iii- atomic structure the determination of the composition of atoms relies heavily on four classic...

III- Atomic Structure

• The determination of the composition of atoms relies heavily on four classic experiments: 1) 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.

2) 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.

3) 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.

4) 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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• 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 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

Dr. Mohamed KhaterModern Physics5

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

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

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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)

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

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• For the H atom, the centripetal force is:

Holding the electron in an orbit r from the positively charged nucleus is provided by the Coulomb electric force:

r

vmmaF cc

2

2

2

04

1

r

eFe


• 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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• 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 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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• 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:

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Energy levels of the hydrogen (Bohr) atom

Graphical Representation of the Energy Levels in Hydrogen (Bohr) Atom

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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

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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• 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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• 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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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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• 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:

• According to the motion of the nucleus, all the energy levels of hydrogen are changed by the fraction:

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m


• 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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iii- atomic structure the determination of the composition of atoms relies heavily on four classic...

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