In the question number 75, if ground state energy of electron is `-13.6eV` then what is the ground state energy of muonic hydrogen atom?

Options

A. `1.8 ke V`
B. `-2.8 ke V`
C. `-3.8 ke V`
D. `4.8 ke V`

Explanation

To find the ground state energy of a muonic hydrogen atom given that the ground state energy of an electron in hydrogen is -13.6 eV, we can use the relationship derived from Bohr's theory of the atom.

### Step-by-Step Solution:

1. **Understand the Relationship**: According to Bohr's theory, the energy of an atom is inversely proportional to the mass of the particle orbiting the nucleus. This means that if we have a different particle (like a muon instead of an electron), we can relate their energies using their masses.

2. **Set Up the Equation**: We can express the relationship as:
\[ \frac{E_{\mu}}{E_e} = \frac{m_{\mu}}{m_e} \]
where:
- \( E_{\mu} \) is the ground state energy of muonic hydrogen,
- \( E_e \) is the ground state energy of electron in hydrogen,
- \( m_{\mu} \) is the mass of the muon,
- \( m_e \) is the mass of the electron.

3. **Substitute Known Values**: We know that:
- \( E_e = -13.6 \, \text{eV} \)
- The mass of the muon is approximately 207 times the mass of the electron, so \( m_{\mu} = 207 \, m_e \).

4. **Plug Values into the Equation**:
\[ \frac{E_{\mu}}{-13.6} = \frac{207 \, m_e}{m_e} \]
The \( m_e \) cancels out:
\[ \frac{E_{\mu}}{-13.6} = 207 \]

5. **Solve for \( E_{\mu} \)**:
\[ E_{\mu} = 207 \times (-13.6) \]
\[ E_{\mu} = -2815.2 \, \text{eV} \]

6. **Convert to Kiloelectron Volts**:
\[ E_{\mu} = -2.8152 \, \text{keV} \quad \text{(since 1 keV = 1000 eV)} \]
We can round this to:
\[ E_{\mu} \approx -2.8 \, \text{keV} \]

### Final Answer:
The ground state energy of the muonic hydrogen atom is approximately \( -2.8 \, \text{keV} \).