If an orbital electron of the hydrogen atom jumps from the groud state to a higher energy state, its orbital value where its velcoity is reduced to half its initial value.. If the radius of the electron orbit in the ground state is `r`, then the radius of the new orbit would be:

Options

A. 2r
B. 4r
C. 8r
D. 16r

Answer

To solve the problem, we need to determine the new radius of the electron's orbit in a hydrogen atom after it jumps to a higher energy state where its velocity is reduced to half of its initial value.

### Step-by-Step Solution:

1. **Understand the Initial Conditions**:
- The electron is initially in the ground state of the hydrogen atom. Let's denote the radius of the electron's orbit in the ground state as \( r \).
- The initial velocity of the electron in the ground state is denoted as \( V \).

2. **Determine the New Velocity**:
- According to the problem, the new velocity \( V' \) after the electron jumps to a higher energy state is half of the initial velocity:
\[ V' = \frac{1}{2} V \]

3. **Relate Velocity to Principal Quantum Number (n)**:
- The velocity of an electron in a hydrogen atom is inversely proportional to the principal quantum number \( n \) of the orbit:
\[ V \propto \frac{1}{n} \]
- Therefore, if the new velocity \( V' \) is half of the initial velocity \( V \), we can set up the following relationship:
\[ \frac{V'}{V} = \frac{1}{2} \implies \frac{1}{n'} = \frac{1}{2} \cdot \frac{1}{n} \]
- This implies:
\[ n' = 2n \]
- For the ground state, \( n = 1 \), thus:
\[ n' = 2 \cdot 1 = 2 \]

4. **Relate Radius to Principal Quantum Number**:
- The radius of the electron's orbit is directly proportional to the square of the principal quantum number \( n \):
\[ r \propto n^2 \]
- Therefore, the radius of the new orbit \( r' \) can be expressed as:
\[ r' = k \cdot n'^2 \]
- Since \( r = k \cdot n^2 \), we can relate the two radii:
\[ r' = k \cdot (2)^2 = k \cdot 4 = 4r \]

5. **Conclusion**:
- The radius of the new orbit \( r' \) is four times the radius of the ground state:
\[ r' = 4r \]

### Final Answer:
The radius of the new orbit would be \( 4r \).