Here, for different values of v ϵ (0,∞), we can have different values for r ϵ (0,∞) - but we know from Bohr's postulates that not all values of r are allowed. So there must be some dependence on a discrete variable (a variable that can take only certain values). This discreteness can be obtained by applying Bohr's third postulate - angular momentum can only be integer multiples of h2π. From mechanics, we know that the angular momentum of a point mass m moving in a circle of radius r with a speed v is equal to mvr. We can write -
Substituting the value of v from equation (2) in equation (1), we obtain the discreteness in the orbital radius that we were looking for –
Cancelling r from both numerators and rearranging,
The term in the brackets is just a bunch of constants. For Z=1, putting all S.I values of these constants gives us the orbital radius r for n=1 in the hydrogen atom, also called the Bohr radius (denoted by a0), numerically equal to 52.9 pm or 0.529 A ̇.