Question

The magnitude of angular momentum, orbit radius and frequency of revolution of electron in hydrogen atom corresponding to quantum number $n$ are $L, r$, and $v$ respectively. Then, according to Bohr's theory of hydrogen atom, is constant for all orbits.

• A. $v r^{2} L$
• B. urL
• C. $v^{2} r L$
• D. v $r L^{2}$

Answer 1

Answer: (B)

For hydrogen atom, $r \propto n^{2}, L \propto n$
Also, $L=m r^{2} \omega=m r^{2} 2 \pi v$
or $v=\frac{L}{2 \pi m r^{2}} $
$\therefore v \propto \frac{n}{\left(n^{2}\right)^{2}}=\frac{1}{n^{3}} $
$\therefore vrL \propto \frac{1}{n^{3}} \cdot n^{2} \cdot n=1$
$\Rightarrow vrL $ is constant for all orbits.
$v r^{2} L \propto \frac{1}{n^{3}} \cdot n^{4} \cdot n=n^{2}$
$v^{2} r L \propto \frac{1}{n^{6}} \cdot n^{2} \cdot n=\frac{1}{n^{3}}$
$v r L^{2} \propto \frac{1}{n^{3}} \cdot n^{2} \cdot n^{2}=n$