topic 3 : electrons in atoms

Topic 3: Electrons in Atoms

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Contents1. ELECTROMAGNETIC RADIATION

2. ATOMIC SPECTRA

3. QUANTUM THEORY

4. THE BOHR ATOM

5. TWO IDEAS LEADING TO A NEW QUANTUM

MECHANICS

6. WAVE MECHANICS

7. QUANTUM NUMBERS AND ELECTRON ORBITALS

8. ELECTRON SPİN : THE 4. QUANTUM NUMBER

9. MULTIELECTRON ATOMS

10.ELECTRON CONFIGURATIONS

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Electromagnetic Radiation Electromagnetic Radiation, is a form of energy

transmission through a vacuum(empty space) or a

medium(such as glass) in which electric and magnetic

fields are propagated as waves.

• Transmits energy through an empty space

• includes visible lights, x-rays, radio waves and optic

waves

• carries certain fundamental characteristics

• It’s velocity is 3,00 x 108 m/s in all of vacuum

environment. (Speed of light)

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Electromagnetic Radiation • Wave, is a disturbance that transmits

energy through a medium .

The distance between the tops of two successive crests ( or the bottoms of two troughs) is called wavelength and designated by the Greek letter lambda “” .

Frequency: is the number of crests or troughs that pass through a given unit of time and designated by the letter “” . The unit is Hz (s-1)

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As the figure shows the radiation component with the magnetic field lies in a plane perpendicular to that of the electric field component.

The wavelength of electromagnetic radiation is shorter for high frequencies(b) and longer for low frequencies (a).

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Wawelength and Frequency

• The relationship between the speed of light (c), the wavelenght () and the frequency () of electromagnetic radiation:

• c = x

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Frequency, Wavelength and Velocity of Electromagnetic Radiation

The SI unit for frequency, s-1, Hertz (Hz), and the basic SI wavelength unit is the meter. However some of the smaller units listed below are also used.

Unit Symbol Length (m)Type of

radiation

Angstrom Å 10-10 X-ray

Nanometer nm 10-9 UV, Visible

Mikrometer 10-6 Infrared

Milimeter mm 10-3 Infrared

Centimeter cm 10-2 Micro wave

Meter m 1 TV, radio

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Electromagnetic Spectrum

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Electromagnetic SpectrumElectromagnetic Spectrum: is a concept that describes the positions of both the forms of radiation founded in the visible region and other forms of electromagnetic radiation indicating the wavelength and frequency ranges.

Visible Region SpectrumIn a medium such as glass, the speed of light is lower than vacuum.As a consequence light is refracted or bent when it passes from one medium to another. Colors are made up of the beams with specific frequency within the capability of human being’s sight.

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Atomic SpectraEach wavelength component of the white light yields an image of the slit in the form of a line. There are so many of these lines that they blend together into an unbroken band of color from red to violet. Therefore, the spectrum of white light is continious. On the other hand, the spectra produced by certain gaseous substances consist of only a limited number of colored lines with dark spaces between them. These discontinious spectra are called atomic spectra or line spectra. Each element has its own distinctive line spectrum.

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Atomic Spectra

The spectrum of white light: When white light is passed through a glass prism,red light is refracted the least and violet light the most. The other colors of the visible spectrum are found between red and violet.

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Atomic Spectra

Hydrogen line spectrum: Only 4 lines(red,greenish blue and two violets at diffrent wavelength) are visible in this spectrum. In addition to these, there are several lines in the ultraviolet region very closed to each other .

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Atomic Spectra

In 1885, Johann Balmer, through trial and error, deduced a formula for the wavelengths of these spectra lines.

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2

1102881,3

22115

n

ns

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Atomic Spectra

• The fact that atomic spectra consist of only limited numbers of well-defined wavelength lines provides a great opportunity to learn about the structure of atoms.

• For example, it suggests that only a limited number of energy values are available to excited gaseous atoms.

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Quantum Theory

Blackbody radiation:The object that emits all type of radiation applied on them is called blackbody . When it is heated, it is observed that every type of wawelength exists at their emission.

Blackbody

As with atomic spectra classical nineteenth century physics could not provide a complete explanation of light emission by heated solids, As a result the quantum theory aroused.

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Quantum Theory• At low temperatures

radiations of low energy (with long wavelength ), and at high temperatures radiations of high energy (with short wavelength) occur. That is the emission of different types of radiation by blackbodies does not depend on the wavelength since according to the wavelength theory the intensity of radiation is proportional to the square of the amplitude.

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Max Planck suggested in 1900 the quantum theory:

• The energy of radiation that a system may possess is limited to a discrete set of values.

