A muan is an unstable elementary partical whose mass `(mu^(-))` can be captured by a hydrogen nucleus (or proton) to from a muonic atom.
a Find the redius of the first Bohr orbit of this atom.
b Find the ionization energy of the atom.

Explanation

To solve the problem of finding the radius of the first Bohr orbit of a muonic atom and its ionization energy, we will follow these steps:

### Part a: Finding the Radius of the First Bohr Orbit

1. **Identify the Masses**:
- The mass of the muon (\(m_\mu\)) is approximately \(207\) times the mass of the electron (\(m_e\)).
- The mass of the proton (\(m_p\)) is approximately \(1836\) times the mass of the electron (\(m_e\)).

\[ m_\mu = 207 m_e \]
\[ m_p = 1836 m_e \]

2. **Calculate the Reduced Mass**:
The reduced mass (\(\mu\)) of the system (muon and proton) is given by the formula:
\[ \mu = \frac{m_\mu m_p}{m_\mu + m_p} \]
Substituting the values:
\[ \mu = \frac{(207 m_e)(1836 m_e)}{207 m_e + 1836 m_e} = \frac{207 \times 1836}{207 + 1836} m_e \]
\[ \mu = \frac{380772}{2043} m_e \approx 186 m_e \]

3. **Use the Bohr Model to Find the Radius**:
The radius of the first orbit (\(r_1\)) in a hydrogen-like atom is given by:
\[ r_n = \frac{n^2 h^2 \epsilon_0}{\pi m e^2} \]
For the first orbit (\(n=1\)):
\[ r_1 = \frac{h^2 \epsilon_0}{\pi m_e e^2} \]
For the muonic atom, the radius is scaled by the ratio of the electron mass to the reduced mass:
\[ r' = \frac{m_e}{\mu} r_1 \]
Substituting the values:
\[ r' = \frac{m_e}{186 m_e} \times 5.29 \times 10^{-11} \text{ m} \approx \frac{1}{186} \times 5.29 \times 10^{-11} \text{ m} \approx 2.85 \times 10^{-13} \text{ m} \]

### Part b: Finding the Ionization Energy

1. **Use the Energy Formula**:
The energy of the first orbit in a hydrogen-like atom is given by:
\[ E_n = -\frac{13.6 \, \text{eV}}{n^2} \]
For the first orbit (\(n=1\)):
\[ E_1 = -13.6 \, \text{eV} \]

2. **Calculate the Ionization Energy for the Muonic Atom**:
The energy for the muonic atom is scaled by the ratio of the reduced mass to the electron mass:
\[ E'_1 = \frac{\mu}{m_e} E_1 \]
Substituting the values:
\[ E'_1 = \frac{186 m_e}{m_e} \times (-13.6 \, \text{eV}) = -186 \times 13.6 \, \text{eV} \approx -2.53 \, \text{eV} \]

### Final Answers:
- **Radius of the first Bohr orbit of the muonic atom**: \(r' \approx 2.85 \times 10^{-13} \text{ m}\)
- **Ionization energy of the muonic atom**: \(E' \approx 2.53 \, \text{eV}\)