if muonic hydrogen atom is an atom in which a negatively charged muon `(mu)` of mass about `207m_e` revolves around a proton, then first bohr radius of this atom is `(r_e=0.53xx10^(-10)m)`

Options

A. `2.56xx10^(-10)m`
B. `2.56xx10^(-11)m`
C. `2.56xx10^(-12)m`
D. `2.56xx10^(-13)m`

Explanation

To find the first Bohr radius of a muonic hydrogen atom, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Bohr Model**:
In the Bohr model of the atom, the radius of the electron's orbit is given by the formula:
\[ r_n = \frac{n^2 h^2}{4 \pi^2 k e^2 m} \]
where \( n \) is the principal quantum number, \( h \) is Planck's constant, \( k \) is Coulomb's constant, \( e \) is the charge of the electron, and \( m \) is the mass of the electron.

2. **Identify the Given Values**:
- The first Bohr radius for a hydrogen atom (with an electron) is given as:
\[ r_e = 0.53 \times 10^{-10} \text{ m} \]
- The mass of the muon is approximately \( 207 m_e \), where \( m_e \) is the mass of the electron.

3. **Establish the Relationship**:
The radius of the orbit in the Bohr model is inversely proportional to the mass of the particle:
\[ r \propto \frac{1}{m} \]
Thus, we can write the relationship between the radii of the muonic hydrogen atom and the normal hydrogen atom as:
\[ r_m = r_e \cdot \frac{m_e}{m_\mu} \]
where \( r_m \) is the radius for the muonic hydrogen atom and \( m_\mu = 207 m_e \).

4. **Substitute the Values**:
Plugging in the values, we get:
\[ r_m = 0.53 \times 10^{-10} \text{ m} \cdot \frac{m_e}{207 m_e} \]
The \( m_e \) cancels out:
\[ r_m = 0.53 \times 10^{-10} \text{ m} \cdot \frac{1}{207} \]

5. **Calculate the Radius**:
Now perform the calculation:
\[ r_m = 0.53 \times 10^{-10} \text{ m} \cdot 0.004831 \approx 2.56 \times 10^{-13} \text{ m} \]

6. **Final Result**:
Therefore, the first Bohr radius of the muonic hydrogen atom is:
\[ r_m \approx 2.56 \times 10^{-13} \text{ m} \]