So we can derive this expression by equating force of attraction on the electron by the nucleus to the centripetal force acting on the electron, i.e: $$ \frac{KZe^2}{r^2}= \frac{mv^2}{r},$$ where $m$ is the mass and $e$ is the charge on the electron, $Z$ is atomic no. of the $H$-like particle, $K$ is the Coulomb constant and $v$ is the tangential velocity.
Using this we can conclude that: $$ v^2=\frac{K Ze^2}{mr}$$ which shows that $$v \varpropto \frac{1}{\sqrt{r}}.$$
But also the Bohr's postulate of angular momentum states that :
$$mvr=\frac{nh}{2 \pi}$$
Which makes it $$v \varpropto \frac{1}{r}.$$
Out of the two relations which is the correct relation ?
Best Answer
A proportionality is meaningful only in a certain context. In the context of a single atom, the quantities $Z,m,e$ are constant and $n,v,r$ are interdependent on each other.
In the context of a single atom, the second proportionality is not correct because the value of $n$ is not constant (different values of $n$ correspond to different orbits). $$v=\frac{nh}{2πmr}⇒v∝\frac{n}{r}$$ The first proportionality is correct in the context of a single atom of one element; however, if you are analyzing many elements, $Z$ is also a variable. It too becomes incorrect.