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 solve the problem of finding the ratio of the wavelengths of the first line of the Balmer series and the first line of the Paschen series, we will follow these steps:

### Step 1: Understand the Series
The Balmer series corresponds to transitions where the final energy level (nf) is 2, and the Paschen series corresponds to transitions where nf is 3. We need to find the first line of each series.

### Step 2: Determine the Initial and Final States
For the first line of the Balmer series:
- nf = 2 (final state)
- ni = 3 (initial state)

For the first line of the Paschen series:
- nf = 3 (final state)
- ni = 4 (initial state)

### Step 3: Use the Rydberg Formula
The Rydberg formula for the wavelength (λ) of the emitted light is given by:
\[ \frac{1}{\lambda} = R \left( \frac{1}{n_f^2} - \frac{1}{n_i^2} \right) \]
where R is the Rydberg constant.

### Step 4: Calculate Wavelength for the Balmer Series
For the first line of the Balmer series:
\[ \frac{1}{\lambda_1} = R \left( \frac{1}{2^2} - \frac{1}{3^2} \right) \]
Calculating the values:
\[ \frac{1}{\lambda_1} = R \left( \frac{1}{4} - \frac{1}{9} \right) = R \left( \frac{9 - 4}{36} \right) = R \left( \frac{5}{36} \right) \]
Thus,
\[ \lambda_1 = \frac{36}{5R} \]

### Step 5: Calculate Wavelength for the Paschen Series
For the first line of the Paschen series:
\[ \frac{1}{\lambda_2} = R \left( \frac{1}{3^2} - \frac{1}{4^2} \right) \]
Calculating the values:
\[ \frac{1}{\lambda_2} = R \left( \frac{1}{9} - \frac{1}{16} \right) = R \left( \frac{16 - 9}{144} \right) = R \left( \frac{7}{144} \right) \]
Thus,
\[ \lambda_2 = \frac{144}{7R} \]

### Step 6: Find the Ratio of Wavelengths
Now, we find the ratio of the wavelengths:
\[ \frac{\lambda_1}{\lambda_2} = \frac{\frac{36}{5R}}{\frac{144}{7R}} = \frac{36}{5R} \cdot \frac{7R}{144} \]
The Rydberg constant (R) cancels out:
\[ \frac{\lambda_1}{\lambda_2} = \frac{36 \cdot 7}{5 \cdot 144} \]
Calculating this gives:
\[ \frac{\lambda_1}{\lambda_2} = \frac{252}{720} = \frac{7}{20} \]

### Final Answer
The ratio of the wavelengths of the first line of the Balmer series to the first line of the Paschen series is:
\[ \frac{\lambda_1}{\lambda_2} = \frac{7}{20} \]