Question

According to Bohr's theory $E_{n}=$ total energy, $K_{n}=$ kinetic energy, $V_{n}=$ potential energy, $r_{n}=$ radius of $n^{\mathrm{th}}$ orbit. Match the items of column I with values of column II

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

• A. $\left(A \rightarrow Q\right) \left(B\rightarrow S\right)\left(C\rightarrow P\right)\left(D \rightarrow S\right)$
• B. $\left(A \rightarrow R\right) \left(B\rightarrow Q\right)\left(C\rightarrow P\right)\left(D \rightarrow S\right)$
• C. $\left(A \rightarrow P\right) \left(B\rightarrow S\right)\left(C\rightarrow Q\right)\left(D \rightarrow R\right)$
• D. $\left(A \rightarrow P\right) \left(B\rightarrow R\right)\left(C\rightarrow S\right)\left(D \rightarrow Q\right)$

Answer 1

Answer: (B)

According to Bohr's theory
Potential energy $\left(V_{n}\right)=-\frac{1}{4 \pi \varepsilon_{0}} \frac{Z e^{2}}{r} ;$
Kinetic energy $\left(K_{n}\right)=\frac{1}{8 \pi \varepsilon_{0}} \frac{Z e^{2}}{r}$
Radius of $n^{\text {th }}$ orbit $\left(r_{n}\right)=\frac{n^{2} h^{2} \varepsilon_{0}}{\pi m Z e^{2}}$
Potential energy $\left(E_{n}\right)=-\frac{m e^{4}}{8 \varepsilon_{0}^{2} h^{2}}\left(\frac{1}{n^{2}}\right)$
$\frac{V_{n}}{K_{n}}=\frac{-\frac{1}{4 \pi \varepsilon_{0}} \frac{Z e^{2}}{r}}{\frac{1}{8 \pi \varepsilon_{0}} \frac{Z e^{2}}{r}}=-2 \quad(\mathrm{~A} \rightarrow \mathrm{R})$
$E_{n} \propto \frac{1}{n^{2}} \propto \frac{1}{r_{n}}$
$r^{n} \rightarrow E_{n}^{x} \quad \therefore \quad x=-1 . \quad(B \rightarrow Q)$
$l$, the orbital quantum number, is connected to the total
angular momentum of the electron. This quantum number is an integer less than $n$, and the total angular momentum of the electron can be calculated using:
Total angular momentum,
$L=[l(l+1)]^{1 / 2} \frac{h}{2 \pi}(l=0,1,2, \ldots, n-1)$
The lowest energy orbital in the hydrogen atom is the $1 s$ orbital, which corresponds with $n=1, l=0$ and $m=0$.
Hence total angular moment in lowest orbital $=0$.
$(\mathrm{C} \rightarrow \mathrm{P})$
$\frac{1}{r^{n}} \propto Z^{y} $ as $r_{n} \propto \frac{1}{Z} $
$ \therefore y=1(D \rightarrow S)$