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. $v^{2}rL$
• 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.
$v{r}^{2}L\propto \frac{1}{n^3}\cdot n^4\cdot n=n^2$
$v^{2}rL\propto \frac{1}{n^6}\cdot n^2\cdot n=\frac{1}{n^3}$
$vr{L}^2\propto \frac{1}{n^3}\cdot n^2\cdot n^2=n$