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Kelvi

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Kelvi

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If the radius of the second Bohr orbit of the He $^+$ ion is $105.8$ pm, what is the radius of the third Bohr orbit of the Li $^{2+}$ ion?

medium

Structure of Atom

2022

chemistry

$158.7$ pm

$15.87$ pm

$1.587$ pm

$158.7$ Å

Explanation

To solve this problem, we will use the formula for the radius of the $n$ -th Bohr orbit: $\\ r_n = \frac{n^2 h^2}{4 \pi^2 m e^2 Z} \\$

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where:

\\

•

n

is the principal quantum number (orbit number),

\\

•

h

is Planck's constant,

\\

•

m

is the electron mass,

\\

•

e

is the elementary charge,

\\

•

Z

is the atomic number of the ion.

\\

The radius of the

n

-th orbit can also be expressed as:

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

where

a_0

is the Bohr radius (approximately

52.9

pm).

\\

For the He

^+

ion in the second orbit:

\\ r_2 = \frac{2^2 a_0}{2} = 105.8

pm

\\

We can solve for

a_0: \\ a_0 = \frac{105.8 \times 2}{4} \\ a_0 = 52.9

pm

\\

Now, for the Li

^{2+}

ion in the third orbit:

\\ r_3 = \frac{3^2 a_0}{3} \\ r_3 = \frac{9 \times 52.9}{3} \\ r_3 = 158.7

pm

\\

Therefore, the radius of the third Bohr orbit of the Li

^{2+}

ion is

158.7

pm.

\\

This corresponds to Option 1.