Structure of Atom

Bohr's Model of Atom and Bohr's Model for Hydrogen Atom

Bohr's Atomic Model :

Angular momentum :
mvr = \frac{nh}{2\pi}
m = mass
v = velocity of electron
r = radius of orbit
h = Planck's constant

Radius of nth orbit
r_{n} = \frac{n^{2}h^{2}}{4\pi^{2}mZ_{e}^{2}}
 e - Charge of nucleus
 Z - Atomic no.of uni- electronic species
simplifying : r_{n} = 0.53 \left(\frac{n^{2}}{Z}\right) Å

Energy of electron in nth orbit of H - atom :
P.E = \frac{-Ze^{2}}{r}

K.E = \frac{1}{2}mv^{2}

Total energy : \frac{-2\pi^{2}mZ^{2}e^{4}}{n^{2}h^{2}}
=13.6 \times \frac{Z^{2}}{n^{2}} \ ev/atom
= 1313 \frac{Z^{2}}{n^{2}} \ kJ/mole
=-313.6 \times \frac{Z^{2}}{n^{2}} \
1 eV = 1.6 × 10-12 erg / atom
       = 1.6 × 10-19 J/atom
       = 23.06

Velocity of electron in nth orbit
According to mvr = \frac{nh}{2\pi}
V_{n} = \frac{2\pi Ze^{2}}{nh}
= 2.18 \times 10^{8}\left(\frac{Z}{n}\right)cm/ sec
V_{n} = 2.18 \times 10^{6}\left(\frac{Z}{n}\right)m/ sec

Rydberg constant (R) :
\frac{1}{R} = \frac{1}{109677} = 9.12 \times 10^{-6}
                       = 912 \times 10^{-8}cm
                       = 912 Å
R = \frac{2\pi^{2}me^{4}}{h^{3}C} ;\frac{1}{\lambda} = R\left[\frac{1}{n_1^2} - \frac{1}{n_2^2}\right]

Advantages of Bohr's Model :

  • Explains the stability of atom
  • Successfully explains the uni-electronic species
  • Coincide the values of velocity, energy, radius with experimental values.

Draw backs :

  • Failed to explain the spectra of multi electron species
  • Failed to explain the fine structure of spectral lines
  • Splitting up of normal spectral lines into several lines.

Wavelength :

λ in terms of kinetic energy
\lambda = \frac{h}{\sqrt{2mKE}}

λ in terms of potential difference(v)
\lambda = \frac{h}{\sqrt{2mev}}
 m - mass of particle
e - charge of electron

λ for charged particle :

for \ e^{-} = \lambda = \frac{12.27}{\sqrt{v}} Å

for \ p^{+} = \lambda = \frac{0.286}{\sqrt{v}} Å

for \ \alpha = \lambda = \frac{0.101}{\sqrt{v}} Å

for \ m = \lambda = \frac{0.286}{\sqrt{KE(in eV)}}

Calculation of (λn) :
λn= Wave length of electron wave in nth orbit species.
\frac{\lambda_{n} = \frac{h}{mv_{n}}}{\lambda_{n} = 3.33\left(\frac{n}{Z}\right) } Å

Circumference of electrons orbit
(C) = 2πrn
       = 2 \times 3.14 \times 0.53 \times 10^{-8}cm\left(\frac{n^{2}}{Z}\right)
      = 3.33\left(\frac{n^{2}}{Z}\right) Å

View the Topic in this Video from 1:15 to 41:48

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1. Number of waves in an orbit = \tt \frac{circumference \ of \ orbit}{wavelength}

2.\tt m\upsilon r=\frac{nh}{2\pi}   (n = 1, 2, 3 .....)
Where, m = mass of electron; ν = velocity of electron ;
r = radius of orbit
n = number of orbit in which electrons are present.

3.\tt \triangle E=E_{2}-E_{1}=hv=\frac{hc}{\lambda}

4.Velocity of an electron in nth Bohr orbit
\tt \left(\upsilon_{n}\right)=2.165\times10^{6}\frac{Z}{n}m/s

5.Radius of nth Bohr orbit
\tt \left(r_{n}\right)=0.53\times10^{-10}\frac{n^{2}}{Z}m=0.53\frac{n^{2}}{Z} Å

6.\tt E_{n}=-2.178\times 10^{-18}\frac{Z^{2}}{n^{2}}J/atom
\tt =-1312\frac{Z^{2}}{n^{2}}kJ/mol
\tt =-13.6\frac{Z^{2}}{n^{2}}eV/atom
\tt \triangle E=-2.178 \times 10^{-18}\left(\frac{1}{n_1^2}-\frac{1}{n_2^2}\right)Z^{2}J/atom
where, n = number of shell; Z=atomic number