• The difference between two of the allowed energies also has a specific value, called quantum of energy.

Quantum Theory

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Quantum Theory• Planck postulated that the energy of a quantum of

electromagnetic radiation is proportional to the frequency of the radiation- the higher the frequency the greater the energy. This is written as the formula below and called as Planck’s equation :

• h: Planck’s constant has a value of 6,626 X 10-34 J.s.

hvE

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The Photoelectric Effect

Quantum Theory

A beam of electrons is produced by shining light on certain metal surfaces. This event is called photoelectric effect, the electrons produced are defined as photo-electrons. This feature was discovered in 1888 by Hertz .

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Quantum Theory• Findings achieved by the photoelectric

experiment:

• The kinetic energy of the ejected electrons rises with the increase in the frequency of the light ; the kinetic energy of the ejected electrons does not depend on the intensity of light.

• If the frequency of the light is below the threshold value (o ) it can not eject any electrons.

• As the intensity of light increases, the number of ejected electrons increase but the kinetic energy of electrons remains unchanged.

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In 1905, Einstein proposed that electromagnetic radiation has particlelike qualities and that particles of light, called photons have a characteristic energy given by Planck’s equation .

When the photons fall on a metal surface, they transfer their energy to the electrons of the metal. However, the emission of the electrons takes place only if the photon’s energy is larger than the minimum energy required by the electrons to leave the metal surface, called Work function.

Quantum TheoryThe Photoelectric Effect

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Quantum TheoryFor the ejection of electrons from a plate of copper an ultraviolet type of radiation or radiation with higher frequency is adequate. Radiation of blue form with lower frequency is enough to eject electrons from potassium. If the supplied energy by a photon is greater than the the work function, the difference between them is transmitted as kinetic energy to the electron to eject it from the metal surface

kfoton EEE 0

Supplied energy Work function kinetic energy of electrons

20 2

1eevmhh

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The planetary atom model of Rutherford had a technical difficulty: The electrons would lose energy collapsing into the nucleus during the electromagnetic radiation. This model is disastrous because it predicts that all atoms are unstable. To overcome this difficulty, Niels Bohr, in 1913, proposed that electrons could only have certain classical motions:

The Bohr Atom

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The Bohr Atom1.The electrons can only travel in certain

circular orbits: At a certain discrete set of distances from the nucleus with specific energies.

2.The electrons has only a fixed set of allowed orbits, called stationary states. As long as an electron remains in a given orbit, its energy is constant and no energy is emitted

3.An electron can pass only from one allowed orbit to another. In such transitions, fixed discrete quantities of energy are involved, in accordance with Planck equation(E= hⱱ)

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The Bohr AtomThe allowed energy states for electrons are defined as n = 1, n=2,n = 3 and continiued similarly. These integers are called the principle quantum number.

The theory allows us to determine the velocities of the electrons in the orbits and meanwhile their kinetic energies.

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The Bohr Atom• When the electron is free of the nucleus,by

convention, it is said to be zero of energy. When the electron is attracted to the nucleus and confined to the orbit n, energy is emitted. The electron energy is indicated with a negative sign to point out that its level declines.

RH= 2,179 X 10-18 J

The energy levels of hydrogen atom

Bohr radius

The orbital radius of Hydrogen atom

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The Bohr Atom If the electron gains an energy of 2,179 x 10-18 J, it moves to the n=∞ orbit, that is, hydrogen atom is ionized. If the electron falls from higher numbered orbits to the orbit n=1 is in the form of ultraviolet light (Lyman series). Electron transitions to the orbit n=2 are called Balmer series. Transitions to the orbit n=3 yield spectral lines in the infrared (Paschen series)

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The Bohr Atom

• Normally the electron in a hydrogen atom is found in the orbit closest to the nucleus (n = 1), this is the lowest allowed energy and called ground state.

• When the electron gains a quantum of energy it moves to a higher level (n = 2 or 3, …) and the atom is in an excited state. When the electron drops from a higher to a lower numbered orbit, a unique quantity of energy is emitted- the difference between the two levels.

