When Z is doubled in an atom, which of the following statements are consistent with Bohr's theory?

Options

A. Energy of a state is doubled
B. Radius of an orbit is doubled
C. Velocity of electrons in an orbit is doubled
D. Radius of an orbit is halved

Answer

To solve the question regarding the effects of doubling the atomic number \( Z \) in an atom according to Bohr's theory, we will analyze the formulas for energy, radius, and velocity of the electron in the hydrogen-like atom.

### Step-by-Step Solution:

1. **Understanding the Energy Formula**:
The energy of the nth orbit for a hydrogen-like atom is given by:
\[ E_n = -\frac{13.6 Z^2}{n^2} \]
Here, \( Z \) is the atomic number and \( n \) is the principal quantum number. When \( Z \) is doubled (i.e., \( Z \rightarrow 2Z \)), the new energy becomes:
\[ E_n' = -\frac{13.6 (2Z)^2}{n^2} = -\frac{13.6 \cdot 4Z^2}{n^2} = 4 \cdot E_n \]
This shows that the energy does not double; it quadruples.

**Hint**: Remember that energy depends on the square of the atomic number \( Z \).

2. **Understanding the Radius Formula**:
The radius of the nth orbit is given by:
\[ r_n = \frac{n^2}{Z} \cdot a_0 \]
When \( Z \) is doubled, the new radius becomes:
\[ r_n' = \frac{n^2}{2Z} \cdot a_0 = \frac{1}{2} r_n \]
This indicates that the radius is halved when \( Z \) is doubled.

**Hint**: The radius is inversely proportional to the atomic number \( Z \).

3. **Understanding the Velocity Formula**:
The velocity of the electron in the nth orbit is given by:
\[ v_n = 2.18 \times 10^6 \cdot \frac{Z}{n} \]
When \( Z \) is doubled, the new velocity becomes:
\[ v_n' = 2.18 \times 10^6 \cdot \frac{2Z}{n} = 2 \cdot v_n \]
This shows that the velocity doubles when \( Z \) is doubled.

**Hint**: Velocity is directly proportional to the atomic number \( Z \).

### Conclusion:
Based on the analysis:

- **Energy**: Quadruples (not doubled)
- **Radius**: Halved (not doubled)
- **Velocity**: Doubled

Thus, the correct statements consistent with Bohr's theory when \( Z \) is doubled are:
- The radius of the orbit is halved.
- The velocity of the electron in the orbit is doubled.

**Final Answers**:
- The radius of the orbit is half.
- The velocity of the electron is double.