Question

Answer the following by appropriately matching the lists based on the information given in the paragraph. Consider Bohr’s model of a one-electron atom where the electron moves around the nucleus. In the following List-I contains some quantities for the $n^{th}$ orbit of the atom and List-II contains options showing how they depend on $n$.

	List 1		List 2
I	Radius	P	$\propto n^{-1}$
II	Angular Momentum	Q	$\propto n^{-2}$
III	Kinetic Energy	R	$\propto n^{-0}$
IV	Potential Energy	S	$\propto n^1$
		T	$\propto n^2$
		U	$\propto n^{\frac{1}{2}}$

Answer 1

Radius:

1. The radius of $n^{th}$ orbit in Bohr's model for a one-electron atom is,
   $r_n=0.529\left(\frac{n^2}{Z}\right)\stackrel{o}{A}$ ($Z=$atomic number)
2. $r_n\propto n^2$

Thus, (I) Radius - (T)$\propto n^2$

Angular Momentum:

1. The angular momentum of the electron in $n^{th}$ orbit is,
   $mv{r}_n=\frac{nh}{2\pi }$ ($h=$Planck's constant)
2. $mv{r}_n\propto n^1$

Thus, (II) Angular momentum - (S)$\propto n^1$

Kinetic Energy:

1. The kinetic energy of the electron in $n^{th}$ orbit is,
   $\begin{array}{rcl}K.E& =& \frac{1}{2}m{v_n}^2;vn=2.18\times {10}^6\times \frac{Z}{n}\\ & =& \frac{1}{2}m{\left(2.18\times {10}^6\times \frac{Z}{n}\right)}^2\\ & & \end{array}$(where $v_n=$ The velocity of the electron in $n^{th}$ orbit)
2. $K.E\propto n^{-2}$

Thus, (III) Kinetic energy - (Q)$\propto n^{-2}$

Potential energy:

1. The potential energy of the electron in $n^{th}$ orbit is,
   $P.E=-2K.E$
2. As we know,
   $$\begin{gathered} K.E\propto n^{-2} \\ \Rightarrow P.E\propto n^{-2} \end{gathered}$$

Thus, (IV) Potential energy - (Q)$\propto n^{-2}$

Hence,

(I) Radius	(T)$\propto n^2$
(II) Angular momentum	(S)$\propto n^1$
(III) Kinetic energy	(Q)$\propto n^{-2}$
(IV) Potential energy	(Q)$\propto n^{-2}$