Let's solve the problem step by step: --- ### **Step 1: Understand the Balmer Series for Hydrogen-Like Atoms** For a hydrogen atom, the wavelength for the Balmer series (transitions to \( n=2 \)) is given by: \[ \frac{1}{\lambda} = R_H \left( \frac{1}{2^2} - \frac{1}{n^2} \right) \] For the second line, \( n = 4 \rightarrow 2 \). #### **Explanation:** This formula gives the wavelength for the transition in hydrogen, where \( R_H \) is the Rydberg constant for hydrogen. --- ### **Step 2: Adjust for the Muonic Hydrogen Atom** For a muonic hydrogen atom, the Rydberg constant is modified because the muon is much heavier than the electron. The Rydberg constant depends on the reduced mass (\( \mu \)): \[ R = R_H \frac{\mu}{m_e} \] For muonic hydrogen, \( \mu \approx m_\mu \) (since \( m_\mu \gg m_p \)), and \( m_\mu = 207 m_e \). #### **Explanation:** The energy levels (and hence the wavelengths) scale with the reduced mass, so for a muon (mass 207 times that of electron), the Rydberg constant is about 207 times larger. --- ### **Step 3: Calculate the Wavelength for the \( 4 \rightarrow 2 \) Transition** For normal hydrogen, the wavelength for \( 4 \rightarrow 2 \) is: \[ \frac{1}{\lambda} = R_H \left( \frac{1}{2^2} - \frac{1}{4^2} \right) = R_H \left( \frac{1}{4} - \frac{1}{16} \right) = R_H \left( \frac{3}{16} \right) \] The normal value for this transition (H-atom) is **486 nm**. For muonic hydrogen, the wavelength will be: \[ \lambda_{\mu} = \frac{\lambda_{e}}{207} \] \[ \lambda_{\mu} = \frac{486\ \text{nm}}{207} \approx 2.35\ \text{nm} \] #### **Explanation:** The wavelength is shorter by a factor of 207 due to the increased mass of the muon. --- ### **Final Answer:** **b) 2.349 nm** This is the wavelength of the second line in the Balmer series for a muonic hydrogen atom.

The concept used involves the Rydberg formula for spectral lines, adjusted for the increased reduced mass in muonic hydrogen. Since the muon is much heavier than the electron, the Rydberg constant scales proportionally, leading to much shorter wavelengths for transitions. Specifically, the wavelength for the \( 4 \rightarrow 2 \) transition in muonic hydrogen is obtained by dividing the normal hydrogen wavelength by the mass ratio (207), resulting in a transition wavelength around 2.35 nm.