(A) The $\mathrm{V}_n$ and $\mathrm{K}_n$ denotes potential energy and kinetic energy of an electron.

The expression for kinetic theory and potential energy are given by Bohr atomic theory as shown below.

$$ \begin{array}{rlrl} & \mathrm{V}_n =-\frac{\mathrm{Ze}^2}{r} \text { and } \mathrm{K}_n=\frac{\mathrm{Ze}^2}{2 r} \\ \text { Hence, } & \frac{\mathrm{V}_n}{\mathrm{~K}_n} =\frac{-\frac{\mathrm{Ze}^2}{r}}{\frac{\mathrm{Ze}^2}{2 r}} \\ & \therefore \frac{\mathrm{~V}_n}{\mathrm{~K}_n} =-2 \end{array} $$

Hence, option A in column I matches with option R in column II.

(B) The $\mathrm{E}_n$ denoted the total energy of an electron which is the sum of potential and kinetic energy of an electron.

The radius of $n^{\text {th }}$ orbit of an atom is denoted by $r_n$.

Where $n$ denoted the number of an orbit.

The radius of $n^{\text {th }}$ orbit is directly proportional to the square of $n$ and the total energy of an electron is inversely proportional to that of square of $n$.

$$ r_n \propto n^2 \text { and } \mathrm{E}_n \propto \frac{1}{n^2} $$

Therefore,

$$ \mathrm{E}_n \propto \frac{1}{r_n} \text { or } r_n \propto \mathrm{E}_n^{-1} $$

The value of $x$ will be,

$$ x=-1 $$

Hence, option B in column I matches with option Q in column II.

(C) According to the Bohr theory, angular momentum for an orbital is given as,

$$ \omega=\sqrt{l(l+1)} \frac{h}{2 \pi} $$

Here, $\omega$ is the angular momentum for an orbital, $l$ is the azimuthal quantum number and $h$ is the Planck's constant.

The lowest atomic orbital in any atom is $1 s$, which has least energy in the atom.

For $1 s$ orbital, the azimuthal quantum number is 0 as the orbital possess spherical shape.

Substituting the value of $l$ in above equation for 1s orbital,

$$ \begin{aligned} & \omega=\sqrt{0(0+1)} \frac{h}{2 \pi} \\ & \omega=0 \end{aligned} $$

Hence, the value of angular momentum for lowest orbital will be zero.

Hence, option C in column I matches with option P in column II.

(D) The Z denotes the atomic number of an element.

The radius of an orbit and atomic number of an element are related by following relation.

$$ \frac{1}{r_n} \propto Z_n $$

If $\frac{1}{r_n} \propto Z^y$, then $y=1$

Hence, option D in column I matches with option $S$ in column II.

Column I		Column II
(A)	$$\frac{V_n}{K_n} = ?$$	(P)	0
(B)	If radius of $$n^{\text{th}}$$ orbit $$\propto E_n^x$$, $$x = ?$$	(Q)	−1
(C)	Angular momentum in lowest orbital	(R)	−2
(D)	$$\frac{1}{r_n} \propto Z^y = ?$$	(S)	1

JEE Advanced Papers