Answer
See the explanation
Work Step by Step
a. Lines \( A \) and \( B \) correspond to electronic transitions from the \( n=4 \) state to the \( n=3 \) state and from the \( n=5 \) state to the \( n=3 \) state, respectively.
b. To calculate the wavelength of line \( A \), we can use the formula for the wavelength of light emitted during an electronic transition:
\[ \frac{1}{\lambda} = R \left( \frac{1}{{n_f}^2} - \frac{1}{{n_i}^2} \right) \]
Where \( R \) is the Rydberg constant, \( n_f \) is the final energy level, and \( n_i \) is the initial energy level.
Given that the wavelength of line \( B \) is \( 142.5 \mathrm{~nm} \) and corresponds to a transition from the \( n=5 \) state to the \( n=3 \) state, we can calculate the wavelength of line \( A \) by considering the transition from the \( n=4 \) state to the \( n=3 \) state.
Plugging in the values, we get:
\[ \frac{1}{\lambda_A} = R \left( \frac{1}{{3}^2} - \frac{1}{{4}^2} \right) \]
Solving for \( \lambda_A \), we find:
\[ \lambda_A = \frac{1}{{R \left( \frac{1}{{3}^2} - \frac{1}{{4}^2} \right)}} \]
the Rydberg constant for one-electron ions, \( R = \frac{Z^2 R_{\infty}}{n^2} \), where \( Z \) is the nuclear charge and \( R_{\infty} \) is the Rydberg constant for hydrogen (\( R_{\infty} = 1.097 \times 10^7 \, \text{m}^{-1} \)).
For the \( n=3 \) to \( n=4 \) transition (line \( A \)), we have:
\[ \lambda_A = \frac{1}{{R \left( \frac{1}{{3}^2} - \frac{1}{{4}^2} \right)}} \]
Substituting the values of \( R \) and solving for \( \lambda_A \):
\[ R = \frac{3^2 \times 1.097 \times 10^7}{3^2} = 1.097 \times 10^7 \, \text{m}^{-1} \]
\[ \lambda_A = \frac{1}{{1.097 \times 10^7 \, \text{m}^{-1} \times \left( \frac{1}{{3}^2} - \frac{1}{{4}^2} \right)}} \]
\[ \lambda_A = \frac{1}{{1.097 \times 10^7 \, \text{m}^{-1} \times \left( \frac{1}{9} - \frac{1}{16} \right)}} \]
\[ \lambda_A = \frac{1}{{1.097 \times 10^7 \, \text{m}^{-1} \times \left( \frac{7}{144} \right)}} \]
\[ \lambda_A = \frac{1}{{1.097 \times 10^7 \, \text{m}^{-1} \times \left( \frac{7}{144} \right)}} \]
\[ \lambda_A \approx 121.5 \, \text{nm} \]
Therefore, the wavelength of line \( A \) is approximately \( 121.5 \, \text{nm} \).
