Question

The Rydberg constant R if $H{e}^{+}$ ions are known to have the wavelength difference between the first (of the longest wavelength) lines of Balmer and Lyman series to 133.7nm is:

• A. $1.096\times {10}^7{m}^{-1}$
• B. $1.096\times {10}^9{m}^{-1}$
• C. $1.096\times {10}^{11}{m}^{-1}$
• D. $1.096\times {10}^{13}{m}^{-1}$

Answer 1

Answer: (A)

$1.096\times {10}^7{m}^{-1}$

Wavelength is given by:

$\frac{1}{\lambda }=R{Z}^2[\frac{1}{n_1^2}-\frac{1}{n_2^2}]$

For the first line of Lyman series $n_1=1$ and $n_2=2$

$\frac{1}{{\lambda }_L}=R{Z}^2[\frac{1}{1^2}-\frac{1}{2^2}]$

${\lambda }_L=\frac{4}{3R{Z}^2}$

For the first line of Balmer series $n_1=2$ and $n_2=3$

$\frac{1}{{\lambda }_B}=R{Z}^2[\frac{1}{2^2}-\frac{1}{3^2}]$

${\lambda }_B=\frac{36}{5R{Z}^2}$

${\lambda }_B-{\lambda }_L=133.7nm=133.7\times {10}^{-9}m$

$\frac{36}{5R{Z}^2}-\frac{4}{3R{Z}^2}=\frac{88}{15R{Z}^2}=133.7\times {10}^{-9}$

$\Rightarrow R=1.096\times {10}^7{m}^{-1}$