[1]

XII Phys Ch - 12 (Atoms)

Chapter - 12 (Atoms)
 Rutherford’s α- particle scattering experiment:
Rutherford and his associates studied the scattering of the
α - particles by a thin gold foil in order to investigate the
structure of the atom. An α-particle is a positively charged
particle having a mass equal to that of helium atom and
positive charge in magnitude equal to twice the charge of
an electron.
Alpha particles are emitted by many radioactive elements.
The scattering of α-particles provide useful information
about the structure of the atom.

Alpha-particles emitted by a 䚰٤妘 엠 radioactive source

were collimated into a narrow beam by their passage
through lead bricks. The beam was allowed to fall on a thin foil of gold of thickness 2.1 × 10 –7 m. The scattered
alpha-particles were observed through a rotatable detector consisting of zinc sulphide screen and a microscope.
 Observations: A graph is plotted between the scattering angle θ and the number of α-particles N (θ),
scattered at ∠θ for a very large number of α-particles.

 Conclusions:
1. Most of the alpha particles pass straight through the gold foil.
2. Only about 0.14% of incident α-particles scatter by more than 10.
3. About one α-particle in every 8000 α-particles deflects by more than 90°.
 Explanation:
In Rutherford’s model, the entire positive charge and most of the mass of the atom are concentrated in the
nucleus with the electrons some distance away. The electrons would be moving in orbital about the nucleus just
as the planets do around the sun. The size of the nucleus comes out to be 10−15 m to 10−14 m. From kinetic
theory, the size of an atom was known to be 10−10 m, about 10000 to 100,000 times larger than the size of the
nucleus. Thus, most of an atom is empty space.
The trajectory of an alpha particle can be computed by Newton’s second law of motion and Coulomb’s law for
electrostatic force of repulsion.
The magnitude of this force is

Where, Ze  Charge of gold nucleus,

2e  Charge on alpha particle
r  Distance between α-particle and the nucleus
 Alpha particle trajectory:
Trajectory traced by an α-particle depends on the impact parameter b of collision. The impact parameter is the
perpendicular distance of the initial velocity vector of the α-particle from the centre of the nucleus.
 [2]
XII Phys Ch - 12 (Atoms)

For large impact parameters, force experienced by the alpha particle is weak because

Hence, the alpha particle will deviate through a much smaller angle. When impact parameter is small, force
experienced is large and hence, the alpha particle will scatter through a large angle.
 Electron orbits:
Let Fc  Centripetal force required to keep a revolving electron in orbit
Fe  Electrostatic force of attraction between the revolving electron and the nucleus
Then, for a dynamically stable orbit in a hydrogen atom,
Fc = Fe

K.E. of electron in the orbit,

From equation (i),

Potential energy of electron in orbit,

Negative sign indicates that revolving electron is bound to the positive nucleus.
∴ Total energy of electron in hydrogen atom

 Atomic Spectra:
Each element emits a characteristic spectrum of radiation. In the excited state, the atoms emit radiations of a
spectrum, which contains certain specific wavelengths only. This spectrum is termed as emission line spectrum
and it consists of bright lines on a dark background. The spectrum emitted by atomic hydrogen is shown in the
figure below.
 [3]
XII Phys Ch - 12 (Atoms)

 Spectral Series:
When the electron in a hydrogen atom jumps from higher energy level to the lower energy level, the difference
of energies of the two energy levels is emitted as a radiation of particular wavelength. It is called a spectral line.
In H-atom, when an electron jumps from the orbit ni to orbit nf, the wavelength of the emitted radiation is given
by,

Where λ is the wavelength,

R is a constant called the Rydberg constant, The value of R is 1.097 × 107 m–1.and
n may have integral values 3, 4, 5, etc.
For transition of the electron between two different energy levels, the spectral lines of different wavelengths are
obtained. These spectral lines are found to fall into a number of spectral series as discussed below.
1. Lyman series: When the electron jumps from any of the outer orbits to the first orbit, the spectral lines
emitted are in the ultraviolet region of the spectrum and they are said to form a series called Lyman series
For Lyman series, nf = 1 and ni = 2, 3, 4, …

2. Balmer series: When the electron jumps from any of the outer orbits to the second orbit, we get a
spectral series called the Balmer series. All the lines of this series in hydrogen have their wavelength in the
visible region.
For Balmer series, nf = 2 and ni = 3, 4, 5, …

3. Paschan series: This series consists of all wavelengths which are emitted when the electron jumps from
outer most orbits to the third orbit. This series is in the infrared region.
For Paschan series, nf = 3 and ni = 4, 5, 6, …

