According to Bohr model of hydrogen atom, the radius of stationary orbit characterized by the principal quantum number n is proportional to
Step 1: Some points about the Bohr model
Step 2: The formula of the Bohr model
Consider a hydrogen-like atom in which an electron of mass and charge revolves in a circular orbit of radius with a velocity round a nucleus of charge . The electrostatic attractive force on the nucleus provides the centripetal force to keep the electron in orbit so that we can write
Here is the permittivity of vacuum and has the value .
From Bohr's quantum condition, we have
where is the angular velocity and is the linear velocity of the electron in the orbit. From this equation we have
Eliminating from both the equations we get
Hence, we get
Thus both the radius of the orbit and the electron velocity depend on the quantum number . The orbit has the smallest radius. For hydrogen , this radius is known as the Bohr radius and is given by
where we have substituted the numerical values of and .
The radii of the orbits are directly proportional to so that the radii of the successive orbits are in the ratios
Step 3: A diagram to represent the Bohr model
Therefore, now according to the Bohr model of the hydrogen atom, the radius of stationary orbit characterized by the principal quantum number n is proportional to .