Download III- Atomic Structure

III- Atomic Structure
• The determination of the composition of atoms relies heavily on
four classic experiments:
1)
2)
3)
4)
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Faraday’s law of electrolysis, which shows that atoms are composed of
positive and negative charges and that atomic charges always consist of
multiples of some unit charge.
Thomson’s determination of e/me . Thomson measured e/me of electrons
from a variety of elements by measuring the deflection of an electron
beam by an electric field.
Millikan’s determination of the fundamental charge, e. By balancing the
electric and gravitational force on individual oil drops, Millikan was able
to determine the fundamental electric charge and to show that charges
always occur in multiples of e.
Rutherford’s scattering of particles from gold atoms, which established
the nuclear model of the atom. Rutherford was able to establish that
most of the mass and all of the positive charge of an atom, +Ze, are
concentrated in a minute volume of the atom with a diameter of about
10-14 m.
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The Particle Nature of Matter
• The explanation of the motion of electrons within the atom
and series of spectral lines emitted by the atom was given by
Bohr. Bohr’s theory was based partly on classical mechanics
and partly on then new quantum ideas. Bohr’s postulates
were:
1. Electrons move about the nucleus in circular orbits determined
by Coulomb’s and Newton’s laws.
2. A spectral line of frequency f is emitted when an electron jumps
from an initial orbit of energy Ei to a final orbit of energy Ef,
where hf=ΔE.
3. The sizes of the electron orbits are determined by requiring the
electron’s angular momentum to be an integral multiple of ћ:
mevr=nћ an d n=1,2,3,…
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The Particle Nature of Matter
• These postulates lead to quantized orbits and quantized energies
for a single electron orbiting a nucleus with charge Ze, given by
where k is the Coulomb’s constant and a0=0.529 Å is the Bohr radius
and Z is the atomic number
• Direct experimental evidence of the quantized energy levels in
atoms is provided by the Franck–Hertz experiment. This experiment
shows that mercury atoms can only accept discrete amounts of
energy from a bombarding electron beam.
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The Particle Nature of Matter
• The atomic structure is responsible for all properties of matter that
shapes the world
• In old times, electrons were believed to orbit the nucleus as
planets do around the sun. However, according to the classical
electromagnetic theory electrons can never have stable orbits!
• In 1913, Bohr resolved this paradox by applying the concepts of the
then recently developed quantum theory to atomic structure.
Despite its many drawbacks and later replacement by a quantum
mechanical description of greater accuracy and usefulness, Bohr’s
model remains a convenient mental picture of the atom
• Bohr’s theory of the hydrogen atom provides a valuable transition
to the more complete and abstract quantum theory of the atom!
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Early Models of the Atom
• In the late 19th century, most scientists accepted the idea that the
atom is the basic building unit of matter. However, they almost
knew nothing about atoms or their structure
• Scientists, nevertheless, were acquainted with the fact that atoms
contain electrons, and since an atom is neutral then there must be
some kind of +ve charges to neutralize electrons
J. J. Thomson, a British physicist, in 1898
suggested that atoms are just positively
charged lumps of matter with electrons
embedded in them like raisins in a cake!
Since Thomson had earlier played a very
important role in discovering the electron, his
model was received by respect
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Dr. Mohamed Khater
Rutherford’s Model
• Ernest Rutherford (a former student of
Thomson) and his students Geiger and
Marsden designed and performed an
experiment in 1911 to verify the theoretical
model proposed by Rutherford
• As they expected to observe, based on
Thomson’s model, most of the α-particles (an
atom that lost 2 e- and so bears a positive
charge of +2e, i.e. it is the He nucleus) go right
through the gold foil with hardly any
deflections because of the very weak electric
forces exerted by the uniformly distributed e• However, there were few particles that were
scattered through very large angles reaching
180˚!
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The α-particles collide with
the screen after scattered
through the gold thin foil,
giving off flashes of light.
Rutherford’s Model
• Rutherford was quite astonished and said: It was an incredible event as if
you fired a 15-inch stone at a piece of tissue paper and it comes back and
hits you!
