Which of the following parameters are the same for all hydrogen like atoms and ions in their ground state?

Options

A. Radius of the orbit
B. Speed of the electron
C. Energy of the atom
D. Orbital angular momentum of the electron

Answer

To determine which parameters are the same for all hydrogen-like atoms and ions in their ground state, we will analyze the given options: radius of the orbit, speed of the electron, energy of the atom, and orbital angular momentum of the electron.

### Step-by-Step Solution:

1. **Radius of the Orbit (r)**:
- The formula for the radius of the orbit in a hydrogen-like atom is given by:
\[ r = r_0 \frac{n^2}{Z} \]
where \( r_0 \) is a constant, \( n \) is the principal quantum number, and \( Z \) is the atomic number.
- For the ground state, \( n = 1 \). Therefore, the radius depends on \( Z \):
\[ r = \frac{r_0}{Z} \]
- Since \( Z \) varies for different hydrogen-like atoms, the radius is not the same for all hydrogen-like atoms.

2. **Speed of the Electron (v)**:
- The speed of the electron in a hydrogen-like atom is given by:
\[ v = v_0 \frac{Z}{n} \]
where \( v_0 \) is a constant.
- For the ground state, \( n = 1 \). Thus, the speed is:
\[ v = v_0 Z \]
- Since \( Z \) varies, the speed of the electron is also not the same for all hydrogen-like atoms.

3. **Energy of the Atom (E)**:
- The energy of the atom in a hydrogen-like atom is given by:
\[ E = -13.6 \, \text{eV} \frac{Z^2}{n^2} \]
- For the ground state, \( n = 1 \):
\[ E = -13.6 \, \text{eV} Z^2 \]
- Since \( Z \) varies, the energy is not the same for all hydrogen-like atoms.

4. **Orbital Angular Momentum (L)**:
- The orbital angular momentum of the electron is given by:
\[ L = n \frac{h}{2\pi} \]
- For the ground state, \( n = 1 \):
\[ L = \frac{h}{2\pi} \]
- This value is constant and does not depend on \( Z \). Therefore, the orbital angular momentum is the same for all hydrogen-like atoms in their ground state.

### Conclusion:
The only parameter that is the same for all hydrogen-like atoms and ions in their ground state is the **orbital angular momentum of the electron**.

### Final Answer:
**Orbital Angular Momentum of the Electron**

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