chemistry: quantum theory and electronic structure of electrons

8/12/2019 Chemistry: Quantum Theory and Electronic Structure of Electrons.

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+

LECTURE OBJECTIVESExplain the dual nature of light and the relationship among its energy,

frequency, and wavelength.

Use Bohrs theory to explain light emission and absorption by gaseous atoms.(Cite experimental evidence that implies that electromagnetic radiation can

display both wave and particle behaviors. Relate particle and wave properties ofmatter using de Broglie's hypothesis.

Introduce the quantum mechanical model of an atom.

Prepared by CVManalo

Quantum Theory and the Electronic Structure of Atoms


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+

Copyright Cengage

Learning. All rights reserved2

Different Colored

Fireworks

Why do we get colors?

Why do different

chemicals give us

different colors?


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+

In fireworks, the energy to excite the electrons comes from the reaction

between the oxidizer and fuel.

The electrons in atoms can be raised to higher-energy orbitals when

the atoms absorb energy.

The excited atoms can then release this excess energy by emitting light

of specific wavelengths, often in the visible region

For a color effect, an element with a colored emission spectrum is

included.


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+ Discussion on the transition from classical physics to

quantum theory. In particular, we become familiar with

properties of waves and electromagnetic radiation and

Plancksformulation of the quantum theory.

Einsteins photoelectric effect is another step toward

the development of the quantum theory. Einsteinsuggested that light behaves like a bundle of particles

called photons.

We then study Bohrs theory for the emission spectrum

of the hydrogen atom: that the energies of an electron inthe atom are quantized and transitions from higher levels

to lower ones account for the emission lines.


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+ Some of the mysteries of Bohrs theory are explained byde Broglie who suggested that electrons can behave

like waves.

The Heisenberg uncertainty principle sets the limits for

measurement of quantum mechanical systems. The

Schrdinger wave equation describes the behavior of

electrons in atoms and molecules.

There are four quantum numbers to describe an electron

in an atom and the characteristics of orbitals in which the

electrons reside.

Electron configuration enables us to keep track of thedistribution of electrons in an atom and understand its

magnetic properties.

Finally, we apply the rules in writing electron confi

gurations to the entire periodic table.


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+

Copyright Cengage

Learning. All rights reserved6

Electromagnetic Radiation

One of the ways that energy travels through

space.

Three characteristics: Wavelength

Frequency

Speed


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+Properties of Waves


wave - as a vibrating disturbance by which energy is transmitted.

- Can illustrate the fundamental properties of a wave water waves.

- regular variation of the peaks and troughs enable us to sense the propagation

of the waves.

Waves are characterized by their length and height and by the number of

waves that pass through a certain point in one second (Figure 7.2)(a) Wavelength and amplitude.

(b) (b) Two waves having different wavelengths and frequencies. The

wavelength of the top wave is three times that of the lower wave, but its

frequency is only one-third that of the lower wave. Both waves have the

same speed and amplitude.


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Properties of Waves

Wavelength(l) is the distance between identical points onsuccessive waves.

Ampl i tudeis the vertical distance from the midline of a

wave to the peak or trough.7.1


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Properties of Waves

Frequency(n) is the number of waves that pass through a

particular point in 1 second (Hz = 1 cycle/s).

The speed (u) of the wave = lx n7.1


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+Properties of Waves


Speed - depends on the type of wave and the nature of the medium through

which the wave is traveling (e.g. air, water, or a vacuum).

speed ( u ) of a wave is the product of its wavelength and its frequency:

u= lx n

Physical dimensions involved:

- The wavelength (l) expresses the length of a wave, or distance/wave.- The frequency ( n ) indicates the number of the waves that pass any

reference point per unit of time, or waves/time.

- Thus, the product of these terms results in dimensions of distance/time,

which is speed:

- Wavelength - expressed in units of meters, centimeters, or nanometers,

- Frequency is measured in hertz (Hz), where 1 Hz = 1 cycle/s

- The word cycle may be left out and the frequency expressed as, forexample,

25/s or 25 s 21 (read as 25 per second).


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+ Electromagnetic Radiation


James Clerk Maxwell (1873) - proposed that visible light consists of electromagnetic

waves.

