The ratio of the longest and shortest wavelengths of the Lyman series is approximately

Options

A. `(4)/(3)`
B. `(9)/(4)`
C. `(9)/(5)`
D. `(16)/(7)`

Explanation

To find the ratio of the longest and shortest wavelengths of the Lyman series, we can follow these steps:

### Step 1: Understand the Lyman Series
The Lyman series consists of spectral lines corresponding to transitions of an electron in a hydrogen atom from higher energy levels (n ≥ 2) to the lowest energy level (n = 1). The wavelengths of these transitions can be calculated using the Rydberg formula.

### Step 2: Identify the Longest Wavelength
The longest wavelength in the Lyman series corresponds to the transition from n = 2 to n = 1. This is because the longest wavelength corresponds to the lowest energy transition.

Using the Rydberg formula:
\[ \frac{1}{\lambda_1} = R \left( \frac{1}{n_1^2} - \frac{1}{n_2^2} \right) \]
For the longest wavelength (n_2 = 2 and n_1 = 1):
\[ \frac{1}{\lambda_1} = R \left( \frac{1}{1^2} - \frac{1}{2^2} \right) = R \left( 1 - \frac{1}{4} \right) = R \left( \frac{3}{4} \right) \]
Thus,
\[ \lambda_1 = \frac{4}{3R} \]

### Step 3: Identify the Shortest Wavelength
The shortest wavelength corresponds to the transition from n = ∞ to n = 1. This is because the shortest wavelength corresponds to the highest energy transition.

Using the Rydberg formula:
\[ \frac{1}{\lambda_2} = R \left( \frac{1}{1^2} - \frac{1}{\infty^2} \right) = R \left( 1 - 0 \right) = R \]
Thus,
\[ \lambda_2 = \frac{1}{R} \]

### Step 4: Calculate the Ratio of Longest to Shortest Wavelength
Now we can find the ratio of the longest wavelength (\(\lambda_1\)) to the shortest wavelength (\(\lambda_2\)):
\[ \frac{\lambda_1}{\lambda_2} = \frac{\frac{4}{3R}}{\frac{1}{R}} = \frac{4}{3} \]

### Final Answer
The ratio of the longest to the shortest wavelengths of the Lyman series is approximately \( \frac{4}{3} \).

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