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The Bohr AtomExcitation Emission

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The Bohr Atom The energy levels of hydrogen atom

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The Bohr Atom

The Bohr’s atom theory makes not only the determination of energy levels of hydrogen atoms but also the ones of the ions with one electron, Example : He+, Li2+

Z: Atomic number

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The Ideas Leading To A New Quantum Mechanics

The Lack of the Bohr’s Atom Theory

The Bohr model does not do a good job of predicting atomic spectra of many electron atoms and the effect of magnetic field on the spectra. After Bohr’s work on hydrogen, two landmark ideas stimulated a new approach to quantum mechanics. We define the concept as modern quantum mechanics composed of the ideas:

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• 1. Wave –Particle DualityTo explain the photoelectric effect Einstein suggested that light has particle like properties,embodied in photons. Other phenomena, however such as the dispersion of light into a spectrum by a prism , are best understood in terms of the wave theory of light. In 1924 Louis de Broglie considering the nature of the light and matter offered a startling proposition: “SMALL PARTICLES MAY AT TİMES DISPLAY WAVELIKE PROPERTIES”

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Wave-Particle Duality

De Broglie’s wavelength

Particle’s momentum Mass Velocity

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2. The Uncertainty Principle of HeisenbergDuring the 1920’s Niels Bohr ve Werner Heisenberg considered hypothetical experiments to establish just how precisely the behaviour of subatomic particles can be determined. The conclusion they reached is that there must be always uncertainties in measurement such that the product of the uncertainty in position(x) and the uncertainty in momentum(p).

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hpx x : position

p: momentum

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The Uncertainty Principle• The significance of this expresssion is that we

cannot measure position and momentum simultaneously. If we design an experiment to locate the position of a particle with great precision, we cannot measure its momentum precisely and vice versa.

• In simpler terms, if we know precisely where a particle is, we cannot also know where it has come from and where it is going. If we know precisely how a particle is moving we can not also know precisely where it is.

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The branch of the physics that deals with the solutions of wave equations is called as wave mechanics or quantum mechanics. Erwin Schrödinger concluded an equation,that can be applicable for the hydrogen atom , by using de Broglie’s function. The acceptable solutions of these wave equations are called wave functions, denoted by the Greek letter (psi) .

Wave Mechanics

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Wave Mechanics• For an electron the situation is more like wave motion

in a short string with fixed ends, a type of wave called a standing wave. We might say that the permitted wavelenghts of a standing wave are quantized. They are related to the length of the string which must be equal to a whole number(n) times one-half the wavelength.

,...3,2,12

nnl

The motion of an electron in the Bohr radius

The total number of nodes=

n+1

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Wave Functions

Schrödinger, concluded the equation below that determines the wave motion of a hydrogen atom.

From the differential equations are resulted the wave functions and the total energy of an electron. Each of these wave functions refers to the energy level of an electron and is in relation to the position of the electron where it can be found.

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Quantum Numbers and Orbitals

• The mathematical procedure producing acceptable wave functions requires the use of the integral parameters, so wave functions are determined according to these integral parameters called quantum numbers .

An orbital represents a region in an atom where an electron is likely to be found.

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Wave Functions of Hydrogen Atom

Since the Schödinger equation can not be solved by the kartesien coordinates, it is solved by being converted into global polar coordinates.

Radial partAngular part

Angular part

Radial part

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Quantum Numbers and Electron Orbitals

In the wave mechanics the electrons in an atom

composed of more than one electron are

distributed in the shells. The shells are

composed of one subshell or many

subshells,the subshells are made up of one

orbital or many orbitals. Each electron of an

atom is defined through three quantum

numbers referring to the shell, subshell and

orbital.

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• Principal Quantum number, n: The energy levels in

atom are divided into the shells represented by the

principle quantum number, “n”. As in the Bohr

quantum theory, it may have only positive, nonzero (n

= 1, 2, 3, …..) integral values. In addition to the

numbers, to indicate the layers, some letters are also

used. The shells are the regions where electrons are

more likely to be found. The greater the n value, the

farer the shell from the nucleus.

• 1 2 3 4 5...

• K L M N O …

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Angular momentum quantum number, l: Energy levels include sub-energy levels. Consequently, shells are seperated into subshells each of which is represented with angular momentum quantum number “l” .This determines the geometrical shape of the electron probability distribution. The number “l” can have all values ranging from 0, 1, 2 to n-1. For n=1 the maximum and unique value of “l” is 0 which means that the level K contains one sub-level. For n=2 , “l” will have 0 and 1 values. Thus, L level is composed of two sub-levels. The total number of sub-levels in a level is equal to the principal quantum number. The sub-shells are indicated as below:

0 1 2 3 4 5 6 …s p d f g h i …

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• To indicate a sub-shell in a shell, the principal quantum number “n” and the angular momentum quantum number are written next to each other . For the second shell (L), the subshells s and p are indicated as 2s (n = 2, l = 0) and 2p (n = 2, l =1 ) .

• Magnetic quantum number, ml: Each subshell is composed of one or more orbitals and each orbit in a sub-shell is defined as magnetic quantum number “ml”. This number may be a positive or negative integer including zero and ranging from – l to +l.

