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MCQMediumJEE 2023Bohr Model & Hydrogen Spectrum

JEE Chemistry 2023 Question with Solution

Which transition in the hydrogen spectrum would have the same wavelength as the Balmer type transition from n=4n = 4 to n=2n = 2 of He+^+ spectrum?

  • A

    n=2 to 1n = 2 \, \text{to} \, 1

  • B

    n=1 to 3n = 1 \, \text{to} \, 3

  • C

    n=1 to 2n = 1 \, \text{to} \, 2

  • D

    n=3 to 4n = 3 \, \text{to} \, 4

Answer

Correct answer:C

Step-by-step solution

Standard Method

Given: The He+^+ transition is from n=4n = 4 to n=2n = 2. We need the hydrogen transition having the same wavelength.

Find: The corresponding transition in the hydrogen spectrum.

Using the Rydberg relation for hydrogen-like species:

1λ(H)=R(1)2(1n12−1n22)\frac{1}{\lambda(H)} = R(1)^2 \left( \frac{1}{n_1^2} - \frac{1}{n_2^2} \right)

and for He+^+:

1λ(He+)=R(2)2(122−142)\frac{1}{\lambda(\text{He}^+)} = R(2)^2 \left( \frac{1}{2^2} - \frac{1}{4^2} \right)

Comparison Method

Given that

λ(H)=λ(He+)\lambda(H) = \lambda(\text{He}^+)

we equate the two expressions:

R(1)2(1n12−1n22)=R(2)2(122−142)R(1)^2 \left( \frac{1}{n_1^2} - \frac{1}{n_2^2} \right) = R(2)^2 \left( \frac{1}{2^2} - \frac{1}{4^2} \right)

On comparing, the solution concludes:

n1=1,n2=2n_1 = 1, \quad n_2 = 2

So the corresponding hydrogen transition is from n=2n = 2 to n=1n = 1.

Therefore, the correct option is C. The solution marks option C as correct, even though the answer key states option (1); this source discrepancy is resolved in favour of the solution.

Common mistakes

  • Students often ignore that He+^+ is a hydrogen-like ion with nuclear charge Z=2Z = 2. This changes the spectral expression. Use the hydrogen-like Rydberg relation with the appropriate factor for He+^+.

  • A common mistake is to reverse the transition direction and read n=1 to 2n = 1 \text{ to } 2 instead of emission from n=2 to 1n = 2 \text{ to } 1. Match the lower and upper levels carefully before selecting the option.

  • Some students select the option from the answer key key without checking the working. Here the solution concludes option C, so the worked solution must be treated.

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