- electrom agnet ic wave has an electric field component and a magnetic field

component.- The two components have the same wavelength and frequency, and hence the

same speed, but they travel or vibrate in mutually perpendicular planes ( Figure).

- provides a mathematical description of the general behavior of light.

- describes how energy in the form of radiation can be propagated through space as

vibrating electric and magnetic fields.Electrom agnet ic radiat ion is the emission and transmission of energy in the form of

electromagnetic waves.

- Electromagnetic waves travel 3.00 x 108m/s or 186,000 miles/s in a vacuum.

- uses the symbol cfor the speed of electromagnetic waves , or the

-speed of l ight

. The wavelength is usually given in nanometers (nm)


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Maxwell (1873), proposed that visible light consists of


Electromagnetic

radiationis the emission

and transmission of energy

in the form of


Speed of light (c) in vacuum = 3.00 x 108m/s

Allelectromagnetic radiation

lx n = c7.1


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+ Electromagnetic Radiation


Various types of electromagnetic radiation differ from one another in wavelength and

frequency (Figure, next slide)

- The long radio waves are emitted by large antennas (e.g. used by broadcastingstations)

- The shorter, visible light waves are produced by the motions of electrons within

atoms and molecules.

- The shortest waves, which also have the highest frequency, are associated with (gamma) rays, which result from changes within the nucleus of the atom.

- the higher the frequency, the more energetic the radiation.

- ultraviolet (UV) radiation, X rays, and rays are high-energy radiation.


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7.1

ELECTROMAGNETIC SPECTRUM
http://localhost/var/www/apps/conversion/tmp/scratch_10//binan/dbs/CAS%20FACULTY%20PORTFOLIO/CAS%20PORTFOLIO%20SY%202010-2011/Faculty%20Portfolio%20-%203rd%20Term/CVMANALO/CHM022/VIDEOS/Putting%20the%20Electromagnetic%20Spectrum%20to%20Use%20[www.keepvid.com].mp4http://localhost/var/www/apps/conversion/tmp/scratch_10//binan/dbs/CAS%20FACULTY%20PORTFOLIO/CAS%20PORTFOLIO%20SY%202010-2011/Faculty%20Portfolio%20-%203rd%20Term/CVMANALO/CHM022/VIDEOS/Putting%20the%20Electromagnetic%20Spectrum%20to%20Use%20[www.keepvid.com].mp4http://localhost/var/www/apps/conversion/tmp/scratch_10//binan/dbs/CAS%20FACULTY%20PORTFOLIO/CAS%20PORTFOLIO%20SY%202010-2011/Faculty%20Portfolio%20-%203rd%20Term/CVMANALO/CHM022/VIDEOS/The%20Electromagnetic%20Spectrum%20[www.keepvid.com].mp4


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+ Mystery #1, Black Body ProblemSolved by Planck in 1900

Max Planck

(1858-1947)

Explained certain aspects of blackbody radiation Blackbodyany object that is a perfect emitter and

a perfect absorber of radiation

Sun and earths surface behaveapproximately as blackbodies

Classical physics assumed that atoms and molecules could

emit (or absorb) any arbitrary amount of radiant energy.


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+

Proposed that energy, like matter, is discontinuous.

Energy is quantized and can only occur in discrete unitsof size hn(packets of energy called Quantum)

Energy (light) is emitted or absorbed in discrete units(quantum).

E = h x n

Plancks constant, h = 6.63 x 10-34Js

Mystery #1, Black Body Problem

Solved by Planck in 1900


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+

7.1

Matter could absorb or emit energy only in the

whole number multiples of the quantity.

So, E = nhn

Where n is an integer (1,2,3)

energy seems to have particulate properties.

Mystery #1, Black Body Problem

Solved by Planck in 1900


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Mystery #2, Photoelectric Effect

Solved by Einstein in 1905

7.2

Photoelectric effect:

the phenomenon in

which electrons are

emitted from the

surface of a metal

when light strikes it.

hn

KE e-

7.2


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Light has both:

1. wave nature

2. particle nature

Photonis a particleof light

Albert Einstein

(1879-1955)

Proposed that a beam of light behave as if itwere composed of a stream of small

particles called PHOTONS

Ephoton = hn= hc /

Where h = Plancks constant

n= frequency of radiation

= wavelength of light

Mystery #2, Photoelectric Effect

Solved by Einstein in 1905

Applicationof Photoelectric Effect


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Applicationof Photoelectric Effect

- Some animals are able to see at night providing them with a distinct advantage

over their prey.

