Question

For atomic model of hydrogen atom given by Niels Bohr, match the following proportionalities.

Column I		Column II
A	Angular momentum	1	$1 / n$
B	Velocity of electron	2	$n^{2}$
C	Radius of electron	3	$1 / n^{2}$
D	Energy of electron	4	$n$

• A. A-(1), B-(2), C-(3), D-(4)
• B. A-(4), B-(3), C-(2), D-(1)
• C. A-(4), B-(1), C-(3), D-(2)
• D. A-(4), B-(1), C-(2), D-(3)

Answer 1

Answer: (D)

As we know, angular momentum $(L)=\frac{n h}{2 \pi}$
$\Rightarrow L \propto n$
Radius of an electron $=r_{n}=\frac{n^{2}}{m}\left(\frac{h}{2 \pi}\right)^{2} \frac{4 \pi \varepsilon_{0}}{e^{2}}$
$\Rightarrow r_{n} \propto n^{2}$
Velocity of electrons
$v_{n}=\frac{1}{n} \frac{e^{2}}{4 \pi \varepsilon_{0}} \cdot \frac{1}{(h / 2 \pi)} v_{n} \propto 1 / n$
Potential energy i.e., $E_{n}=-\frac{m e^{4}}{8 n^{2} \varepsilon_{0}^{2} h^{2}}$
$\therefore E_{n} \propto 1 / n^{2}$