In a Bohr atom the electron is replaced by a particle of mass 150 times the mass of the electron and the same charge . If `a_(0)` is the radius of the first Bohr orbit of the orbital atom, then that of the new atom will be

Options

A. `150a_(0)`
B. `sqrt(150)a_(0)`
C. `(a_(0))/(sqrt(150))`
D. `(a_(0))/(150)`

Answer

To solve the problem, we need to find the radius of the first Bohr orbit for a new atom where the electron is replaced by a particle of mass 150 times that of the electron, while keeping the charge the same.

### Step-by-Step Solution:

1. **Understand the Formula for the Radius of the Bohr Orbit**:
The radius of the nth Bohr orbit is given by the formula:
\[ r_n = \frac{h^2}{4 \pi^2 m e^2} n^2 \]
where:
- \( h \) is Planck's constant,
- \( m \) is the mass of the electron,
- \( e \) is the charge of the electron,
- \( n \) is the principal quantum number (for the first orbit, \( n = 1 \)).

2. **Identify the Radius of the First Bohr Orbit**:
For the first Bohr orbit (\( n = 1 \)), the radius \( r_1 \) can be denoted as \( a_0 \):
\[ a_0 = \frac{h^2}{4 \pi^2 m e^2} \]

3. **Determine the Mass of the New Particle**:
The new particle has a mass \( m' = 150m \) (150 times the mass of the electron).

4. **Calculate the Radius of the New Atom**:
Using the formula for the radius of the nth Bohr orbit, for the new particle we have:
\[ r' = \frac{h^2}{4 \pi^2 (150m) e^2} n^2 \]
For the first orbit (\( n = 1 \)):
\[ r' = \frac{h^2}{4 \pi^2 (150m) e^2} \]

5. **Relate the New Radius to the Original Radius**:
We can express \( r' \) in terms of \( a_0 \):
\[ r' = \frac{1}{150} \cdot \frac{h^2}{4 \pi^2 m e^2} = \frac{a_0}{150} \]

6. **Conclusion**:
Therefore, the radius of the new atom will be:
\[ r' = \frac{a_0}{150} \]

### Final Answer:
The radius of the new atom will be \( \frac{a_0}{150} \).

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