Atoms & Nuclei

JEE Main level, numerical answer, medium

131 of 239

Question 131 of 239

The first three spectral lines of HH-atom in the Balmer series are given λ1,λ2,λ3{\lambda }_{1},{\lambda }_{2},{\lambda }_{3} considering the Bohr atomic model, the wave lengths of first and third spectral lines (λ1λ3)\left(\dfrac{{\lambda }_{1}}{{\lambda }_{3}}\right) are related by a factor of approximately xx ×10−1\times {10}^{-1}. The value of xx, to the nearest integer, is ________.

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Answer: 15

1λ1=Rz2(122−132)\dfrac{1}{{\lambda }_{1}}=R{z}^{2}\left(\dfrac{1}{{2}^{2}}-\dfrac{1}{{3}^{2}}\right)
1λ1=Rz2536\dfrac{1}{{\lambda }_{1}}=R{z}^{2}\dfrac{5}{36} ...(i)...(i)
For
3rd{3}^{rd} line
1λ3=Rz2(122−152)\dfrac{1}{{\lambda }_{3}}=R{z}^{2}\left(\dfrac{1}{{2}^{2}}-\dfrac{1}{{5}^{2}}\right)
1λ3=Rz221100\dfrac{1}{{\lambda }_{3}}=R{z}^{2}\dfrac{21}{100} ...(ii)...(ii)
(ii)+(i)(ii)+(i)
λ1λ3=21100×365=1.512=15.12×10−1\dfrac{{\lambda }_{1}}{{\lambda }_{3}}=\dfrac{21}{100}\times \dfrac{36}{5}=1.512=15.12\times {10}^{-1}
x≈15x\approx 15

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