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In Bohr’s model of the hydrogen atom, the radius of the first orbit of an electron is ${r_0}$. Then the radius of the third orbit is?
A) $\dfrac{{{r_0}}}{9}$
B) ${r_0}$
C) $3{r_0}$
D) $9{r_0}$

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Last updated date: 19th Jul 2024
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Answer
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Hint: Bohr proposed that electrons orbited the nucleus in specific orbits or shells with a fixed radius. According to Bohr’s model, an electron would absorb energy in the form of photons to get excited to a higher energy level. The Bohr model derives a radius for the nth excited state of hydrogen-like atoms.

Formula Used:
We will be using the formula of Radius of the nth orbit of an electron. It is given by ${r_n} = {r_0}\dfrac{{{n^2}}}{Z}$.
Given: In Bohr’s model of the hydrogen atom, the radius of the first orbit is ${r_0}$ and to find the radius of the third orbit.

Complete step by step solution:
The radius of the nth orbit of an electron is given by ${r_n} = {r_0}\dfrac{{{n^2}}}{Z}$
Where, ${r_n}$- Radius of the nth Orbit
${r_0}$- Radius of the first Orbit
Z - Atomic number
\[n\] - Number of orbits
For the hydrogen atom, the Atomic number $Z = 1$
$ \Rightarrow {r_n} = {r_0}{n^2}$
So, for the third orbit $n = 3$
Thus the radius of the third orbit ${r_3} = {r_0} \times {3^2} = 9{r_0}$
The radius of the third orbit is calculated as $9{r_0}$.
Answer is Option (D), $9{r_0}$.

Note: In $1913$, Niel Bohr introduced the atomic hydrogen model. Bohr proposed that electrons travel in specific orbits, shells around the nucleus. The Bohr model is used to describe the structure of hydrogen energy levels. It is the first atomic model to explain the radiation spectra of atomic hydrogen.
Limitation of the Bohr model of the Hydrogen Atom:
It couldn’t explain why spectral lines are more intense.
It doesn’t work well for complex atoms.
Heisenberg’s uncertainty principle contradicts Bohr’s idea of electrons existing in specific orbits with a known velocity and radius.