If `lambda_(1) and lambda_(2)` are the maximum wavelength limits of Lyman and Balmer series of H atom, `(lambda_1)/(lambda_2)` will be .

Options

A. `(5)/(27)`
B. `(27)/(5)`
C. `(5)/(36)`
D. `(3)/(4)`.

Explanation

To find the ratio of the maximum wavelengths of the Lyman and Balmer series for the hydrogen atom, we will follow these steps:

### Step 1: Understanding the Lyman Series
The Lyman series corresponds to electronic transitions in a hydrogen atom where the electron falls to the first energy level (n=1) from higher energy levels (n=2, 3, ...). The maximum wavelength (λ₁) occurs when the transition is from n=2 to n=1.

### Step 2: Calculate λ₁ for the Lyman Series
Using the Rydberg formula:
\[ \frac{1}{\lambda_1} = R \left( \frac{1}{n_1^2} - \frac{1}{n_2^2} \right) \]
For the Lyman series, \( n_1 = 1 \) and \( n_2 = 2 \):
\[ \frac{1}{\lambda_1} = R \left( \frac{1}{1^2} - \frac{1}{2^2} \right) = R \left( 1 - \frac{1}{4} \right) = R \left( \frac{3}{4} \right) \]
Thus,
\[ \lambda_1 = \frac{4}{3R} \]

### Step 3: Understanding the Balmer Series
The Balmer series corresponds to transitions where the electron falls to the second energy level (n=2) from higher levels (n=3, 4, ...). The maximum wavelength (λ₂) occurs when the transition is from n=3 to n=2.

### Step 4: Calculate λ₂ for the Balmer Series
Using the Rydberg formula:
\[ \frac{1}{\lambda_2} = R \left( \frac{1}{n_1^2} - \frac{1}{n_2^2} \right) \]
For the Balmer series, \( n_1 = 2 \) and \( n_2 = 3 \):
\[ \frac{1}{\lambda_2} = R \left( \frac{1}{2^2} - \frac{1}{3^2} \right) = R \left( \frac{1}{4} - \frac{1}{9} \right) \]
Finding a common denominator (36):
\[ \frac{1}{\lambda_2} = R \left( \frac{9}{36} - \frac{4}{36} \right) = R \left( \frac{5}{36} \right) \]
Thus,
\[ \lambda_2 = \frac{36}{5R} \]

### Step 5: Calculate the Ratio \( \frac{\lambda_1}{\lambda_2} \)
Now we can find the ratio of the maximum wavelengths:
\[ \frac{\lambda_1}{\lambda_2} = \frac{\frac{4}{3R}}{\frac{36}{5R}} = \frac{4}{3} \cdot \frac{5}{36} = \frac{20}{108} = \frac{5}{27} \]

### Final Answer
The ratio \( \frac{\lambda_1}{\lambda_2} \) is \( \frac{5}{27} \).

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