- Same advantage used by the military forces by using night vision technology.

- All types of night vision equipment are electro-optical devices that amplify existinglight. A lens collects light and focuses it on an image intensifier.

- The image intensifier is based on the photoelectric effectmaterials that give offelectrons when light is shone on them.

- Night vision intensifiers use semiconductor-based materials to produce large

numbers of electrons for a given input of photons. The

emitted electrons are then directed onto a screen coveredwith compounds that phosphoresce (glow when struck by

electrons). While television tubes use various phosphors to

produce color pictures, night vision devices use phosphors

that appear green, because the human eye can distinguish

more shades of green than any other color. The viewing

screen shows an image that otherwise would be invisibleto the naked eye during nighttime viewing.

Current night vision devices use gallium arsenide

(GaAs)based intensifiers that can amplify input light asmuch as 50,000 times. These devices are so sensitive they

can use starlight to produce an image. It is also now pos-


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+Summary of the Works of Planck and Einstein

Energy is quantized. It can occur only in discrete units

called quanta.

Electromagnetic radiation, which was previously thought

to exhibit only wave properties, also exhibit particulate

properties, thus the dual nature of light.


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+



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De Broglie (1924) reasoned that

e-is both particle and wave.

Why is e-energy quantized?

7.4

u = velocity of e- mass = mass of e-

2r = nl l=h

mass x u

Louis de Broglie(1892-1987)


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l= h/mu

l= 6.63 x 10-34/ (2.5 x 10-3x 15.6)

l= 1.7 x 10-32m = 1.7 x 10-23nm

What is the de Broglie wavelength (in nm)

associated with a 2.5 g Ping-Pong ball

traveling at 15.6 m/s?

m in kgh in Js u in (m/s)

7.4


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+Properties of Matter

Large pieces of matterpredominantly exhibit particulate

propertiesbecause their is so small that it is notobservable.

Very small pieces of mattersuch as photons exhibit

predominantly wave properties.

Those with intermediate mass, such as electrons, show

clearly both particulate and wave properties.


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E = h x n

E = 6.63 x 10-34(Js) x 3.00 x 10 8(m/s) / 0.154 x 10-9(m)

E = 1.29 x 10 -15J

E = h x c /l

7.2

When copper is bombarded with high-energy

electrons, X rays are emitted. Calculate the energy

(in joules) associated with the photons if the

wavelength of the X rays is 0.154 nm.


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+Nuclear Atom Model (Ernest Rutherford)

+

-

-

-

-

-

-


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+ The Bohr Model

(Niels Bohr)


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+Bohrs Theory of the Hydrogen Atom - Emission spectra of atoms

Emiss ion spectraeither continuous or line spectra of

radiation emitted by substances.- emission spectrum of a substance can be seen by

energizing a sample of material either with thermal energy or

other form of energy (e.g. high-voltage electrical discharge).

- A red-hot or white-hot iron bar removed from a high-temperature source produces a characteristic visible glow w/c

is a portion of its emission spectrum that is sensed by eye.

The warmth of the same iron bar represents another portion of

its emission spectrumthe infrared region.

- Common feature of the emission spectra of the sun and of a

heated solid is that both are continuous meaning all

wavelengths of visible light are represented in the spectra

(see the visible region, next slide).Prepared by CVManalo

The Visible Spectrum


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The Visible Spectrum

White light consist of a number of lightwaves with different wavelengths.

A continuous spectrum is produced.

f f


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+Bohrs Theory of the Hydrogen Atom - Emission spectra of atoms

- Gas phase emission spectra of atoms do not show a

continuous spread of wavelengths from red to violet; rather,the atoms produce bright lines in different parts of the visible

spectrum.

- These l ine spectra are the light emission only at specific

wavelengths. Figure 7.6 (next slide) show a diagram of adischarge tube used to study emission spectra,

- Figure 7.7 shows the color emitted by hydrogen

atoms in a discharge tube.


