question_answer

Energy levels A, B, C of a certain atom corresponding to increasing values of energy i.e. \[{{E}_{A}}<{{E}_{B}}<{{E}_{C}}\]. If \[{{\lambda }_{1}},\ {{\lambda }_{2}},\ {{\lambda }_{3}}\]are the wavelengths of radiations corresponding to the transitions C to B, B to A and C to A respectively, which of the following statements is correct            [AIIMS 1995; CBSE PMT 1990, 2005]

A)            \[{{\lambda }_{3}}={{\lambda }_{1}}+{{\lambda }_{2}}\]    

B)            \[{{\lambda }_{3}}=\frac{{{\lambda }_{1}}{{\lambda }_{2}}}{{{\lambda }_{1}}+{{\lambda }_{2}}}\]

C)            \[{{\lambda }_{1}}+{{\lambda }_{2}}+{{\lambda }_{3}}=0\]               

D)            \[\lambda _{3}^{2}=\lambda _{1}^{2}+\lambda _{2}^{2}\]

Correct Answer: B

Solution :

                   Let the energy in A, B and C state be EA. EB and EC, then from the figure              \[{{\nu }_{\text{Balmer}}}=\frac{c}{{{\lambda }_{\max }}}=Rc\left[ \frac{1}{{{(2)}^{2}}}-\frac{1}{{{(3)}^{2}}} \right]=\frac{5RC}{36}\]or \[\frac{hc}{{{\lambda }_{1}}}+\frac{hc}{{{\lambda }_{2}}}=\frac{hc}{{{\lambda }_{3}}}\]            \[\Rightarrow \,{{\lambda }_{3}}=\frac{{{\lambda }_{1}}{{\lambda }_{2}}}{{{\lambda }_{1}}+{{\lambda }_{2}}}\]