• Now since an α-particle is 8000 times heavier than the electron and
those used in this experiment had high speed of 2×107 m/s, it was clear
that powerful forces were needed to cause such extraordinary
deflections
Rutherford, therefore, was able to suggest his model of the
atom as being composed of a tiny nucleus containing +ve
charges and nearly all its mass is concentrated in it. The
electrons are located some distance away. With this
picture, i.e. an atom being largely empty space, it is easy to
see why most α particles go right through the thin foil.
when an α-particle happens to come near a nucleus the
intense electric field there scatters it through a large angle,
depending on the nuclear charge (i.e. atomic number)
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Rutherford’s Atom: The Planetary Model
• Rutherford was tempted to propose that
electrons move in orbital motion around the tiny
(≈10-4 of the atom), massive and positively
charged nucleus as planets around sun do, since
electrons can not be stationary against the huge
electric force pulling them toward the nucleus
• For the H atom, the centripetal force is:
v2
Fc  mac  m
r
Holding the electron in an orbit r from the positively charged nucleus
is provided by the Coulomb electric force:
e2
Fe 
40 r 2
1
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Rutherford’s Atom: The Planetary Model
• The condition for a dynamically stable orbiting is: Fc = Fe
From which the electron velocity:
• The total energy of the electron in the H atom is: Etot = KE+PE. The
electrical PE of an electron is equal to the work done on the
electron by the electrical force of the nucleus -Fer (compare this
with mechanical potential energy due to the work done by the
agent and the gravitational force mgh and –mgh, respectively)
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Rutherford’s Atom: The Planetary Model
• Substituting for the velocity of the electron in the last equation:
Which is the total energy of the hydrogen atom as calculated by
Rutherford. The minus sign of the electron energy indicates that it is
always “slaved” to the nucleus (i.e. it is doing work in the benefit of
the nucleus for free!) Only when the atomic electron does not
follow a closed orbit, i.e. knocked out the atom, its Etot becomes > 0
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The Failure of Rutherford’s Model and Classical Physics!
• To complete his analysis, Rutherford applied
principles of classical mechanics (Newton’s laws
of motion) and classical electricity (Coulomb’s
law). However, classical electromagnetic theory
(the other pillar of classical physics along with
classical mechanics) tells us that when an
electric charge is accelerated it undergoes a
curved path and em radiation must be released
accordingly. Therefore, this electron loses
energy constantly and spiral into the nucleus
within a very brief time (<< 1s)
• However, in reality the electron does not fall to
the nucleus and therefore the atom as a whole
does not collapse.
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The Bohr Atomic Model
• The first theory of the atom to meet success in 1913! Bohr received
the Nobel prize in physics in 1922 for formulating this theory
• We start with the de Broglie wavelength formula for an electron is λ
= h/mv. Note that we put γ = 1 because velectron<<1 in this
treatment)
• But we have:
→
• By substituting 5.3×10-11 m for the radius r of the electron orbit in H
atom, we find the electron wavelength to be:
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The Bohr Atomic Model
• The de Broglie wavelength is exactly equal to the circumference of
the electron orbit 2πr = 33×10-11 m. This means that “the orbit of
an electron in a hydrogen atom corresponds to one complete
electron wave”
• This fact provided the clue to Bohr to construct a theory of the
atom: “an electron can only circle a nucleus if its orbit contains an
integral number of de Broglie wavelengths” → nλ = 2πrn, where n =
1, 2, 3, ….and called the quantum number of the orbit
• Substitute for the value of λ in the last equation:
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The Bohr Atomic Model
• And so the possible electron orbits are those whose radii are given
by:
• So the radius of the innermost orbit of the hydrogen atom (Bohr
radius) can be calculated by substituting for the constants and n = 1:
a0 = r1 = 5.29×10-11 m
and the other radii can then be calculated as: rn = n2a0
• The electron energy En is then given in terms of the orbit radius rn
as:
Energy levels of
the hydrogen
(Bohr) atom
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Graphical Representation of the Energy Levels in Hydrogen
(Bohr) Atom
The electron does not have enough
energy to escape from the atom (-ve
energy levels).
The levels are quantized, i.e. only these
energies are allowed. An analogy may be
a person on a ladder who can stand only
on its step but not in between.
