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Atoms and Nuclei question

2025 · 4 Apr · Shift 1 · Q60

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Atoms and Nuclei
2025 · 4 Apr · Shift 1 · Q60

Atoms and Nuclei question

2025 · 4 Apr · Shift 1 · Q60

JEE MainPhysicsAtoms and NucleiMCQ+4 / −1

Considering the Bohr model of hydrogen like atoms, the ratio of the radius of

5^{\text {th }}

orbit of the electron in

\mathrm{Li}^{2+}

and

\mathrm{He}^{+}

is

A
$\frac{3}{2}$
B
$\frac{2}{3}$
C
$\frac{4}{9}$
D
$\frac{9}{4}$

View written solutionFree

Correct answer: B

Bohr radius formula for hydrogen-like atoms

For a hydrogen-like ion, the radius of the $n^{\text{th}}$ orbit is

r_n = \frac{n^2 a_0}{Z}

where:

$n$ = principal quantum number
$a_0$ = Bohr radius
$Z$ = atomic number

Write the radius for each ion in the $5^{\text{th}}$ orbit

Since both are in the $5^{\text{th}}$ orbit, $n=5$ for both.

For $\mathrm{Li}^{2+}$ , $Z=3$

r_{\mathrm{Li}^{2+}} = \frac{5^2 a_0}{3} = \frac{25a_0}{3}

For $\mathrm{He}^{+}$ , $Z=2$

r_{\mathrm{He}^{+}} = \frac{5^2 a_0}{2} = \frac{25a_0}{2}

Find the required ratio

\frac{r_{\mathrm{Li}^{2+}}}{r_{\mathrm{He}^{+}}} = \frac{\frac{25a_0}{3}}{\frac{25a_0}{2}} = \frac{25a_0}{3} \cdot \frac{2}{25a_0} = \frac{2}{3}

Match with the options

\frac{2}{3}

So the correct option is B.

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