Question

Imagine that the electron in a hydrogen atom is replaced by a muon (μ). The mass of muon particle is 207 times that of an electron and charge is equal to the charge of an electron. The ionization potential of this hydrogen atom will be :

• A. 13.6 eV
• B. 27.2 eV
• C. 331.2 eV
• D. 2815.2 eV

Hint:

Ionization energy depends directly on the reduced mass. Since the muon is much heavier, it orbits closer and is much harder to remove.

Answer 1

The Correct Option is D

The problem at hand involves determining the ionization potential of a hydrogen atom when its electron is replaced with a muon. The muon is a particle similar to an electron but with a significantly greater mass. Let's solve this step-by-step:

Background Concepts:

• The ionization potential of hydrogen, when it has an electron, is 13.6 eV.
• The ionization potential or energy level of a hydrogen-like atom is calculated using the formula: E = - \frac{m Z^2 e^4}{8 \varepsilon_0^2 h^2 n^2}, where:
  • m is the mass of the orbiting particle (electron or muon).
  • Z is the atomic number (for hydrogen, Z = 1).
  • e is the charge of the electron (same for the muon).
  • \varepsilon_0 is the permittivity of free space.
  • h is Planck's constant.
  • n is the principal quantum number (for the ground state, n = 1).

Steps to Solution:

1. When an electron in a hydrogen atom is replaced by a muon, the formula for ionization potential remains the same, but the muon mass is used instead of the electron mass.
2. The mass of the muon is given as 207 times the mass of the electron (m_\mu = 207 m_e).
3. Thus, the ionization energy formula for the muonic hydrogen atom becomes: E_{\mu} = - \frac{207 m_e e^4}{8 \varepsilon_0^2 h^2}.
4. Since E = 13.6 \text{ eV} for the electron-based atom, substituting the muon mass gives:
   E_{\mu} = 207 \times 13.6 \text{ eV}.
5. Calculate the new ionization potential:
   E_{\mu} = 207 \times 13.6 = 2815.2 \text{ eV}.
6. Therefore, the ionization potential of the hydrogen atom when the electron is replaced by a muon is 2815.2 eV.

Conclusion:

Thus, the correct answer is that the ionization potential of the hydrogen atom when an electron is replaced by a muon is 2815.2 eV.