## 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 n ^{th} 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}}

Simplifying

=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}} \ k.cal/mole

1 eV = 1.6 × 10^{-12} erg / atom

= 1.6 × 10^{-19} J/atom

= 23.06 k.cal/mole

**Velocity of electron in n ^{th} 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 (λ _{n}) :**

λ

_{n}= Wave length of electron wave in n

^{th}orbit species.

\frac{\lambda_{n} = \frac{h}{mv_{n}}}{\lambda_{n} = 3.33\left(\frac{n}{Z}\right) } Å

**Circumference of electrons orbit**

(C) = 2πr_{n}

= 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