Using Bohr's postulates of the atomic model, derive the expression for radius of th electron orbit. Thus, obtain the expression of Bohr's radius.

Explanation

We know that when an electron revolves in a stable orbit, the centripetal force is provided by electrostatic force of attraction acting on it due to a proton present in the nucleus.
`therefore " " (m v_(n)^(2))/(r_(n)) = (1)/(4pi in_(0)) .(e^(2))/(r_(n)^(2)) or v_(n) = (nh)/( 2pi m r_(n))" ".....(i)`
and from Bohr.s quantum condition , we have .
`m v_(n) r_(n) =(n h)/(2 pi) or v_(n) =(n h)/( n h)/(4 pi in_(0) m r_(n))" "........(ii) `
Squaring (ii) and then equating it with (i) , we get .
`(n^(2)h^(2))/(4 pi^(2) m^(2) r_(n)^(2)) = (e^(2))/(4 pi in_(0) m.r_(n)) rArr r_(n) = (n^(2) h^(2))/(4pi^(2) m^(2)) xx (4 pi in_(0) m)/(e^(2)) = ( in_(0)h^(2))/( pi m e^(2)) = n^(2)`
In stable orbit of hydrogen atom n = 1 and then radius of 1st orbit is called Bohr.s radius `a_(0)` . Obviously ` a_(0) = (in_(0) h^(2))/(pi m e^(2))`