Ionization energy of a hydrogen-like ion A is greater than that of another hydrogen like ion B . Let r, u, E and L represent the radius of the orbit , speed of the electron , total energy of the electron and angular momentum of the electron respectively (for the same n ). In ground state :

Options

A. `r_(A) gt r_(B)`
B. `u_(A) gt u_(B)`
C. `E_(A) lt E_(B)`
D. `L_(A) gt L_(B)`

Answer

To analyze the given problem about the hydrogen-like ions A and B, we need to understand the relationships between ionization energy, radius, speed, total energy, and angular momentum in the context of quantum mechanics.

### Step-by-Step Solution:

1. **Understanding Ionization Energy**:
- The ionization energy (IE) of a hydrogen-like ion is given by the formula:
\[ IE = 13.6 \, \text{eV} \times Z^2 \]
- Since it is stated that the ionization energy of ion A is greater than that of ion B, we can conclude:
\[ IE_A > IE_B \implies Z_A^2 > Z_B^2 \implies Z_A > Z_B \]
- Thus, the atomic number \( Z \) of ion A is greater than that of ion B.

2. **Angular Momentum (L)**:
- The angular momentum of an electron in a hydrogen-like atom is quantized and given by:
\[ L = n \frac{h}{2\pi} \]
- For both ions A and B in the ground state (n=1), the angular momentum will be the same:
\[ L_A = L_B \]
- Therefore, angular momentum does not depend on the atomic number \( Z \) for the same principal quantum number.

3. **Radius of the Orbit (r)**:
- The radius of the electron's orbit in a hydrogen-like atom is given by:
\[ r = \frac{n^2}{Z} \cdot a_0 \]
- Since \( Z_A > Z_B \), the radius for ion A will be less than that for ion B:
\[ r_A < r_B \]

4. **Speed of the Electron (u)**:
- The speed of the electron in a hydrogen-like atom is given by:
\[ u = \frac{Z e^2}{4 \pi \epsilon_0 h} \cdot \frac{1}{n} \]
- Since \( Z_A > Z_B \), the speed of the electron in ion A will be greater than that in ion B:
\[ u_A > u_B \]

5. **Total Energy (E)**:
- The total energy of the electron in a hydrogen-like atom is given by:
\[ E = -\frac{Z^2 \cdot 13.6 \, \text{eV}}{n^2} \]
- Since \( Z_A > Z_B \), the total energy for ion A will be more negative than that for ion B:
\[ E_A < E_B \]

### Summary of Relationships:
- **Angular Momentum**: \( L_A = L_B \)
- **Radius**: \( r_A < r_B \)
- **Speed**: \( u_A > u_B \)
- **Total Energy**: \( E_A < E_B \)