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Question: The radius of the first Bohr orbit is \({a_0}\)​ . The \[{n^{th}}\] orbit has a radius: A) \(n{a_0...

The radius of the first Bohr orbit is a0{a_0}​ . The nth{n^{th}} orbit has a radius:
A) na0n{a_0}
B) a0n\dfrac{{{a_0}}}{n}
C) n2a0{n^2}{a_0}
D) a0n2\dfrac{{{a_0}}}{{{n^2}}}

Explanation

Solution

The radius of Bohr’s first orbit is given, we know Bohr gave the model for hydrogen atom. We can use the formula given by him for the radius of nth{n^{th}} orbit. From there, we can find the required value of radius.
The known value of first orbit’s radius of hydrogen atom is 0.529A∘0.529\mathop A\limits^\circ

Complete step by step answer:
According to Bohr’s model of hydrogen atom, the radius of the orbit for nth{n^{th}} orbit is given as:
rn=ε0n2h2πmZe2{r_n} = \dfrac{{{\varepsilon _0}{n^2}{h^2}}}{{\pi mZ{e^2}}} where,
n is the principal quantum number of the orbit, h is Planck’s constant, e is charged on an electron, r is the radius and Z is atomic number.
All the other quantities on R.H.S in the above equation are constant except n and Z. so, radius will be directly proportional to these (in the form as in the equation)
⇒rn∝n2Z\Rightarrow {r_n} \propto \dfrac{{{n^2}}}{Z}
The value of ε0h2πme2\dfrac{{{\varepsilon _0}{h^2}}}{{\pi m{e^2}}} is 0.529A∘0.529\mathop A\limits^\circ
⇒rn=0.529×n2Z\Rightarrow {r_n} = 0.529 \times \dfrac{{{n^2}}}{Z}
The radius of first orbit of hydrogen atom has known value of 0.529A∘0.529\mathop A\limits^\circ and the question, it is given to be a0{a_0}, so the equation becomes:
⇒rn=a0×n2Z\Rightarrow {r_n} = {a_0} \times \dfrac{{{n^2}}}{Z}
The atomic number of hydrogen atoms is 1 so the value of Z also becomes 1. The value of radius in nth{n^{th}} orbit will be given by:
⇒rn=n2a0(∵Z=1)\Rightarrow {r_n} = {n^2}{a_0}\left( {\because Z = 1} \right)

So, the correct answer is “Option C”.

Note:
The formula of Bohr’s radius used here is applicable to isoelectronic to hydrogen i.e. having the same number of electrons as that in hydrogen atom.
We generally denote Bohr’s radius by a0{a_0} only or sometimes by rbohr{r_{bohr}}.
A∘\mathop A\limits^\circ used in the question is the units called angstrom and the radius of orbits in atoms are generally measured in this unit only because it is a very small quantity compared to metres. (1A∘=10−11m)\left( {1\mathop A\limits^\circ = {{10}^{ - 11}}m} \right)