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JEE Main & Advanced Physics Nuclear Physics And Radioactivity Question Bank Critical Thinking

  • question_answer
    Hydrogen (H), deuterium , singly ionized helium \[(H{{e}^{+}})\]and doubly ionized lithium \[(Li)\] all have one electron around the nucleus. Consider n =2 to n = 1 transition. The wavelengths of emitted radiations are \[{{\lambda }_{1}},\ {{\lambda }_{2}},\ {{\lambda }_{3}}\]and \[{{\lambda }_{4}}\] respectively. Then approximately                                [KCET 1994]

    A)             \[{{\lambda }_{1}}={{\lambda }_{2}}=4{{\lambda }_{3}}=9{{\lambda }_{4}}\]

    B)            \[4{{\lambda }_{1}}=2{{\lambda }_{2}}=2{{\lambda }_{3}}={{\lambda }_{4}}\]

    C)            \[{{\lambda }_{1}}=2{{\lambda }_{2}}=2\sqrt{2}{{\lambda }_{3}}=3\sqrt{2}{{\lambda }_{4}}\]

    D)            \[{{\lambda }_{1}}={{\lambda }_{2}}=2{{\lambda }_{3}}=3\sqrt{2}{{\lambda }_{4}}\]

    Correct Answer: A

    Solution :

               Using \[\Delta E\propto {{Z}^{2}}\] (∵ \[{{n}_{1}}\] and \[{{n}_{2}}\] are same)                    Þ \[\frac{hc}{\lambda }\propto {{Z}^{2}}\] Þ \[\lambda {{Z}^{2}}=\text{constant}\]                    Þ \[{{\lambda }_{1}}Z_{1}^{2}={{\lambda }_{2}}Z_{2}^{2}={{\lambda }_{3}}Z_{3}^{2}={{\lambda }_{4}}Z_{^{4}}^{2}\]                    Þ \[{{\lambda }_{1}}\times 1={{\lambda }_{2}}\times {{1}^{2}}={{\lambda }_{3}}\times {{2}^{2}}={{\lambda }_{4}}\times {{3}^{3}}\]                    Þ \[{{\lambda }_{1}}={{\lambda }_{2}}=4{{\lambda }_{3}}=9{{\lambda }_{4}}\].                             


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