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According to Bohr model of hydrogen atom, the radius of stationary orbit characterized by the principal quantum number n is proportional to


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Solution

Step 1: Some points about the Bohr model

  1. It is created by Neil and Rutherford in 1913.
  2. The structure of the system is like a solar system.
  3. An electron moves around the nucleus in a circular orbit.
  4. It emits energy in the form of light.
  5. It postulates that electrons orbit the nucleus at fixed energy levels.

Step 2: The formula of the Bohr model

Consider a hydrogen-like atom in which an electron of mass m and charge -e revolves in a circular orbit of radius r with a velocity v round a nucleus of charge +Ze. The electrostatic attractive force on the nucleus Ze24πε0r2 provides the centripetal force mv2r to keep the electron in orbit so that we can write

14πε0.Ze2r2=mv2r which gives

v2=14πε0.Ze2mr

Here ε0 is the permittivity of vacuum and has the value 10-9/36πF/m.

From Bohr's quantum condition, we have

L=mr2ω=mvr=nh

where ω is the angular velocity and v is the linear velocity of the electron in the orbit. From this equation we have

v=nhmr

Eliminating v from both the equations we get

14πε0.Ze2r2=nhmr2 which gives

r=4πε0n2h2mZe2

Hence, we get

v=Ze24πε0nh

Thus both the radius of the orbit and the electron velocity depend on the quantum number n. The orbit n=1 has the smallest radius. For hydrogen Z=1, this radius is known as the Bohr radius and is given by

a0=4πε0h2me2=0.529×10-10m=0.529A

where we have substituted the numerical values of ε0,m,e and h.

The radii of the orbits are directly proportional to n2 so that the radii of the successive orbits are in the ratios 1:4:9:16:....

rn2

Step 3: A diagram to represent the Bohr model

Bohr Model of the Atom - Overview and Examples

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 n2.


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