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JEE Main 2023 · Chemistry · Atomic Structure · medium

The wavelength of the first line of the Balmer series of hydrogen is λ1, and that of the second line of the Lyman series of He^+ is λ2. The ratio…

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The wavelength of the first line of the Balmer series of hydrogen is λ1\lambda_1, and that of the second line of the Lyman series of He+\mathrm{He^+} is λ2\lambda_2. The ratio λ1/λ2\lambda_1/\lambda_2 is
  1. A. 1285\frac{128}{5}
  2. B. 5128\frac{5}{128}
  3. C. 645\frac{64}{5}
  4. D. 3245\frac{32}{45}
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Answer: (A) 1285\frac{128}{5}

Use 1λ=RZ2(1n12−1n22)\frac1\lambda=RZ^2\left(\frac1{n_1^2}-\frac1{n_2^2}\right). First Balmer line of H (Z=1Z=1, 3→23\to2): 1λ1=R(14−19)=5R36\frac{1}{\lambda_1}=R\left(\frac14-\frac19\right)=\frac{5R}{36}. Second Lyman line of He+\mathrm{He^+} (Z=2Z=2, 3→13\to1): 1λ2=4R(1−19)=32R9\frac1{\lambda_2}=4R\left(1-\frac19\right)=\frac{32R}{9}. Therefore λ1λ2=32/95/36=32×369×5=1285\frac{\lambda_1}{\lambda_2}=\frac{32/9}{5/36}=\frac{32\times36}{9\times5}=\frac{128}{5}. The second Lyman line is 3→13\to1 (not 2→12\to1, which is the first).
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