Find the ratio of series limit wavelength of Balmer series to wavelength of first time line of paschen series.

Explanation

To find the ratio of the series limit wavelength of the Balmer series to the wavelength of the first line of the Paschen series, we will use the Rydberg formula for hydrogen-like atoms:

\[ \frac{1}{\lambda} = RZ^2 \left( \frac{1}{n_f^2} - \frac{1}{n_i^2} \right) \]

where:
- \( \lambda \) is the wavelength,
- \( R \) is the Rydberg constant,
- \( Z \) is the atomic number,
- \( n_f \) is the final energy level,
- \( n_i \) is the initial energy level.

### Step 1: Calculate the series limit wavelength of the Balmer series

The Balmer series corresponds to transitions from higher energy levels down to \( n_f = 2 \). The series limit occurs when \( n_i \) approaches infinity (\( n_i \to \infty \)).

Using the Rydberg formula for the Balmer series:

\[ \frac{1}{\lambda_{\text{Balmer}}} = RZ^2 \left( \frac{1}{2^2} - \frac{1}{\infty^2} \right) \]

This simplifies to:

\[ \frac{1}{\lambda_{\text{Balmer}}} = RZ^2 \left( \frac{1}{4} \right) \]

Thus, the wavelength for the series limit of the Balmer series is:

\[ \lambda_{\text{Balmer}} = \frac{4}{RZ^2} \]

### Step 2: Calculate the wavelength of the first line of the Paschen series

The Paschen series corresponds to transitions from higher energy levels down to \( n_f = 3 \). The first line of the Paschen series occurs when \( n_i = 4 \).

Using the Rydberg formula for the Paschen series:

\[ \frac{1}{\lambda_{\text{Paschen}}} = RZ^2 \left( \frac{1}{3^2} - \frac{1}{4^2} \right) \]

Calculating the values:

\[ \frac{1}{\lambda_{\text{Paschen}}} = RZ^2 \left( \frac{1}{9} - \frac{1}{16} \right) \]

Finding a common denominator (144):

\[ \frac{1}{\lambda_{\text{Paschen}}} = RZ^2 \left( \frac{16 - 9}{144} \right) = RZ^2 \left( \frac{7}{144} \right) \]

Thus, the wavelength for the first line of the Paschen series is:

\[ \lambda_{\text{Paschen}} = \frac{144}{7RZ^2} \]

### Step 3: Find the ratio of the two wavelengths

Now, we can find the ratio of the series limit wavelength of the Balmer series to the wavelength of the first line of the Paschen series:

\[ \text{Ratio} = \frac{\lambda_{\text{Balmer}}}{\lambda_{\text{Paschen}}} = \frac{\frac{4}{RZ^2}}{\frac{144}{7RZ^2}} \]

The \( RZ^2 \) cancels out:

\[ \text{Ratio} = \frac{4}{144} \cdot \frac{7}{1} = \frac{4 \cdot 7}{144} = \frac{28}{144} = \frac{7}{36} \]

### Conclusion

Thus, the required ratio of the series limit wavelength of the Balmer series to the wavelength of the first line of the Paschen series is:

\[ \frac{7}{36} \]