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 \[{{\lambda }_{1}},\,\,{{\lambda }_{2}}\,and\,{{\lambda }_{3}}\] are the wave- lengths of radiations corresponding to the transitions C to B, B to A and C to A respectively, which of the following statements is correct?  

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

Correct Answer: B

Solution :

Let the energy in A, B and C state be\[{{E}_{A}},\,\,{{E}_{B}}\,and\,\,{{E}_{C}}\], then from the figure \[\left( {{E}_{C}}-{{E}_{B}} \right)+\left( {{E}_{B}}-{{E}_{A}} \right)=\left( {{E}_{C}}-{{E}_{A}} \right)\] \[\frac{hc}{{{\lambda }_{1}}}+\frac{hc}{{{\lambda }_{2}}}=\frac{hc}{{{\lambda }_{3}}}\] \[\Rightarrow \,\,\,\,\,\,\,{{\lambda }_{3}}=\frac{{{\lambda }_{1}}{{\lambda }_{2}}}{{{\lambda }_{1}}+{{\lambda }_{2}}}\]