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Atoms and Nuclei question

2019 · 8 Apr · Shift 1 · Q51
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Atoms and Nuclei question

2019 · 8 Apr · Shift 1 · Q51

JEE MainPhysicsAtoms and NucleiMCQ+4 / −1
Radiation coming from transitions n = 2 to n = 1 of hydrogen atoms fall on He+ ions in n = 1 and n = 2 states. The possible transition of helium ions as they absorb energy from the radiation is :
  1. A
    n = 1 →\to→ n = 4
  2. B
    n = 2 →\to→ n = 5
  3. C
    n = 2 →\to→ n = 4
  4. D
    n = 2 →\to→ n = 3
View written solutionFree

Correct answer: C

  1. Energy of radiation emitted by hydrogen for the transition n=2→1n=2 \to 1n=2→1

For hydrogen, En=−13.6n2 eVE_n=-\frac{13.6}{n^2}\,\text{eV}En​=−n213.6​eV

So the photon emitted in the transition 2→12 \to 12→1 has energy ΔE=13.6(1−14)=13.6⋅34=10.2 eV\Delta E = 13.6\left(1-\frac{1}{4}\right)=13.6\cdot \frac{3}{4}=10.2\,\text{eV}ΔE=13.6(1−41​)=13.6⋅43​=10.2eV

Thus, the incident radiation has energy: Eγ=10.2 eVE_\gamma=10.2\,\text{eV}Eγ​=10.2eV

  1. Energy levels of He+\text{He}^+He+ ion

He+\text{He}^+He+ is a hydrogen-like ion with atomic number Z=2Z=2Z=2.

Hence, En=−13.6Z2n2=−54.4n2 eVE_n=-\frac{13.6Z^2}{n^2}=-\frac{54.4}{n^2}\,\text{eV}En​=−n213.6Z2​=−n254.4​eV

A transition from nin_ini​ to nfn_fnf​ requires absorption energy ΔE=54.4(1ni2−1nf2)\Delta E = 54.4\left(\frac{1}{n_i^2}-\frac{1}{n_f^2}\right)ΔE=54.4(ni2​1​−nf2​1​) with nf>nin_f>n_inf​>ni​.

  1. Check each option

Option A: n=1→4n=1 \to 4n=1→4

=54.4\cdot \frac{15}{16}=51.0\,\text{eV}$$ This is not $10.2\,\text{eV}$. So, **A is incorrect**. ### Option B: $n=2 \to 5$ $$\Delta E = 54.4\left(\frac{1}{4}-\frac{1}{25}\right) =54.4\left(\frac{21}{100}\right) =11.424\,\text{eV}$$ This is not $10.2\,\text{eV}$. So, **B is incorrect**. ### Option C: $n=2 \to 4$ $$\Delta E = 54.4\left(\frac{1}{4}-\frac{1}{16}\right) =54.4\cdot \frac{3}{16} =10.2\,\text{eV}$$ This exactly matches the incident photon energy. So, **C is correct**. ### Option D: $n=2 \to 3$ $$\Delta E = 54.4\left(\frac{1}{4}-\frac{1}{9}\right) =54.4\cdot \frac{5}{36} \approx 7.56\,\text{eV}$$ This is not $10.2\,\text{eV}$. So, **D is incorrect**. --- 4. **Conclusion** The only possible transition of $\text{He}^+$ that can absorb the hydrogen $2\to1$ radiation is $$n=2 \to 4$$ Therefore, the correct option is **C**.
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