4. Brackett series: The series obtained by the transition of the electron from outer most orbit to the forth
orbit is called Brackett series. The wavelengths of these lines are in the infrared region.
For Brackett series, nf = 4 and ni = 5, 6, 7, …

5. Pfund series: The lines of the series are obtained when the electron jumps from any state ni = 6, 7... to
nf =5. This series also lies in the infrared region.
For Pfund series, nf = 5 and ni = 6, 7, 8, …
 [4]
XII Phys Ch - 12 (Atoms)
 Bohr's Model of Hydrogen Atom:
Bohr suggested a new model for the atom as Rutherford’s atomic model was unstable. He introduced the concept
of stationary orbits.
 Postulates of Bohr’s Atom Model:
1. In a hydrogen atom, an electron revolves in certain stable orbit without the emission of radiant energy
around the nucleus. These are the stationary (orbits) states of the atom have definite total energy.
2. The electrons revolve around the nucleus only in those orbits for which the angular momentum is the
h
integral multiple of . Where, h is the Planck’s constant (= 6.6 × 10–34 J s). Thus the angular momentum
2π
(L) of the orbiting electron is quantised. i.e. L = nh/2π
3. An electron might make a transition from one of its specified non-radiating orbits to another of lower energy.
When it does so, a photon is emitted having energy equal to the energy difference between the initial and
final state.
The frequency of the emitted photon is then given by , hν = Ei − Ef
where Ei and Ef are the energies of the initial and final states and Ei > Ef .
 Total Energy of electron in nth orbital of hydrogen atom:

Substitution of values of h, m, ε0 and e gives r1 = 5.29 × 10–11 m.

Total energy:
 [5]
XII Phys Ch - 12 (Atoms)
The negative sign of the total energy of an electron moving in an orbit means that the electron is bound with the
nucleus.

 Line Spectra of Hydrogen Atom:

When an electron in a hydrogen atom jumps from the higher level to the lower energy level, the difference of
energies of the two energy levels is emitted as a radiation of particular wavelength.

Where,

It is called Rydberg’s constant. Its value is 1.09678 × 107 m−1.

The different spectral series are as follows:

The various lines in the atomic spectra are produced when electrons jump from higher energy state to a lower
energy state and photons are emitted. These spectral lines are called emission lines. But when an atom absorbs a
photon that has precisely the same energy needed by the electron in a lower energy state to make transitions to a
higher energy state, the process is called absorption.
 Energy Levels – (Hydrogen Atom):
The Energies of the electrons in an atom can have only certain values. These values are called energy levels of
the atoms. The energy of an atom is the least (largest negative value) when its electron is revolving in an orbit
closest to the nucleus i.e., the one for which n = 1. For n = 2, 3, ... the absolute value of the energy E is smaller,
hence the energy is progressively larger in the outer orbits. The lowest state
of the atom, called the ground state.By Bohr’s theory, the total energy of an electron in the nth orbit of hydrogen
atom is given by –
13.6
En   2 eV
n
By substituting the values of n we can find the energy of an electron of hydrogen atom.
 [6]
XII Phys Ch - 12 (Atoms)

 De Broglie’s Explanation of Bohr’s Second Postulate of Quantization:

De-Broglie’s hypothesis that electron has a wavelength λ = h/p gave an explanation for Bohr’s quantised orbits
by bringing in the wave particle duality. Orbits correspond to circular standing waves in which the
circumference of the orbits equal whole number of wavelength.
nh
For stable orbit the angular momentum of revolving electron, L  , n = 1, 2 , 3 …
2
h
According to de-Broglie's hypothesis  
p
If the speed of the electron is much less than the speed of light, the momentum p = mvn.
h
Thus  
mvn
For an electron moving in n circular orbit of radius rn, the total distance is the circumference of the orbit,
th

= 2πrn.
Thus 2 rn  n n = 1, 2, 3...
nh
 2 rn 
mvn
nh
 mvn rn 
2
This is the quantum condition proposed by Bohr for the angular
momentum of the electron.
Thus de Broglie hypothesis provided an explanation for Bohr’s second
postulate for the quantisation of angular momentum of the orbiting
electron.

 Limitations of Bohr’s theory:

 Bohr’s model is applicable only to hydrogenic (single electron) atoms. It cannot be extended to even two
electron atoms.
 While the Bohr’s model correctly predicts the frequencies of the light emitted by hydrogenic atoms, the
model is unable to explain the relative intensities of the frequencies in the spectrum.
Bohr’s model is unable to account for the intensity variations.
 It is silent about wave properties of electrons.