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Structure of Atom question

2020 · 4 Sep · Shift 2 · Q19
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Structure of Atom question

2020 · 4 Sep · Shift 2 · Q19

JEE MainChemistryStructure of AtomMCQ+4 / −1
The shortest wavelength of HHH atom in the Lyman series is λ\lambdaλ 1. The longest wavelength in the Balmar series of He+He^+He+ is :
  1. A
    5λ19{{5{\lambda _1}} \over 9}95λ1​​
  2. B
    36λ15{{36{\lambda _1}} \over 5}536λ1​​
  3. C
    27λ15{{27{\lambda _1}} \over 5}527λ1​​
  4. D
    9λ15{{9{\lambda _1}} \over 5}59λ1​​
View written solutionFree

Correct answer: D

  1. Use the Rydberg formula

For a hydrogen-like species,

1λ=RZ2(1n12−1n22)\frac{1}{\lambda}=RZ^2\left(\frac{1}{n_1^2}-\frac{1}{n_2^2}\right)λ1​=RZ2(n12​1​−n22​1​)

where ZZZ is the atomic number.

  1. Find λ1\lambda_1λ1​: shortest wavelength of H atom in Lyman series

For hydrogen, Z=1Z=1Z=1.

In the Lyman series, transitions end at n1=1n_1=1n1​=1. The shortest wavelength corresponds to the largest energy gap, i.e. n2=∞n_2=\inftyn2​=∞.

So,

1λ1=R(1−0)=R\frac{1}{\lambda_1}=R\left(1-0\right)=Rλ1​1​=R(1−0)=R

Hence,

λ1=1R\lambda_1=\frac{1}{R}λ1​=R1​
  1. Find the longest wavelength in Balmer series of He+He^+He+

For He+He^+He+, Z=2Z=2Z=2.

In the Balmer series, transitions end at n1=2n_1=2n1​=2. The longest wavelength corresponds to the smallest energy gap, so take the nearest higher level:

n2=3n_2=3n2​=3

Thus,

1λ=R(2)2(122−132)\frac{1}{\lambda}=R(2)^2\left(\frac{1}{2^2}-\frac{1}{3^2}\right)λ1​=R(2)2(221​−321​) 1λ=4R(14−19)\frac{1}{\lambda}=4R\left(\frac{1}{4}-\frac{1}{9}\right)λ1​=4R(41​−91​) 1λ=4R(9−436)=4R⋅536\frac{1}{\lambda}=4R\left(\frac{9-4}{36}\right)=4R\cdot \frac{5}{36}λ1​=4R(369−4​)=4R⋅365​ 1λ=20R36=5R9\frac{1}{\lambda}=\frac{20R}{36}=\frac{5R}{9}λ1​=3620R​=95R​

So,

λ=95R\lambda=\frac{9}{5R}λ=5R9​

But since

λ1=1R\lambda_1=\frac{1}{R}λ1​=R1​

we get

λ=95λ1\lambda=\frac{9}{5}\lambda_1λ=59​λ1​
  1. Match with the options
λ=9λ15\boxed{\lambda=\frac{9\lambda_1}{5}}λ=59λ1​​​

So the correct option is:

D

  1. Comparison with stored answer

Stored correct answer: D

Our derived answer: D

They match.

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