Radii of the Bohr Orbits

Setting up the two conditions together. Section 1.4 gave two conditions that any allowed Bohr orbit of a hydrogen-like atom (nuclear charge $+Ze$, one orbiting electron) must simultaneously satisfy: the force-balance condition (Postulate 1) and the angular-momentum quantisation condition (Postulate 2). Solving these two equations together, for a given $n$, pins down the exact radius $r_n$ of that orbit.

From the force-balance condition:

From the quantisation condition, $mvr=nh/2\pi$, so $v = \dfrac{nh}{2\pi m r}$, and hence

Combining the two expressions for $v^2$:

So the radius of the $n$th permitted orbit of a hydrogen-like atom of atomic number $Z$ is

Numerical value and the Bohr radius. Substituting the known values of $\epsilon_0$, $h$, $m$ (electron mass) and $e$ for the case $Z=1$, $n=1$ (the innermost orbit of ordinary hydrogen) gives $r_1 = 0.529\times10^{-10}\ \text{m} = 0.529$ Å (Numerical 1 carries out this substitution in full). This particular value is called the Bohr radius, usually written $a_0$, and it sets the natural size scale for the hydrogen atom. Since $r_n\propto n^2/Z$, every other allowed orbit's radius can be written directly in terms of it:

$r_n = \frac{0.529\,n^2}{Z}\ \text{Å} \qquad \text{(hydrogen: } r_n = n^2a_0\text{)}$ …