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Principal quantum number

n

Orbital quantum number

l

Sub-shell

Magnetic quantum number

ml

The number

of orbitals in the sub-shell

3 0 3s 0 1

1 3p -1,0,1 3

2 3d -2,-1,0,1,2 5

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Quantum numbers and Electron Orbitals

The shells and sub-shells of Hydrogen atom

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s orbitalss orbital: Spherically symmetric

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p orbitals

p orbital: Electron density is in form of a dumbbell.Two lobes are seperated by a nodal plane in which charge density drops to zero.

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d orbital

d orbital: There are 5 different type of d orbitals. Their orientations vary respectively.

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Electron Spin-The fourth Quantum Number

Stern-Gerlach experimentAg atoms vaporized in the oven are collimated into a beam by the slit and the beam is passed through a non-uniform magnetic field. The beam splits in two with two opposite directions ( A spinning unpaired electron behaves as two magnets with opposite pole directions.

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Electron SpinSpin magnetic quantum number, ms: An electron generates a magnetic field because of its spin on its axis. As a result of this action ( spin at one direction, and at the opposite direction) the spin magnetic quantum number may have values : ms=+1/2 ve ms=-1/2.

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Multielectron Atoms• Schrödinger developed his wave equation for the hydrogen atom. For multielectron atoms a new factor arises: mutual repulsions between electrons. Because exact electron positions are not known,electron repulsions can only be approximated.

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Multielectron Atoms• In multielectron atoms the attractive force of nucleus for a given electron increases as the nuclear charge rises, which leads to a decrease of the energy level of an orbital. Hence, multielectron atoms have different orbital energies

The orbitals with different energy levels

The orbital energy level decreases with rising nucleus charge

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The electron configuration of an atom is a designation of how electrons are distributed among various orbitals.

Rules for Assigning Electrons in Orbitals

1. Electrons occupy orbitals in a way that minimizes the energy of the atom.

Electron Configurations

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Electron Configurations• The diagram shows the order in which

electrons occupy orbitals in these shells, first 1s then 2s and 2p and so on. The order of the filling of orbitals has been established by experiment, principally through spectroscopy and it is the order that we must follow in assigning electron configurations to the element. Except for a few elements the order in which the orbitals fill in:

1s, 2s, 2p, 3s, 3p, 4s, 3d, 4p, 5s, 4d, 5p, 6s, 4f, 5d, 6p, 7s, 5f, 6d, 7p

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Electron Configuration

2. No two electrons in an atom may have all four quantum numbers alike (Pauli exclusion principle).

3. When orbitals of identical energy are available, electrons initially occupy these orbitals singly. As a result of this rule, known as Hund’s rule an atom tends to have as many unpaired electrons as possible. The electrons do this by seeking out empty orbitals of similar energy in preference to pairing up with other electrons in half-filled orbitals.

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Notation of Electron Configuration

Since the atomic number of Carbon element is 6, in all of three indications there are 6 electrons. The electrons are those with parallel spins which occupy different orbitals in the same sub-shell singly.

spdf notation (condensed)

spdf notation (expanded)

orbital diagram

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The Aufbau ProcessAufbau means “constructing or building” and what we do is assign electron configurations to the elements in order of increasing atomic number.

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The Electron Configuration of some elements(C, N, Ne, Na)

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Valence Electrons• Electrons that are added to the electronic shell of

highest principal quantum number(the outermost or valence shell) are called valence electrons. The electron configuration of Na is written below with the neon core ( 1s2s2p6 ) and for the other third period elements only the valence-shell electron configuration is shown.

• Na Mg Al Si P S Cl Ar• [Ne]3s1 3s2 3s23p13s23p23s23p3 3s23p4 3s23p5

3s23p6

• 6C: [He]2s22p224Cr: [Ar]4s13d5

53I: [Kr]4d105s25p5

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The elements of the third period end with Argon. After argon instead of 3d the next sub-shell to fill is 4s.

The 19. electron of potassum occupies 4s instead of 3d orbital since 4s has lower energy level.

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Example: Write out the electron configuration of 38Sr, [38Sr]+2 and 26Fe ,[26Fe]+2 in the condensed spdf notation ?

38Sr: 1s22s22p63s23p64s23d104p65s2 (according to the order of orbital energy levels)

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Solution• 38Sr: 1s22s22p63s23p63d104s24p65s2 (according to

the increasing principal quantum number “n” )

• 38Sr: [Kr]5s2 (according to the order with the indication of noble gas core electron configuration )

• [38Sr]+2: 1s22s22p63s23p63d104s24p6

• 26Fe: 1s22s22p63s23p64s23d6

• [26Fe]+2: 1s22s22p63s23p63d6

topic 3 : electrons in atoms

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