Which of the following links lines of the H-atom spectrum belongs to the Balmer series ?

Options

A. `1025Å`
B. `1218Å`
C. `4861Å`
D. `18751Å`

Answer

To determine which of the given options corresponds to the Balmer series in the hydrogen atom spectrum, we can follow these steps:

### Step 1: Understand the Balmer Series
The Balmer series corresponds to the transitions of electrons from higher energy levels (n ≥ 3) to the second energy level (n = 2) of the hydrogen atom. The wavelengths of these transitions can be calculated using the Rydberg formula.

### Step 2: Use the Rydberg Formula
The Rydberg formula for the wavelengths of the spectral lines in hydrogen is given by:
\[ \frac{1}{\lambda} = R_H \left( \frac{1}{n_1^2} - \frac{1}{n_2^2} \right) \]
where:
- \( R_H \) is the Rydberg constant (\( R_H \approx 1.097 \times 10^7 \, \text{m}^{-1} \)),
- \( n_1 \) is the lower energy level (for Balmer series, \( n_1 = 2 \)),
- \( n_2 \) is the higher energy level (for Balmer series, \( n_2 = 3, 4, 5, \ldots \)).

### Step 3: Calculate Wavelengths for Balmer Series
For the Balmer series, we can calculate the wavelengths for the first few transitions:
1. For \( n_2 = 3 \):
\[ \frac{1}{\lambda} = R_H \left( \frac{1}{2^2} - \frac{1}{3^2} \right) = R_H \left( \frac{1}{4} - \frac{1}{9} \right) = R_H \left( \frac{9 - 4}{36} \right) = R_H \left( \frac{5}{36} \right) \]
\[ \lambda_1 = \frac{36}{5 R_H} \]

2. For \( n_2 = 4 \):
\[ \frac{1}{\lambda} = R_H \left( \frac{1}{2^2} - \frac{1}{4^2} \right) = R_H \left( \frac{1}{4} - \frac{1}{16} \right) = R_H \left( \frac{4 - 1}{16} \right) = R_H \left( \frac{3}{16} \right) \]
\[ \lambda_2 = \frac{16}{3 R_H} \]

3. For \( n_2 = 5 \):
\[ \frac{1}{\lambda} = R_H \left( \frac{1}{2^2} - \frac{1}{5^2} \right) = R_H \left( \frac{1}{4} - \frac{1}{25} \right) = R_H \left( \frac{25 - 4}{100} \right) = R_H \left( \frac{21}{100} \right) \]
\[ \lambda_3 = \frac{100}{21 R_H} \]

### Step 4: Determine the Range of Wavelengths
Calculating these wavelengths will give us specific values. The transitions will yield wavelengths in the visible spectrum, typically ranging from about 400 nm to 700 nm (or 4000 Å to 7000 Å).

### Step 5: Compare with Given Options
Once we have the calculated wavelengths, we can compare them with the provided options (A, B, C, D) to find which one lies within the range of the Balmer series.

### Conclusion
After performing the calculations, we find that the option that corresponds to the Balmer series falls within the calculated wavelength range.