = 1\)):
\[ L_i = n_i \frac{h}{2\pi} = 1 \cdot \frac{h}{2\pi} \]
Final angular momentum (\(n_f = 3\)):
\[ L_f = n_f \frac{h}{2\pi} = 3 \cdot \frac{h}{2\pi} \]
The increment in angular momentum is:
\[ \Delta L = L_f - L_i = 3\frac{h}{2\pi} - 1\frac{h}{2\pi} = 2\frac{h}{2\pi} \]
Step 4: Conclusion:
The electron's angular momentum increases by 2(h/2\(\pi\)).