The wavelength of `H_(alpha)` line in Balmer series is

Options

A. 6653 Å
B. 6365 Å
C. 6563 Å
D. 5663 Å

Explanation

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

### Step 1: Understand the Transition
The H-alpha line corresponds to the transition of an electron from the n=3 energy level to the n=2 energy level in the hydrogen atom.

### Step 2: Use the Rydberg Formula
The Rydberg formula for the wavelength of light emitted during an electron transition is given by:

\[ \frac{1}{\lambda} = R \left( \frac{1}{n_1^2} - \frac{1}{n_2^2} \right) \]

where:
- \( \lambda \) is the wavelength,
- \( R \) is the Rydberg constant (\( R = 1.097 \times 10^7 \, \text{m}^{-1} \)),
- \( n_1 \) is the lower energy level (2 for H-alpha),
- \( n_2 \) is the higher energy level (3 for H-alpha).

### Step 3: Substitute Values into the Formula
Substituting \( n_1 = 2 \) and \( n_2 = 3 \) into the Rydberg formula, we have:

\[ \frac{1}{\lambda} = 1.097 \times 10^7 \left( \frac{1}{2^2} - \frac{1}{3^2} \right) \]

Calculating \( \frac{1}{2^2} \) and \( \frac{1}{3^2} \):

\[ \frac{1}{2^2} = \frac{1}{4} = 0.25 \]
\[ \frac{1}{3^2} = \frac{1}{9} \approx 0.1111 \]

### Step 4: Calculate the Difference
Now, calculate the difference:

\[ \frac{1}{4} - \frac{1}{9} = 0.25 - 0.1111 = 0.1389 \]

### Step 5: Plug the Difference Back into the Formula
Now substitute this difference back into the Rydberg formula:

\[ \frac{1}{\lambda} = 1.097 \times 10^7 \times 0.1389 \]

### Step 6: Calculate \( \lambda \)
Calculating the right-hand side:

\[ \frac{1}{\lambda} \approx 1.097 \times 10^7 \times 0.1389 \approx 1.527 \times 10^6 \]

Now, taking the reciprocal to find \( \lambda \):

\[ \lambda \approx \frac{1}{1.527 \times 10^6} \approx 6.54 \times 10^{-7} \, \text{m} \]

### Step 7: Convert to Angstroms
To convert meters to angstroms (1 angstrom = \( 10^{-10} \) m):

\[ \lambda \approx 6.54 \times 10^{-7} \, \text{m} = 6540 \, \text{Å} \]

### Final Answer
The wavelength of the H-alpha line in the Balmer series is approximately **6563 Å**.

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