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Energy levels and atomic structures
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Structure of Atom
An atom is the smallest constituent unit of ordinary matter that has the properties of
a element. Every solid, liquid, gas, and plasma is composed of neutral or ionized atoms.
Atoms are very small; typical sizes are around 100 pm (a ten-billionth of a meter, in the
short scale).
 An atom consists of a small, dense nucleus at the center, surrounded by electrons
which orbit the nucleus.
 The nucleus contains more than 99% of the mass of an atom, but concentrates in an
extremely small volume
 A nucleus contains two types of particles: protons and neutrons
 A proton has a positive electric change, equal and opposite to that of an electron.
 A neutron, about the same mass of a proton, has no electric charge.
 An atom has no net electric charge
Development of Atomic Models
Thomson model
In the nineteenth century, Thomson described the atom as a ball of positive charge
containing a number of electrons.
Rutherford’s Atomic Model
In the early twentieth century, Rutherford showed that most of an atom's mass is
concentrated in a small, positively charged region called the nucleus.
Rutherford proposed that an atom has a positively charged core (nucleus) surrounded by
the negative electrons.
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Bohr Atomic model
After Rutherford's discovery, Bohr proposed that electrons travel in definite orbits around
the nucleus.
Quantum mechanical model
Modern atomic theory described the electronic structure of the atom as the probability of
finding electrons within certain regions of space.
 Bohr Model of the Atom
Fundamental postulates:
The Danish physicist Niels Bohr, who first presented this model of the atom, based it on
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fundamental postulates.
(1) Electrons move around the nucleus in circular non-radiating orbits - called “stationary
states”. However, they are not at rest!
(2) An atom only emits or absorbs electromagnetic radiation when an electron
makes a transition from one state to another.
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(3) Only certain stationary states are allowed: those where the orbital angular momentum
of the electron is given by
where n is an integer ≥1 (n = 1, 2, 3 … etc.)
and h is Planck's constant.
This is known as the quantization of angular momentum. The equation implies that an
integer number of wavelengths fit round the orbit:
whence
since,
, (de Broglie relation)

Sizes Of Allowed Orbits
Classically, for an orbiting electron (mass m, charge -e and speed v), the centripetal force
is balanced by the electric (Coulomb) force:
 Energies of Allowed Orbits
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 The minus sign means energy is required to remove the electron
 This energy is called the binding energy.
Substituting for r from Equation above gives the energies of other allowed states:
 Quantum Numbers And Energy Levels
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 Normally, the electron is in the lowest state (n = 1), the ground state.
 It can gain energy from electromagnetic radiation, collisions with other atoms, etc. and
be promoted into one of the higher levels, i.e. to an excited state of the atom.
 Note: the energy gained must equal the energy required.
 If the electron acquires energy > 13.6 eV, it is liberated from the atom altogether.
 The atom is ionized.
 Energy Conservation And Spectral Lines
An electron in an excited state normally returns very quickly to its ground state, either
directly or via an intermediate state
 When the electron moves from its initial Ei to its final Ef state, a photon is emitted (or
absorbed, if Ei < Ef .)
 The energy of the photon emitted (or absorbed) is given by energy conservation
 For hydrogen
[
]
:
where ni and nf (with ni > nf) are the quantum numbers of the initial and final states .
Possible decays from an n = 3 excited atomic state:
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All these transitions correspond to discrete photon energies. A series of sharp spectral
lines are produced. All these lines are in the ultra-violet region of the emission
spectrum. They are called the Lyman Series.
Electrons from states with ni > 2 can return, initially, to the first-excited state (nf = 2),
emitting one photon and then to the ground state emitting a second photon, with Eph =
10.2 eV, which is part of the Lyman series. Photons from transitions to the first-excited,
(n = 2) state of the hydrogen atom form another series of spectral lines. This series is in
the visible part of the spectrum from yellow, for the lowest energy, to violet for the
highest energies. It is called the Balmer Series.
Problems with the Bohr Model
 The Bohr model applies only to one electron atoms.
 The Bohr model doesn’t account for the observed spectra of multielectron elements or
ions.
 The movement of electrons in atoms is much less clearly defined than Bohr allowed.
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Energy levels and atomic structures
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The uncertainty principle (1927)
 The Heisenberg’s uncertainty principle says that you cannot determine the position and
momentum of an electron at the same time.
 only probability of finding an electron with a given energy a given space.
(
)(
)
⁄
(
)(
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⁄
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properties of light
 Light has different colours.
 Light can travel through a vacuum.
 Light can be reflected and refracted,
Classification of Electromagnetic Radiation
Dual properties of Light: (1) waves and (2) particles
 Light is an electromagnetic radiation wave, e.g, Young’s double slit experiment
 Light is also a particle-like packet of energy - photon
 Light particle is called photon
 The energy of phone is related to the wavelength of light
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 Light has a dual personality; it behaves as a stream of particle like photons, but each
photon has wavelike properties
 Particles” of light are called photons
 Each photon has a particular energy

is Planck’s constant
this relationship between the energy of a photon and the frequency or wavelength of the
wave.
An Equation For Matter Waves?
De Broglie postulated that every particles has an associated wave of wavelength.
An equation for matter waves is :
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where m is the mass of the particle and v its speed. Equation is known as the de Broglie
relation and the wavelength λ of the matter wave is called de Broglie wavelength.
The dual aspect of matter is evident in the de Broglie relation. On the left hand side of
Eq, λ is the attribute of a wave while on the right hand side the momentum p is atypical
attribute of a particle. Planck’s constant h relates the two attributes.
Wave-Particle Duality Of Light
Evidence for wave-nature of light
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Diffraction and interference
Evidence for particle-nature of light
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Photoelectric effect
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Compton effect
Wave-Particle Duality Of Matter
Based on the idea that light and all other electromagnetic Radiation may be considered
a particle or a wave nature, Louis de Broglie suggested that the same kind of duality
must be applicable to matter
If electromagnetic radiation behaves as a particle, de Broglie reasoned, why couldn’t a
particle in motion, such as an electron, behave as a wave?
:
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Wave-Particle Duality Of Light
The phenomena of reflection, refraction, interference and diffraction can all be
explained using the idea of light as a wave motion. Furthermore, the fact that light can
be polarised indicates that the waves are transverse.
The photoelectric effect however, requires an explanation which considers light and all
other electromagnetic radiation as a particle motion (i.e. consisting of discrete packets of
energy called photons).
Note
All physical entities can be described as waves or particles. The two models are linked
by the following relationships :
Wave functions
Schrödinger: Replace the precise trajectory of particles by a wave function
mathematical function that varies with position.
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physical interpretation of wave functions. Probability of finding a particle in a region is
proportional to .
Quantum mechanics acknowledges the wave-particle duality of matter by supposing that,
a particle is distributed through space like a wave, rather than traveling along a definite
path. The wave that in quantum mechanics replaces the classical concept of particle
trajectory is called a wave function, (“psi”).
 Wave functions , are mathematical descriptions of the motion of electron waves as
they vary with location and with time.
is the probability density. To calculate the probability that a particle is in a small
region in space multiply y2 by the volume of the region.
(
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