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 double
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 relationships between \( Z \), energy, radius, and velocity of the electron in the orbit.

### Step-by-Step Solution:

1. **Understanding the Energy Relation**:
According to Bohr's theory, the energy \( E \) of an electron in a hydrogen-like atom is given by the formula:
\[ E \propto -Z^2 \]
If \( Z \) is doubled (i.e., \( Z' = 2Z \)), then:
\[ E' \propto -(2Z)^2 = -4Z^2 \]
This indicates that the energy becomes four times more negative (or four times the magnitude) compared to the original energy. Therefore, the energy does not double; it quadruples.

**Hint**: Remember that energy is proportional to the square of the atomic number \( Z \).

2. **Understanding the Radius Relation**:
The radius \( r \) of the electron's orbit is given by:
\[ r \propto \frac{1}{Z} \]
If \( Z \) is doubled, then:
\[ r' \propto \frac{1}{2Z} = \frac{1}{2}r \]
This means the radius becomes half of its original value.

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

3. **Understanding the Velocity Relation**:
The velocity \( v \) of the electron in the orbit is given by:
\[ v \propto Z \]
If \( Z \) is doubled, then:
\[ v' \propto 2Z \]
This indicates that the velocity doubles.

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

### Summary of Findings:
- Energy does not double; it quadruples.
- The radius of the orbit becomes half.
- The velocity of the electron doubles.

### Conclusion:
From the analysis:
- The correct statements consistent with Bohr's theory when \( Z \) is doubled are:
- The radius of the orbit is half.
- The velocity of the electron is doubled.

Thus, the correct options are:
- **C**: The radius of an orbit is half.
- **D**: The velocity is doubled.