A hydrogen atom and a `Li^(2+)` ion are both in the second excited state. If `l_H` and `l_(Li)` are their respective electronic angular momenta, and `E_H and E_(Li)` their respective energies, then
(a) `l_H gt l_(Li) and |E_H| gt |E_(Li)|`
(b) `l_H = l_(Li) and |E_H| lt |E_(Li)|`
(C ) `l_H = l_(Li) and |E_H| gt |E_(Li)|`
(d) `l_H lt l_(Li) and |E_H| lt|E_(Li)|`

Options

A. `l_(H)gtl_(Li)` and `|E_(H)|gt|E_(Li)|`
B. `l_(H)=l_(Li)` and `|E_(H)|gt|E_(Li)|`
C. `l_(H)=l_(Li)` and `|E_(H)|lt|E_(Li)|`
D. `l_(H)lt l_(Li)` and `|E_(H)|lt|E_(Li)|`

Explanation

To solve the problem, we need to analyze the angular momentum and energy of a hydrogen atom and a lithium ion (`Li^(2+)`) both in the second excited state.

### Step-by-Step Solution:

1. **Identify the Atomic Numbers**:
- For hydrogen (H), the atomic number \( Z_H = 1 \).
- For lithium ion (`Li^(2+)`), the atomic number \( Z_{Li} = 3 \).

2. **Determine the Principal Quantum Number (n)**:
- Both the hydrogen atom and the lithium ion are in the second excited state. The principal quantum number \( n \) for the second excited state is \( n = 3 \) for both atoms.

3. **Angular Momentum Calculation**:
- The angular momentum \( L \) of an electron in an atom is given by the formula:
\[ L = n \cdot \hbar \]
where \( \hbar \) is the reduced Planck's constant.
- Since both hydrogen and lithium are in the same excited state (n = 3), their angular momenta will be:
\[ L_H = L_{Li} \quad \text{(since both have } n = 3\text{)} \]
- Therefore, \( l_H = l_{Li} \).

4. **Energy Calculation**:
- The energy of an electron in a hydrogen-like atom is given by:
\[ E_n = -\frac{Z^2 \cdot 13.6 \, \text{eV}}{n^2} \]
- For hydrogen:
\[ E_H = -\frac{1^2 \cdot 13.6 \, \text{eV}}{3^2} = -\frac{13.6 \, \text{eV}}{9} \approx -1.51 \, \text{eV} \]
- For lithium ion (`Li^(2+)`):
\[ E_{Li} = -\frac{3^2 \cdot 13.6 \, \text{eV}}{3^2} = -\frac{9 \cdot 13.6 \, \text{eV}}{9} = -13.6 \, \text{eV} \]
- Comparing the magnitudes of the energies:
\[ |E_H| \approx 1.51 \, \text{eV} < |E_{Li}| = 13.6 \, \text{eV} \]

5. **Conclusion**:
- From the calculations, we find:
- Angular momentum: \( l_H = l_{Li} \)
- Energy: \( |E_H| < |E_{Li}| \)

Thus, the correct option is (b) \( l_H = l_{Li} \) and \( |E_H| < |E_{Li}| \).