The shortest wavelength in the balmer series is `(R=1.097xx10^7m^-1)`

Options

A. 200 nm
B. 256.8 nm
C. 300 nm
D. 364.6 nm

Explanation

To find the shortest wavelength in the Balmer series, we can use the Rydberg formula for hydrogen, which is given by:

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

Where:
- \( \lambda \) is the wavelength,
- \( R \) is the Rydberg constant (\( R = 1.097 \times 10^7 \, \text{m}^{-1} \)),
- \( n_f \) is the final energy level,
- \( n_i \) is the initial energy level.

### Step 1: Identify the final and initial energy levels
For the Balmer series, the final energy level \( n_f \) is 2. The initial energy level \( n_i \) can take values from 3 to infinity (as it approaches infinity, we get the shortest wavelength).

### Step 2: Write the formula for the shortest wavelength
To find the shortest wavelength, we take the limit as \( n_i \) approaches infinity:

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

As \( n_i \to \infty \), \( \frac{1}{n_i^2} \to 0 \). Thus, the equation simplifies to:

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

### Step 3: Solve for \( \lambda \)
Now, we can find \( \lambda \):

\[ \lambda = \frac{4}{R} \]

### Step 4: Substitute the value of \( R \)
Substituting the value of the Rydberg constant:

\[ \lambda = \frac{4}{1.097 \times 10^7} \]

### Step 5: Calculate \( \lambda \)
Now, calculate the value:

\[ \lambda = \frac{4}{1.097 \times 10^7} \approx 3.644 \times 10^{-7} \text{ m} \]

### Step 6: Convert to nanometers
To convert meters to nanometers, multiply by \( 10^9 \):

\[ \lambda \approx 3.644 \times 10^{-7} \text{ m} \times 10^9 \approx 364.4 \text{ nm} \]

### Final Answer
The shortest wavelength in the Balmer series is approximately **364.4 nm**.

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