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Atoms and Nuclei question

2023 · 13 Apr · Shift 1 · Q66
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Atoms and Nuclei question

2023 · 13 Apr · Shift 1 · Q66

JEE MainPhysicsAtoms and NucleiNumerical+4 / −1
The radius of 2nd 2^{\text {nd }}2nd  orbit of He+\mathrm{He}^{+}He+ of Bohr's model is r1r_{1}r1​ and that of fourth orbit of Be3+\mathrm{Be}^{3+}Be3+ is represented as r2r_{2}r2​. Now the ratio r2r1\frac{r_{2}}{r_{1}}r1​r2​​ is x:1x: 1x:1. The value of xxx is ‾\underline{\hspace{2cm}}​.
Numerical answer
View written solutionFree

Correct answer: 2

  1. In Bohr's model, the radius of the nthn^{\text{th}}nth orbit for a hydrogen-like ion is
rn=a0n2Zr_n = a_0\frac{n^2}{Z}rn​=a0​Zn2​

where:

  • a0a_0a0​ is the Bohr radius,
  • nnn is the orbit number,
  • ZZZ is the atomic number.
  1. For He+\mathrm{He}^+He+:
  • Z=2Z=2Z=2
  • orbit number n=2n=2n=2

So,

r1=a0222=a042=2a0r_1 = a_0\frac{2^2}{2} = a_0\frac{4}{2} = 2a_0r1​=a0​222​=a0​24​=2a0​
  1. For Be3+\mathrm{Be}^{3+}Be3+:
  • Z=4Z=4Z=4
  • orbit number n=4n=4n=4

So,

r2=a0424=a0164=4a0r_2 = a_0\frac{4^2}{4} = a_0\frac{16}{4} = 4a_0r2​=a0​442​=a0​416​=4a0​
  1. Now the ratio is
r2r1=4a02a0=2\frac{r_2}{r_1} = \frac{4a_0}{2a_0} = 2r1​r2​​=2a0​4a0​​=2

Thus,

r2r1=2:1\frac{r_2}{r_1} = 2:1r1​r2​​=2:1

So, x=2x=2x=2.

  1. Comparison with stored answer:
  • Derived answer: 222
  • Stored correct answer: 222

They match.

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