The ratio of shortest wavelength lines in Lyman , Balmer and Paschen series is 1:4:x. Calculate the value of x ?

Explanation

To solve the problem of finding the value of \( x \) in the ratio of the shortest wavelength lines in the Lyman, Balmer, and Paschen series, we will follow these steps:

### Step 1: Understand the Series and Their Shortest Wavelengths
- The Lyman series corresponds to transitions to the first energy level (n=1).
- The Balmer series corresponds to transitions to the second energy level (n=2).
- The Paschen series corresponds to transitions to the third energy level (n=3).

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

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

where \( R \) is the Rydberg constant, \( n_1 \) is the lower energy level, and \( n_2 \) is the higher energy level.

### Step 3: Calculate the Shortest Wavelength for Each Series

**For the Lyman Series (n=1):**
- The shortest wavelength occurs when \( n_2 \) approaches infinity.
- Thus, using the Rydberg formula:

\[ \frac{1}{\lambda_L} = R \left( \frac{1}{1^2} - 0 \right) = R \]

So, \( \lambda_L = \frac{1}{R} \).

**For the Balmer Series (n=2):**
- The shortest wavelength occurs when \( n_2 \) approaches infinity.
- Thus:

\[ \frac{1}{\lambda_B} = R \left( \frac{1}{2^2} - 0 \right) = \frac{R}{4} \]

So, \( \lambda_B = \frac{4}{R} \).

**For the Paschen Series (n=3):**
- The shortest wavelength occurs when \( n_2 \) approaches infinity.
- Thus:

\[ \frac{1}{\lambda_P} = R \left( \frac{1}{3^2} - 0 \right) = \frac{R}{9} \]

So, \( \lambda_P = \frac{9}{R} \).

### Step 4: Establish the Ratios
From the calculations, we have:
- \( \lambda_L = \frac{1}{R} \)
- \( \lambda_B = \frac{4}{R} \)
- \( \lambda_P = \frac{9}{R} \)

The ratios of the wavelengths are:
\[ \lambda_L : \lambda_B : \lambda_P = \frac{1}{R} : \frac{4}{R} : \frac{9}{R} = 1 : 4 : 9 \]

### Step 5: Relate to Given Ratio
The problem states that the ratio is \( 1 : 4 : x \). From our calculations, we found that the ratio is \( 1 : 4 : 9 \).

Thus, we can conclude that:
\[ x = 9 \]

### Final Answer
The value of \( x \) is \( 9 \).

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