Question

In Paschen series, wavelength of first line is '$\lambda_1$' and for Brackett series, wavelength of first line is '$\lambda_2$' then ratio $\frac{\lambda_1}{\lambda_2}$ is ______.

• A. $\frac{7}{400}$
• B. $\frac{9}{144}$
• C. $\frac{81}{175}$
• D. $\frac{108}{509}$

Hint:

Series bases: Lyman ($n=1$), Balmer ($n=2$), Paschen ($n=3$), Brackett ($n=4$), Pfund ($n=5$). The "first line" or "$H_\alpha$ line" is always from $n+1$ down to $n$.

Answer 1

The Correct Option is C

Step 1: Understanding the Question:
We must apply the Rydberg equation for hydrogen emission spectra to find the wavelengths of the first lines of the Paschen and Brackett series and compute their ratio.

Step 2: Detailed Explanation:
The Rydberg formula for wavelength $\lambda$ is:
$\frac{1}{\lambda} = R \left( \frac{1}{n_1^2} - \frac{1}{n_2^2} \right)$
1. Paschen Series (first line, $\lambda_1$):
For the Paschen series, the lower level is $n_1 = 3$.
The "first line" corresponds to the smallest energy jump, which is from $n_2 = 4$.
$\frac{1}{\lambda_1} = R \left( \frac{1}{3^2} - \frac{1}{4^2} \right)$
$\frac{1}{\lambda_1} = R \left( \frac{1}{9} - \frac{1}{16} \right)$
$\frac{1}{\lambda_1} = R \left( \frac{16 - 9}{144} \right) = R \left( \frac{7}{144} \right)$
So, $\lambda_1 = \frac{144}{7R}$.
2. Brackett Series (first line, $\lambda_2$):
For the Brackett series, the lower level is $n_1 = 4$.
The "first line" corresponds to the smallest energy jump, which is from $n_2 = 5$.
$\frac{1}{\lambda_2} = R \left( \frac{1}{4^2} - \frac{1}{5^2} \right)$
$\frac{1}{\lambda_2} = R \left( \frac{1}{16} - \frac{1}{25} \right)$
$\frac{1}{\lambda_2} = R \left( \frac{25 - 16}{400} \right) = R \left( \frac{9}{400} \right)$
So, $\lambda_2 = \frac{400}{9R}$.
3. Calculate the ratio $\frac{\lambda_1{\lambda_2}$:}
$\text{Ratio} = \frac{ \frac{144}{7R} }{ \frac{400}{9R} }$
$\text{Ratio} = \frac{144}{7R} \times \frac{9R}{400}$
The Rydberg constant ($R$) cancels out:
$\text{Ratio} = \frac{144 \times 9}{7 \times 400}$
Divide 144 and 400 by 16:
$\frac{144}{16} = 9$ and $\frac{400}{16} = 25$.
$\text{Ratio} = \frac{9 \times 9}{7 \times 25}$
$\text{Ratio} = \frac{81}{175}$

Step 3: Final Answer:
The ratio is $\frac{81}{175}$, matching option (c).