The ratio of wavelengths of the `1st` line of Balmer series and the `1st` line of Paschen series is

Options

A. `20:7`
B. `7:20`
C. `7:4`
D. `4:7`

Answer

To find the ratio of wavelengths of the first line of the Balmer series and the first line of the Paschen series, we can follow these steps:

### Step 1: Understand the Series
The Balmer series corresponds to transitions where the final energy level (n_l) is 2, while the Paschen series corresponds to transitions where the final energy level (n_l) is 3.

### Step 2: Determine the Wavelength for the Balmer Series
For the first line of the Balmer series, the transition is from n_h = 3 to n_l = 2. Using the formula for the wavelength:
\[ \frac{1}{\lambda_B} = RZ^2 \left( \frac{1}{n_L^2} - \frac{1}{n_H^2} \right) \]
Substituting \(n_L = 2\) and \(n_H = 3\):
\[ \frac{1}{\lambda_B} = RZ^2 \left( \frac{1}{2^2} - \frac{1}{3^2} \right) = RZ^2 \left( \frac{1}{4} - \frac{1}{9} \right) \]
Calculating the fractions:
\[ \frac{1}{4} - \frac{1}{9} = \frac{9 - 4}{36} = \frac{5}{36} \]
Thus,
\[ \frac{1}{\lambda_B} = \frac{5RZ^2}{36} \]
This gives:
\[ \lambda_B = \frac{36}{5RZ^2} \]

### Step 3: Determine the Wavelength for the Paschen Series
For the first line of the Paschen series, the transition is from n_h = 4 to n_l = 3. Using the same formula:
\[ \frac{1}{\lambda_P} = RZ^2 \left( \frac{1}{3^2} - \frac{1}{4^2} \right) = RZ^2 \left( \frac{1}{9} - \frac{1}{16} \right) \]
Calculating the fractions:
\[ \frac{1}{9} - \frac{1}{16} = \frac{16 - 9}{144} = \frac{7}{144} \]
Thus,
\[ \frac{1}{\lambda_P} = \frac{7RZ^2}{144} \]
This gives:
\[ \lambda_P = \frac{144}{7RZ^2} \]

### Step 4: Calculate the Ratio of Wavelengths
Now, we can find the ratio of the wavelengths:
\[ \frac{\lambda_B}{\lambda_P} = \frac{\frac{36}{5RZ^2}}{\frac{144}{7RZ^2}} = \frac{36 \cdot 7}{5 \cdot 144} \]
Simplifying this:
\[ = \frac{252}{720} = \frac{7}{20} \]

### Conclusion
The ratio of the wavelengths of the first line of the Balmer series to the first line of the Paschen series is:
\[ \frac{7}{20} \]