Question

If ${\lambda }_1$ and ${\lambda }_2$ are the wavelengths of the third member of Lyman and the first member of the Paschen series respectively, then the value of ${\lambda }_1:{\lambda }_2$ is

• A. $1:3$
• B. $1:9$
• C. $7:135$
• D. $7:108$

Answer 1

The correct option is C

$7:135$

Step 1: Given data

${\lambda }_1$ is the wavelength of the third member of the Lyman series.

${\lambda }_2$ is the wavelength of the first member of the Paschen series.

We have to find the ratio ${\lambda }_1:{\lambda }_2$

Step 2: Formula and calculation

We know,

Rydberg's equation, $\frac{1}{\lambda }=R\left(\frac{1}{n_f^2}-\frac{1}{n_i^2}\right)$

where,

$R\to$Rydberg's constant

$n_i\to$Initial energy level

$n_f\to$Final energy level

$\lambda \to$Wavelength

Step 3: Calculation

For the third member of the Lyman series,

For Lyman series, $n_f=1$ and $n_i=2,3,4,...$

Since we have to do for the third member of the Lyman series, we take the third value of $n_i$, i.e., $n_i=4$

$$\begin{gathered} \therefore n_i=4 \\ {n}_f=1 \\ \lambda ={\lambda }_1 \end{gathered}$$

$$\begin{gathered} \frac{1}{{\lambda }_1}=R\left(\frac{1}{1^2}-\frac{1}{4^2}\right) \\ \Rightarrow \frac{1}{{\lambda }_1}=R\left(\frac{16-1}{16}\right) \\ \Rightarrow {\lambda }_1=\frac{16}{15R}\to \left(1\right) \end{gathered}$$

For the first member of the Paschen series,

For Paschen series, $n_f=3$ and $n_i=4,5,6,...$

Since we have to do for first member of Paschen series, we take the first value of $n_i$, i.e., $n_i=4$

$$\begin{gathered} \therefore n_i=4 \\ {n}_f=3 \\ \lambda ={\lambda }_2 \end{gathered}$$

$$\begin{gathered} \frac{1}{{\lambda }_2}=R\left(\frac{1}{3^2}-\frac{1}{4^2}\right) \\ \Rightarrow \frac{1}{{\lambda }_2}=R\left(\frac{16-9}{144}\right) \\ \Rightarrow {\lambda }_2=\frac{144}{7R}\to \left(2\right) \end{gathered}$$

Calculating ratio using both the equations (1) and (2),

$$\begin{gathered} \frac{{\lambda }_1}{{\lambda }_2}=\frac{16}{15R}\times \frac{7R}{144} \\ \Rightarrow \frac{{\lambda }_1}{{\lambda }_2}=\frac{7}{135} \\ \Rightarrow {\lambda }_1:{\lambda }_2=7:135 \end{gathered}$$

Therefore, the value of ${\lambda }_1:{\lambda }_2$ is $7:135$.

Hence, option C is correct.