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.

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\[{{E}_{A}},\,\,{{E}_{B}}\,and\,\,{{E}_{C}}\] then from the figure \[{{v}_{Balmer}}\,\,=\,\,\frac{c}{{{\lambda }_{\max }}}\,\,=\,\,Rc\,\left[ \frac{1}{{{(2)}^{2}}}-\frac{1}{{{(3)}^{2}}} \right]\,\,=\,\,\frac{5\,RC}{36}\] \[or\,\,\,\,\,\frac{hc}{{{\lambda }_{1}}}+\frac{hc}{{{\lambda }_{2}}}=\frac{hc}{{{\lambda }_{3}}}\,\,\,\,\,\,\,\,\,\,\,\,\,\,{{\lambda }_{3}}\,=\,\frac{{{\lambda }_{1}}{{\lambda }_{2}}}{{{\lambda }_{1}}+{{\lambda }_{2}}}\]