As the principal quantum number
increases, the corresponding energy
values En approaches closer to 0.
In the limit n = ∞, E∞ = 0, and the electron
is no longer bound to the nucleus. The
energy of the electron then starts to be
+ve which means that it becomes free and
has no quantum conditions to fulfill.
The ionization energy of the atom = -E1 =
+13.6 eV
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Atomic Excitation: Origin of Line Spectra
• According to Bohr’s model, electrons cannot exist in an atom except
in certain specific energy levels
• Upon exciting the atom, an electron jumps from a lower energy
state to a higher one. This electron cannot stay in the excited state
forever, so after a brief period of time (≈ 10-8 s) it drops to the initial
energy state producing emission of a single photon whose energy is
equal to the energy difference between the two energy states
involved
• As was experimentally confirmed, this emitted energy is given off all
at once in the form of a photon rather in some gradual manner. This
observation fits in well with the Bohr’s concept
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Atomic Excitation: Origin of Line Spectra
• Take the quantum number of the initial (higher energy) state as ni,
and the quantum number of the final (lower energy) state as nf so:
Initial Energy – Final Energy = +ve quantity = Photon Energy → Ei –
Ef = hν, where ν is the frequency of the emitted photon
Note that – E1 is a + ve quantity because E1 is a – ve quantity! The
frequency of the released photon in this transition is therefore:
Recalling that λ = c/ν → 1/λ = ν/c, therefore:
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Atomic Excitation: Origin of Line Spectra
• This is the theoretical formula of the hydrogen atom spectrum,
which suggests that the radiation emitted by excited hydrogen
atoms should contain certain wavelengths only
• From this equation, five hydrogen spectra series were formulated:
The Lyman series (UV):
The Balmer series (VIS):
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Atomic Excitation: Origin of Line Spectra
The Paschen series (IR):
The Brackett series (IR):
The Pfund series (IR):
• Note that in case of the electron gets enough energy to go out the
atom, we still use ∞ as ni not as nf. Therefore, nf ALWAYS takes the
lower value and ni takes the higher value to keep the RHS +ve.
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Spectral Series of Hydrogen
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Nuclear Effect on Wavelengths of Spectral Lines
• So far we have assumed that the
hydrogen nucleus (i.e. the proton)
remains stationary while the
orbital electron revolves around it
• What actually happens is that
both proton and electron revolve
around their common center of
mass, which of course is very close
to the nucleus due to its greater
mass
• A system of this kind is equivalent
to a single particle of mass m’ that
revolves around the position of
the heaver particle
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Nuclear Effect on Wavelengths of Spectral Lines
• If m is the electron mass and M is the nuclear mass, then m’ is:
• m’ is called the reduced mass of the electron because its value is
less than m, and it has to be taken into account. The energy levels
of the hydrogen atom, corrected for nuclear motion, is therefore
given by:
m
• According to the motion of the nucleus, all the energy levels of
hydrogen are changed by the fraction:
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Nuclear Effect on Wavelengths of Spectral Lines
• This value corresponds to an increase in En by 0.055% (i.e. En
becomes less negative), which means a decrease in wavelength
(shorter λ)
• The notion of reduced mass led the American chemist Urey in 1932
to discover the element deuterium (1H2), a stable abundant isotope
of 1H1 whose nucleus contains one proton and one neutron. About
one hydrogen atom in 6400 is deuterium
• Because of the greater nuclear mass, the spectral lines of
deuterium are all shifted slightly to wavelengths shorter than those
of ordinary hydrogen. Urey noted that, for example, the Hα spectral
line of hydrogen occurs at 656.3 nm while that for deuterium
happens at 656.1 nm
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The Franck-Hertz Experiment
• A series of experiments that confirmed Bohr’s theory of the
hydrogen atom, based on atoms excitation by collisions with
energetic electrons (1914)
• These experiments demonstrated that atomic energy levels indeed
exist, and, furthermore, they exactly correspond to those suggested
by atomic line spectra
• Franck & Hertz found that, for example, an electron energy of 4.9
eV was required to excite the 253.6 nm spectral line in Hg vapour; a
photon of 253.6 nm light has an energy of just 4.9 eV
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