`H_(α)` line of Balmer series is 6500 `Å` . The wave length of `H_(gamma)` is

Options

A. 4815 `Å`
B. 4341`Å`
C. `7800Å`
D. `3800Å`

Answer

To find the wavelength of the H-gamma line in the Balmer series, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Balmer Series**: The Balmer series corresponds to electronic transitions in hydrogen where the electron falls to the n=2 energy level from higher energy levels (n=3, 4, 5, ...). The H-alpha line corresponds to the transition from n=3 to n=2, while the H-gamma line corresponds to the transition from n=5 to n=2.

2. **Identify the Given Information**: We know that the wavelength of the H-alpha line (n=3 to n=2) is given as 6500 Å (angstroms).

3. **Use the Rydberg Formula**: The formula to calculate the wavelength of the spectral lines in hydrogen is:
\[ \frac{1}{\lambda} = R_z^2 \left( \frac{1}{n_1^2} - \frac{1}{n_2^2} \right) \]
where \( R \) is the Rydberg constant (approximately 109677 cm⁻¹ for hydrogen), \( n_1 \) is the lower energy level, and \( n_2 \) is the higher energy level.

4. **Set Values for H-gamma**: For H-gamma, \( n_1 = 2 \) and \( n_2 = 5 \). Plugging these values into the Rydberg formula gives:
\[ \frac{1}{\lambda_{\gamma}} = R \left( \frac{1}{2^2} - \frac{1}{5^2} \right) \]

5. **Calculate the Right Side**:
- Calculate \( \frac{1}{2^2} = \frac{1}{4} = 0.25 \)
- Calculate \( \frac{1}{5^2} = \frac{1}{25} = 0.04 \)
- So, \( \frac{1}{2^2} - \frac{1}{5^2} = 0.25 - 0.04 = 0.21 \)

6. **Substitute Rydberg Constant**: Now substituting the Rydberg constant:
\[ \frac{1}{\lambda_{\gamma}} = 109677 \times 0.21 \]
\[ \frac{1}{\lambda_{\gamma}} = 23030.17 \, \text{cm}^{-1} \]

7. **Find Wavelength**: To find \( \lambda_{\gamma} \):
\[ \lambda_{\gamma} = \frac{1}{23030.17} \, \text{cm} \]
\[ \lambda_{\gamma} \approx 0.0000434 \, \text{cm} \]

8. **Convert to Angstroms**: Since \( 1 \, \text{cm} = 10^{10} \, \text{Å} \):
\[ \lambda_{\gamma} \approx 0.0000434 \times 10^{10} \, \text{Å} = 4340 \, \text{Å} \]

### Final Answer:
The wavelength of the H-gamma line is approximately **4340 Å**.

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