What is the shortest wavelength present in the Paschen series of spectral lines?

Options

A. 720 nm
B. 790 nm
C. 800 nm
D. 820 nm

Explanation

To find the shortest wavelength present in the Paschen series of spectral lines, we can follow these steps:

### Step 1: Understand the Paschen Series
The Paschen series corresponds to electronic transitions in a hydrogen atom where the final energy level (n_final) is 3. The initial energy level (n_initial) can be any integer greater than 3 (n_initial = 4, 5, 6,...).

### Step 2: Use the Rydberg Formula
The wavelength (λ) of the emitted light can be calculated using the Rydberg formula:
\[ \frac{1}{\lambda} = R \left( \frac{1}{n_{final}^2} - \frac{1}{n_{initial}^2} \right) \]
where R is the Rydberg constant, approximately \( R = 1.097 \times 10^7 \, \text{m}^{-1} \).

### Step 3: Set n_final and n_initial
For the shortest wavelength in the Paschen series:
- Set \( n_{final} = 3 \)
- Set \( n_{initial} \) to the smallest value greater than 3, which is 4.

### Step 4: Substitute Values into the Rydberg Formula
Substituting the values into the Rydberg formula:
\[ \frac{1}{\lambda} = R \left( \frac{1}{3^2} - \frac{1}{4^2} \right) \]

Calculating \( \frac{1}{3^2} \) and \( \frac{1}{4^2} \):
\[ \frac{1}{3^2} = \frac{1}{9}, \quad \frac{1}{4^2} = \frac{1}{16} \]

Now, calculate \( \frac{1}{9} - \frac{1}{16} \):
Finding a common denominator (which is 144):
\[ \frac{1}{9} = \frac{16}{144}, \quad \frac{1}{16} = \frac{9}{144} \]
Thus,
\[ \frac{1}{9} - \frac{1}{16} = \frac{16}{144} - \frac{9}{144} = \frac{7}{144} \]

### Step 5: Calculate Wavelength
Now substituting back into the formula:
\[ \frac{1}{\lambda} = R \left( \frac{7}{144} \right) \]
\[ \lambda = \frac{144}{7R} \]

Substituting the value of R:
\[ \lambda = \frac{144}{7 \times 1.097 \times 10^7} \]

Calculating the denominator:
\[ 7 \times 1.097 \times 10^7 \approx 7.679 \times 10^7 \]
Thus,
\[ \lambda \approx \frac{144}{7.679 \times 10^7} \approx 1.875 \times 10^{-9} \, \text{m} \]

### Step 6: Convert to Nanometers
To convert meters to nanometers (1 m = \( 10^9 \) nm):
\[ \lambda \approx 1.875 \times 10^{-9} \times 10^9 \, \text{nm} = 187.5 \, \text{nm} \]

### Conclusion
The shortest wavelength present in the Paschen series is approximately **187.5 nm**.