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 $(L)=\frac{nh}{2\pi }$
$\Rightarrow L\propto n$
Radius of an electron $=r_n=\frac{n^2}{m}{\left(\frac{h}{2\pi }\right)}^2\frac{4\pi {\epsilon }_0}{e^2}$
$\Rightarrow r_n\propto n^2$
Velocity of electrons
$v_n=\frac{1}{n}\frac{e^2}{4\pi {\epsilon }_0}\cdot \frac{1}{(h/2\pi )}{v}_n\propto 1/n$
Potential energy i.e., $E_n=-\frac{m{e}^4}{8n^2{\epsilon }_0^2{h}^2}$
$\therefore E_n\propto 1/n^2$