and John Venables, Dept of Physics and Astronomy, Arizona State University, Tempe, Arizona

**
The BOHR ATOM**

*"Anyone who is not shocked by quantum theory
has not understood it"
- Neils Bohr*

In 1913 Neils Bohr proposed his model of atom
which superceded Rutherford's atomic model. Though the planetary model proposed
by Rutherford was widely accepted, it fell short on many counts. The nuclear
atom proposed by Rutherford was unstable. According to classical theories this
atom should collapse. It also failed to explain the discrete spectral lines
of elements. Bohr's model of atom could successfully explain the stability of
atom by introducing ** Quantization**.
It could also explain the Hydrogen spectra. Bohr obtained the value of radius
of hydrogen atom and its energy, both of which agree well with experimental
results. Was this a coincidence!? Bohr's atomic theory formed the basis for
the

**Bohr's
Postulates **

(a) The electron revolves in circular orbits around the nucleus which are restricted by the quantization of angular momentum i.e. they revolve in orbits where the angular momentum of electron is an integral multiple of h/2π, where h is Planck's constant.

In these orbits of special
radius electron does not radiate energy as expected from Maxwell's laws. These
orbits are called * stationary states*.
This is called as

*
source :
http://www.upscale.utoronto.ca/GeneralInterest/Harrison/BohrModel/BohrModel.html*

(b) The energy of the
atom has a definite value in a stationary orbit. The electron can jump from
one stationary orbit to another. If it jumps from an orbit of higher energy
E_{2} to an orbit of lower energy E_{1}, it emits a photon.
The energy of the photon is E_{2}-E_{1}.The wavelength of the
emitted radiation is given by the Einstein - Planck equation.

E_{2}-E_{1}= hν
= hc/λ

The electron can also absorb energy from some source and jump from a lower energy level to a higher energy level as shown in the following figure.

*source :
http://www.tannerm.com/bohratom.htm*

*The figure above shows the various ways of how an electron
can reach ground level after being excited to the third energy level n=3.The
total energy that the electron emits as photon is hv _{30}=hv_{32}+hv_{10}=hv_{32}+hv_{20}=hv_{32}+hv_{32}+hv_{21}+hv_{10.}*

**Energy
of a Hydrogen Atom**

The above postulates
can be used to calculate allowed energies of the atom for different allowed
orbits of the electron. The theory developed should be applicable to hydrogen
atoms and ions having just one electron. Thus, within the Bohr atom framework,
it is valid for He^{+}, Li^{++}, Be^{3+} etc. Let us
consider the case of an ion with the charge of nucleus being Ze and an electron
moving with a constant speed v along a circle of radius r with the center at
the nucleus. The force acting on the electron is that due to Coulomb attraction
and is equal to

F
= Ze^{2}/4πε_{0}r^{2}

The acceleration of the
electron is towards the center and has a magnitude v^{2}/r. If m is
the mass of the electron, from Newton's law we obtain

Ze^{2}/4πε_{0}r^{2}
= mv^{2}/r

Using Bohr's angular
momentum quantization rule for the value n, the
* Principal quantum number*, we obtain
both the velocity v, and the radius r as:

v
= Ze^{2}/2e_{0}hn
r
= ε_{0}h^{2}n^{2}/πmZe^{2} ...(i)

We see that the allowed
radii are proportional to n^{2}. For each value of n, we have an allowed
orbit. For n=1, we have the first orbit (smallest radius) , for n=2, we have
the second orbit and so on.

The kinetic energy of the electron in the nth orbit is

K.E
= mv^{2}/2 = mZ^{2}e^{4}/8ε_{0}^{2}h^{2}n^{2}
...(ii)

The potential energy of the atom is

P.E
= -Ze^{2}/4πε_{0}r = -mZ^{2}e^{4}/4ε_{0}^{2}h^{2}n^{2}...(iii)

We have taken the potential energy to be zero when the nucleus and the electron are widely separated. The total energy of the atom is

E
= K.E+P.E
= -mZ^{2}e^{4}/8ε_{0}^{2}h^{2}n^{2}
...(iv)

Equations (i) to (iv)
give various parameters of the atom when the electron is in the nth orbit .The
atom is also said to be in the * nth energy state
*in this case. In deriving the expression for the total energy E, we have
considered the kinetic energy of the electron and the potential energy of the
electron-nucleus pair. It is assumed that of the acceleration of the nucleus
is negligible on account of its large mass (i.e. that the reduced mass of the
system is the same as the electron mass- this can be corrected easily later).

**Radii
of different Orbits**

From equation (i) the radius of the smallest circle allowed to the electron is (n=1)

r_{1}
= ε_{0}h^{2}/πmZe^{2}

For hydrogen atom Z=1
and substituting the values of other constants we get r_{1}=0.0529..nm.This
length is called the * Bohr radius *and
is a convenient unit for measuring lengths in atomic physics. It is generally
denoted by the symbol a

r_{n}
= n^{2}a_{0}.

For a hydrogen-like ion with Z protons in the nucleus

r_{n}
= n^{2}a_{0}/Z. ...(v)

**Ground
and Excited states**

From equation (iv) the total energy of the atom in the state n=1 is

E_{1}
= -mZ^{2}e^{2}/8ε_{0}^{2}h^{2}

For hydrogen atom Z=1and
substituting the values of the constants E_{1}=-13.6 eV. This is the
energy when the electron revolves in the smallest allowed orbit r=a_{0}
i.e. the one with radius around 0.053nm. We also see from equation (iv) that
energy of an electron is proportional to 1/n^{2}.Thus ,

E
_{n} = E_{1}/n^{2} = -13.6/n^{2}
(eV) ...(vi)

The energy in the state
n=2 is E_{2}=E_{1}/4=-3.4 eV. In the state n=3 it is E_{1}/9
= -1.5 eV etc. The lowest energy corresponds to the smallest circle. Note that
the energy is negative and hence a larger magnitude means lower energy. The
zero of energy corresponds to the state where the electron and the nucleus are
widely separated. The state of atom with the lowest energy is called is
* ground state.* The states with higher
energies are called

**Limitations
of Bohr's Model **

Bohr's atomic model was ultimately
not successful. It defied all attempts at improvement over the ten-year period
following the original publication, and all 'obvious' improvements
lead nowhere; the only 'success' was with the hydrogen atom and similar atoms
i.e. those with one electron like He^{+}, Li^{++} etc. Bohr's atomic
model attributes a planetary motion to electrons which means that electrons move around
the nucleus in defined circular orbits. This is not the modern view. The electron
distribution around the nucleus of an atom is described by a probability distribution,
giving rise to 'electron clouds' rather than discrete circular orbits. What survives
into the * new Quantum Theory* is the need for
single-valued wave functions, and the fact that, for hydrogen-like atoms, Bohr's model
identified various dimensional parameters correctly; the rest is

*"Every word I utter
is to be understood not as an affirmation but as a question."*

*
-Neils Bohr*

Return to Background Information home page

**Related Pages :**

*
http://www.colorado.edu/physics/2000/quantumzone/ - Contains applets for
spectral lines of elements.*

*
http://www.dauger.com/orbitals - This web page has 3D animations of orbitals.
*

*
http://www.phys.virginia.edu/classes/252/Bohr_Atom/Bohr_Atom.html - More
about history of Bohr's atom.*

*
http://www.colorado.edu/physics/2000/quantumzone/bohr.html - Interactive
applets to understand Bohr's atom.*

**References used:**

H.C.Verma - Concepts of Physics-2

Stephen Gasiorowicz - Quantum Physics