Calculation of radius of orbit:



Derivation:



Electrons revolves in orbit.

Centripetal force acting on electron away from centre & force of attraction towards centre.


For electron to revolve in same orbit.




[
= 1 in CGS unit] ....................... (i)


mvr = n     ....................................................... (ii)


v =

Putting value of v from eq. (ii) in eq. (i)



   .................................................... (iii)


r₀  =

= 0.529 A⁰ Bohr radius

For H-like atom, like He⁺
,






Calculation energy of electron:




Total energy of electron (E) = K.E. + P.E.


=mv²-






Dumb Question

: Why P.E. is -  ?




Ans: P.E. is work done when electron moves from_{to r}.


P.E. =






Dumb Question
: Why Force is -ve ?




Ans: Force is work attractive. So, it is taken as -ve.


From eq. (i)   mv² =

E =-


= -






Dumb Question
: What does -ve sign signify ?




Ans: -ve sign show's that electron is bound to that orbit & atom.


E = -


Substituting value of r from eq. (iii) ..................



Calculation of velocity of electron in any orbit :




Substituting value of r from (ii) in (i)


v =


For H-like atom,

v_n  = x 2.188 x 10⁸ cms ^-1

For H-atom, putting z = 1

v_n =    cms^-1  From eq. (i)

v²  =









Calculation of no. of revolutions of electron in an orbit per sec
:
mvr = n


v =


No. of rev./sec =





No. of revolutions per sec =
=






Calculation of no. of waves in any orbit
:




No. of waves in any orbit =






= De Broglie relation.




Waver no.
: It is reciprocal of wavelength.


For H-atom    (wave no.)
  = R

RRydberg constant



R = 1.097 x 10^-7

For Lyman Series n₁
= 1,    n₂
= 2, 3, 4, .....................................


For Balmer Series n₁= 2,    n₂ = 3, 4, 5, .....................................


For Paschen Series n₁= 3,    n₂ = 4, 5, 6, .....................................


For Brackett Series n₁= 4,    n₂ = 5, 6, 7, .....................................


For Pfund Series n₁ = 5,    n₂ = 6, 7, 8, .....................................


For H-like atom,



= Rz²
z - atomic No.

when n₂
in Redbergis formula is

i.e. n₂=






De Broglie Relation
:




Matters have dual nature of particle & wave If assumed as wave, its energy.


E = h
Plank's quantum theory .................................... (i)


If assumed as particle, its energy.


E = mc²
Einstein Eq. .................................................. (ii)


Equating (1) & (2)


h= mc²

sinc  =


h= mc²

=

Deg broglie pointed that this eq. can be applicable to any particle.



=

=



= de broglie wavelength.