Calculate the radius of the third Bohr orbit ( $n = 3$ ) of a Li $^{2 +}$ ion (nuclear charge $Z = 3$ ). Use the Bohr radius $a_{0} = 0.529 \overset{˚}{A}$ .

medium 3 marks Class 12 · Physics · Atoms

Answer

1.59\,

\overset{˚}{A}

\;(=159\,

pm

)

The correct answer is 1.59\, $\overset{˚}{A}$ \;(=159\, $pm$ ) — Round to two significant figures: $r_{3} \approx 1.59 \overset{˚}{A}$ , which is $159 pm$ .

Step-by-step solution

The Bohr radius for an orbit $n$ in a hydrogen‑like ion is $r_{n} = a_{0} \frac{n ^{2}}{Z}$ .
Substitute $n = 3$ , $Z = 3$ and $a_{0} = 0.529 \overset{˚}{A}$ : $r_{3} = 0.529 \overset{˚}{A} \times \frac{3 ^{2}}{3} = 0.529 \overset{˚}{A} \times 3$ .
Calculate the product: $0.529 \times 3 = 1.587 \overset{˚}{A}$ .
Round to two significant figures: $r_{3} \approx 1.59 \overset{˚}{A}$ , which is $159 pm$ .

Common mistake: Using

r_{n} = a_{0} n^{2} Z

instead of dividing by

Z

, which overestimates the radius.

Concept tested: Bohr radius for hydrogen‑like ions

Published 15 September 2026 · How this solution was prepared and checked

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Calculate the radius of the third Bohr orbit (n=3) of a Li2+ ion (nuclear charge Z=3). Use the Bohr radius a0​=0.529A˚.

Step-by-step solution

Calculate the radius of the third Bohr orbit ( $n = 3$ ) of a Li $^{2 +}$ ion (nuclear charge $Z = 3$ ). Use the Bohr radius $a_{0} = 0.529 \overset{˚}{A}$ .