A `mu-meson `("charge -e , mass = 207 m, where `m` is mass of electron") can be captured by a proton to form a hydrogen - like ''mesic'' atom. Calculate the radius of the first Bohr orbit , the binding energy and the wavelength of the line in the Lyman series for such an atom. The mass of the proton is `1836` times the mass of the electron. The radius of the first Bohr orbit and the binding energy of hydrogen are `0.529 Å` and `13.6 e V` , repectively. `Take `R = 1.67 xx 109678 cm^(-1)`

Explanation

To solve the problem of calculating the radius of the first Bohr orbit, the binding energy, and the wavelength of the line in the Lyman series for a mesic atom formed by a mu-meson and a proton, we will follow these steps:

### Step 1: Calculate the Reduced Mass
The reduced mass (μ) of the system can be calculated using the formula:
\[ \mu = \frac{m_1 m_2}{m_1 + m_2} \]
where \(m_1\) is the mass of the mu-meson and \(m_2\) is the mass of the proton.

Given:
- Mass of mu-meson = \(207m\) (where \(m\) is the mass of the electron)
- Mass of proton = \(1836m\)

Substituting the values:
\[ \mu = \frac{(207m)(1836m)}{207m + 1836m} = \frac{(207)(1836)m^2}{2043m} = \frac{380772m^2}{2043m} \approx 186m \]

### Step 2: Calculate the Radius of the First Bohr Orbit
The radius of the first Bohr orbit (r) can be calculated using the formula:
\[ r = r_0 \frac{m_e}{\mu} \]
where \(r_0\) is the radius of the first Bohr orbit for hydrogen, which is given as \(0.529 \, \text{Å}\).

Substituting the values:
\[ r = 0.529 \, \text{Å} \cdot \frac{m}{186m} = \frac{0.529}{186} \, \text{Å} \approx 0.00284 \, \text{Å} \]

### Step 3: Calculate the Binding Energy
The binding energy (E) can be calculated using the formula:
\[ E = -\frac{\mu e^4}{2 \hbar^2} R \]
where \(R\) is the Rydberg constant given as \(1.67 \times 10^9 \, \text{cm}^{-1}\).

Using the known binding energy of hydrogen (\(E_H = 13.6 \, \text{eV}\)):
\[ E = E_H \cdot \frac{\mu}{m_e} = 13.6 \, \text{eV} \cdot \frac{186m}{m} = 13.6 \times 186 \, \text{eV} \approx 2530.56 \, \text{eV} \]

### Step 4: Calculate the Wavelength of the Line in the Lyman Series
For the Lyman series, the wavelength (λ) can be calculated using the formula:
\[ \frac{1}{\lambda} = R \mu \left(1 - \frac{1}{n^2}\right) \]
For the first line in the Lyman series, \(n = 2\):
\[ \frac{1}{\lambda} = R \mu \left(1 - \frac{1}{2^2}\right) = R \mu \left(1 - \frac{1}{4}\right) = R \mu \cdot \frac{3}{4} \]

Substituting the values:
\[ \frac{1}{\lambda} = \left(1.67 \times 10^9 \, \text{cm}^{-1}\right) \cdot (186m) \cdot \frac{3}{4} \]
Calculating λ:
\[ \lambda = \frac{4}{3} \cdot \frac{1}{R \mu} \approx \frac{4}{3} \cdot \frac{1}{1.67 \times 10^9 \cdot 186} \approx 653.6 \, \text{Å} \]

### Final Answers
- **Radius of the first Bohr orbit**: \(0.00284 \, \text{Å}\)
- **Binding energy**: \(-2530.56 \, \text{eV}\)
- **Wavelength of the line in the Lyman series**: \(653.6 \, \text{Å}\)