Radius of orbit

source : Hydrogen Atom

## Radius of orbit –

*Consider an electron of charge ‘e’ revolving around nucleus of charge ‘Ze’,where*

*Z = atomic number*

*e = charge on proton*

*suppose ‘m’ is the mass of the electron and ‘r’ is the radius of the orbit and ‘v’ is the tangential velocity of the revolving electron.*

*The electrostatic force of attraction between the nucleus and the electron ( according to Coulomb law )*

*= KZe x e/r ^{2} = KZe^{2}/r^{2}*

*where , K = 1/4 πε_{0 = }constant*

*Numerical value of K in SI unit = 9 x 10 ^{9} N m^{2}/C^{2} but in CGS units K = 1*

*Centrifugal force acting on the electron= mv ^{2}/r*

*According to Bohr, these two opposing forces must be balancing each other exactly to keep the electron in orbit,*

*KZe ^{2}/r^{2} = mv^{2}/r*

*For hydrogen , Z = 1 ,hence,*

*KZe ^{2}/r^{2} =mv^{2}/r*

*KZe ^{2} /r = mv^{2} ———— eq 1*

*According to one postulate of Bohr theory ( Electron can revolve in those orbits whose angular momentum is ‘h /2 ***π ‘ or its integral multiple).**

**π ‘ or its integral multiple).**

*mvr = nh/2 π *

*v = nh/2 πmr ——–eq 2*

*putting the value of ‘v’ in eq 1*

*KZe ^{2} /r = m(nh/2πmr)^{2}*

*r = n*^{2}h^{2}/4**π**^{2}me^{2}KZ

^{2}h

^{2}/4

**π**

^{2}me^{2}KZ*For hydrogen , Z = 1 ,hence,*

*r = n*^{2}h^{2}/4**π**^{2}me^{2}K

^{2}h

^{2}/4

**π**

^{2}me^{2}K*h = 6.625 x 10 ^{-34} , π = 3.14 , m = 9.1 x 10^{-31}, e = 1.6 x 10^{-19} , K = 9 x 10^{9} Nm^{2}/C^{2}*

*r = n ^{2} x 6.625 x 10^{-34} x 6.625 x 10^{-34} / 4 x 3.14 x 3.14 x 9.1 x 10^{-31} x 1.6 x 10^{-19} x 1.6 x 10^{-19} x 9 x 10^{9}*

*r = 0.529 x 10*^{-10} x n^{2} metre

^{-10}x n

^{2}metre

*radius ‘r’ = 0.529 x n*^{2} Å

^{2}Å

*r= 0.529 x 10*^{-8} x n^{2} cm.

^{-8}x n

^{2}cm.

*The value of ‘r’ for first orbit is called Bohr radius. We can calculate the values of ‘r’ for n = 2 , 3 , 4 etc. orbits of H- atom in excited state.*

*For n=2 , Then radius of orbit*

*r= 0.529 x 10*^{-8} x n^{2} cm = 0.529 x 10^{-8} x 2 x 2 = 2.12 x 10^{-8} cm.

^{-8}x n

^{2}cm = 0.529 x 10

^{-8}x 2 x 2 = 2.12 x 10

^{-8}cm.

*For n=3 ,*

*r= 0.529 x 10*^{-8} x n^{2} cm = 0.529 x 10^{-8} x 3 x 3 = 4.76 x 10^{-8} cm.

^{-8}x n

^{2}cm = 0.529 x 10

^{-8}x 3 x 3 = 4.76 x 10

^{-8}cm.

*For n=4 ,*

*r= 0.529 x 10*^{-8} x n^{2} cm = 0.529 x 10^{-8} x 4 x 4 = 8.46 x 10^{-8} cm.

^{-8}x n

^{2}cm = 0.529 x 10

^{-8}x 4 x 4 = 8.46 x 10

^{-8}cm.