Question

If ${\lambda }_1$ and ${\lambda }_2$ are the wavelengths of the third member of Lyman and 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 $7:135$

For Lyman series :

$n_1=1,n_2=4$

$\frac{1}{{\lambda }_1}=R{Z}^2(\frac{1}{1^2}-\frac{1}{4^2})$

$\frac{1}{{\lambda }_1}=R{Z}^2\left(\frac{16-1}{1\times 16}\right)$

$\frac{1}{{\lambda }_1}=R{Z}^2\left(\frac{15}{1\times 16}\right)$

${\lambda }_1=\frac{16}{15R{Z}^2}$

For paschen series :

$n_1=3,n_2=4$

$\frac{1}{{\lambda }_2}=R{Z}^2(\frac{1}{3^2}-\frac{1}{4^2})$

$\frac{1}{{\lambda }_2}=R{Z}^2\left(\frac{16-9}{9\times 16}\right)$

$\frac{1}{{\lambda }_2}=R{Z}^2\left(\frac{7}{9\times 16}\right)$

${\lambda }_2=\frac{9\times 16}{7R{Z}^2}$

So ,

$\frac{{\lambda }_1}{{\lambda }_2}=\frac{\frac{16}{15R{Z}^2}}{\frac{9\times 16}{7R{Z}^2}}$

$\frac{{\lambda }_1}{{\lambda }_2}=\frac{16\times 7}{15\times 9\times 16}$

$\frac{{\lambda }_1}{{\lambda }_2}=\frac{7}{135}$

$\frac{{\lambda }_1}{{\lambda }_2}=7:135$