Atomic Theory |

1.
Rutherford's
Model of the atom |
2.
The Bohr Atom |

__Rutherford's Model of the atom:__

Rutherford proposed the nuclear model of
the atom on the basis of his experimental results of the scattering of alpha-particles by
the atoms. In his experiment alpha-particles emitted with speeds of about 2 x 10^{7}
m/s struck a thin gold foil several thousand atomic layers thick. Most of the
alpha-particles pass undeflected through the foil, but some were scattered at some angle.
According to Rutherford's model of the atom, almost the entire mass and the total positive
charge of the atom are confined within a very small part at the center of the atom. This
part is known as the **nucleus** of the atom and has a radius less than 10^{-12}
cm which is small compared to the radius of the atom (approx. 10^{-8} cm). The
electron in the atom revolves in orbits around this central core. The radii of these
orbits determine the **atomic radii**.

Please on the link below to see a simulation of Rutherford's alpha-particle scattering:

Simulation of Rutherford's alpha-particle scattering

**Drawback of Rutherford's
model:**

According to electromagnetic theory an
accelerated particle loses energy by the emission of electromagnetic radiation. So, the
electrons revolving in this atomic orbits will loose energy by emitting e.m. radiation and
thus eventually will collapse onto the positively charged nucleus of the atom. *Thus
there cannot be any stable atom*! This is the main drawback of Rutherford's simple
picture of the atom.

Bohr solved this using quantum mechanical concepts.

__The Bohr atom:__

Bohr's theory of atom is based on __ three__
fundamental postulates:

(a) The electron can revolve in a number of
specified circular orbits, known as **stationary
orbits**, around the central positively
charged nucleus. In each stationary orbit, the electron possesses an orbital angular
momentum

**p = nh/(2p) = n
hbar**,

where **n=1,2,3**,... is an
integer known as the **quantum number** and **hbar = h/2****p** (**h**
is the Planck's constant).

(b) In a stationary orbit the electron **does
not** radiate electromagnetic energy.

(c) When an electron makes a transition
from an initial stationary orbit of higher energy **E _{i}** to a
final stationary orbit of lower energy

**hf = E _{i} -
E_{f} ,**

where **f **is the frequency
of radiation. The above condition is known as **Bohr's
frequency condition**.

If the energy **E _{i}**
of the initial orbit is less than the energy

**Radius and Energy of the Bohr
orbit:**

Consider the mass and charge of the
electron to be **m** and **-e** and the nuclear charge is **+Ze**.
Also we assume that the nuclear mass (**M**) is infinitely large
compared to the electron mass. Equating the electrostatic attraction between the nucleus
and the electron to the centrifugal force we get:

**mv ^{2}/r =
Ze^{2}/r^{2}.
Eq. (1)**

The angular velocity **w** of
the revolving electron is related to the velocity as **v = w r**. We can
rewrite Bohr's quantum condition as follows:

**p = mw ^{2}r
= mvr = n hbar. Eq. (2)**

**r** is the **radius**
of the electron orbit. From Eqs. (1) and (2) we get -

**r = r _{n} =
n^{2} hbar^{2}/(mZe^{2}),
v = v_{n} = n
hbar/mr_{n} = Ze^{2}/(n hbar**).

The **kinetic** and **potential**
energies of the electrons are:

**T = mv ^{2}/2
= Ze^{2}/2r, V = -Ze^{2}/r**.

The **total energy** is,

**E = T + V = -Ze ^{2}/r**,

=> **E _{n} = -mZ^{2} e^{4}/(2n^{2}
hbar^{2}).
Eq. (3) **

**r _{n}** is the
radius of the

Eq. (3) gives a set of discrete energy
levels for the electron, the energy being the lowest for n=1 which is known as the **ground state** of the electron.

The energy emitted in the form of radiation
when an electron makes a transition from an orbit with **n = n _{i}**
to an orbit with

**hf = E _{i} -
E_{f} = (mZ^{2}e^{4}/2 hbar^{2})(1/n_{f}^{2}
- 1/n_{i}^{2}).
Eq. (4)**

The energy needed to ionize an atom (**n _{f}
-> infinity**) which is initially in the ground state (

**I = mZ ^{2} e^{4}/
(2hbar^{2}).**

For hydrogen (Z=1), this gives **I ~
13.6 eV**.

Defining **f'= f/c = 1/****l **as the **wave number**, we get from Eq. (4) for the transition **n _{i}
-> n_{f},**

**f' = (mZ ^{2}e^{4}/4pc hbar^{2})(
1/n_{f}^{2} - 1/n_{i}^{2}) = R Z^{2} (1/n_{f}^{2}
- 1/n_{i}^{2})**,

for the emission of radiation (**n _{f}
< n_{i}**). For absorption

Note that **R = me ^{4}/4 p c hbar^{2} = 109, 737 cm^{-1}**
is a

For a given value of the final state
quantum number **n _{f}**, transitions from different initial states
give rise to lines of a particular

In the case of hydrogen (**Z=1**)
a number of these series have been identified:

**Lyman series:****n _{f}**=
1,

**f' = R _{H}
(1 - 1/n_{i}^{2} )**

**Balmer series:**** n _{f}**=
2,

**f' = R _{H}
(1/4 - 1/n_{i}^{2} )**

**Paschen series:**** n _{f}**= 3,

**f' = R _{H}
(1/9 - 1/n_{i}^{2} )**

**Brackett series:**** n _{f}**= 4,

**f' = R _{H}
(1/16 - 1/n_{i}^{2} )**

The different spectral series of hydrogen experimentally measured are in excellent agreement with Boh'r theory. The experimental verification of Bohr's theory constitues another triumph of the quantum hypothesis.

To view the **energy levels**
of the **Hydrogen atom** please
below -

Please on the hyperlink below to see a simulation of the Bohr atom:

**Some cool sites on the Bohr
atom:**

- http://www.colorado.edu/physics/2000/quatumzone/bohr.html
- http://heppc19.phys.nwu.edu/~anderson/java/vpl/atomic/hydrogen.html
- http://home.a-city.de/walter.fendt/physengl/bohrengl.htm

**© Kingshuk Majumdar (2000)**