The ratio of the largest to shortest wavelength in Balmer series of hydrogen spectra is,

Options

A. `(25)/(9)`
B. `(17)/(6)`
C. `(9)/(5)`
D. `(5)/(4)`

Explanation

To find the ratio of the largest to shortest wavelength in the Balmer series of hydrogen spectra, we can follow these steps:

### Step 1: Understand the Rydberg Formula
The wavelength (\( \lambda \)) for the Balmer series can be expressed using the Rydberg formula:
\[ \frac{1}{\lambda} = R \left( \frac{1}{2^2} - \frac{1}{n^2} \right) \]
where \( R \) is the Rydberg constant, \( n \) is an integer (3, 4, 5, ...), and \( 2^2 = 4 \).

### Step 2: Identify the Wavelengths
From the formula, we can rearrange it to find \( \lambda \):
\[ \lambda = \frac{1}{R \left( \frac{1}{4} - \frac{1}{n^2} \right)} \]
- **Shortest Wavelength (\( \lambda_{\text{min}} \))**: This occurs when \( n \) is at its minimum value, which is 3.
- **Longest Wavelength (\( \lambda_{\text{max}} \))**: This occurs when \( n \) approaches infinity.

### Step 3: Calculate the Shortest Wavelength
For \( n = 3 \):
\[ \lambda_{\text{min}} = \frac{1}{R \left( \frac{1}{4} - \frac{1}{3^2} \right)} = \frac{1}{R \left( \frac{1}{4} - \frac{1}{9} \right)} \]
Calculating the term inside the parentheses:
\[ \frac{1}{4} = \frac{9}{36}, \quad \frac{1}{9} = \frac{4}{36} \quad \Rightarrow \quad \frac{1}{4} - \frac{1}{9} = \frac{9 - 4}{36} = \frac{5}{36} \]
Thus,
\[ \lambda_{\text{min}} = \frac{1}{R \cdot \frac{5}{36}} = \frac{36}{5R} \]

### Step 4: Calculate the Longest Wavelength
For \( n \to \infty \):
\[ \lambda_{\text{max}} = \frac{1}{R \left( \frac{1}{4} - 0 \right)} = \frac{1}{R \cdot \frac{1}{4}} = \frac{4}{R} \]

### Step 5: Find the Ratio of Largest to Shortest Wavelength
Now we can find the ratio:
\[ \text{Ratio} = \frac{\lambda_{\text{max}}}{\lambda_{\text{min}}} = \frac{\frac{4}{R}}{\frac{36}{5R}} = \frac{4}{R} \cdot \frac{5R}{36} = \frac{20}{36} = \frac{5}{9} \]

### Conclusion
Thus, the ratio of the largest to shortest wavelength in the Balmer series of hydrogen spectra is:
\[ \frac{5}{9} \]