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7.3

Line Emission Spectrum of Hydrogen Atoms

Only certain energies are allowed for the electron in the hydrogen atom

Energy of the electron in the hydrogen atom is quantized


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Figure 7.6

(a)Experiment for studying the emission spectra of atoms and

molecules.

- Studied gas is in a discharge tube containing two electrodes.

As electrons flow from the negative electrode to

the positive electrode, they collide with the gas. This collision

process eventually leads to the emission of light by the atoms(or molecules). The emitted light is separated into its

components by a prism. Each component color is focused at a

definite position, according to its wavelength, and forms a

colored image of the slit on the photographic plate.The colored

images are called spectral lines.

(b) The line emission spectrum of hydrogen atoms


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Light is emitted by excitedgaseous hydrogen atoms.

Atomic Spectra

The spectra produced by certain gaseous substancesconsist of only a limited number of colored lines with dark

spaces between them.

This discontinuous spectra are called atomic or linespectra.

Line Spectrum of


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+Line Spectrum of

Hydrogen

Each element has its owndistinctive line spectrum- a

kind of atomic fingerprint.

Helium and Neon

Spectra


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- Every element has a unique emission spectrum.

- The characteristic lines in atomic spectra can be

used in chemical analysis to identify unknown

atoms (same as fingerprints to ID people).

- When the lines of the emission spectrum of a

known element exactly match that of the unknown

sample, the identity of the sample is established.

- Figure 7.8 (net slide) shows the emission spectra

of several elements.


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7.3

Bohrs Theory of the Hydrogen Atom Emission spectra of the H


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+Bohrs Theory of the Hydrogen Atom - Emission spectra of the H

atom

In 1913 Bohr presented a theoretical explanation of the

emission spectrum of the hydrogen atom

Bohrs treatment is very complex and no longer consideredto be correct in all its details.

We will consider only Bohrs important assumptions and finalresults, which accounts for the spectral lines.

Scientists believed that an atom is an entity in which electrons whirled around


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+ Scientists believed that an atom is an entity in which electrons whirled around

the nucleus in circular orbits at high velocities.

Model similar to the motions of the planets around the sun.

In the hydrogen atom, the electrostatic attraction between the positive solarproton and the negative planetary electron pulls the electron inward and thatthis force is balanced exactly by the outward acceleration due to the circular

motion of the electron.

In classical physics, an electron moving in an orbit of a hydrogen atom would

experience an acceleration toward the nucleus by radiating away energy in theform of electromagnetic waves. Thus, such an electron would quickly spiral into

the nucleus and annihilate itself with the proton.

To explain why this does not happen, Bohr postulated that the electron is

allowed to occupy only certain orbits of specific energies.

the energies of the electron are quantized. An electron in any of the allowed

orbits will not spiral into the nucleus and therefore will not radiate energy. Bohr

attributed the emission of radiation by an energized hydrogen atom to the

electron dropping from a higher-energy allowed orbit to a lower one and emitting

a quantum of energy (a photon) in the form of light ( Figure 7.9 ).


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+

Figure 7.9 The emission process in an excited

hydrogen atom, according to Bohrs theory.

An electron originally in a higher energy orbit ( n =3)

falls back to a lower-energy orbit ( n=2). As a result, a

photon with energy hv is given off. The value of hv isequal to the difference in energies of the two orbits

occupied by the electron in the emission process.

For simplicity, only three orbits are shown.


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1. The e- moves in circular orbits about thenucleus.

2. The e- has only a fixed set of allowed orbits.

As long as an e- remains in a given orbit, its

energy is constant and no energy is emitted.

3. An e- can pass only from one allowed orbit

to another . In such transitions, fixed discrete

quantities of energy (quanta) are involved, in

accordance with Plancks equation, E = hn.

Bohr showed that the energies that an

electron in hydrogen atom can occupy aregiven by:

Bohrs Model of the Atom (1913)

En= -RH( )

1

n2

n(principal quantum number) = 1,2,3,

RH(Rydberg constant) = 2.18 x 10-18J 7.3

1


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En= -RH( )

1

n2

7.3

The negative sign in the equation signifies that the energy of the

electron in the atom is lower than the energy of a free electron (an

electron that is far from the nucleus)

energy of a free electron is arbitrarily assigned a value of zero (n equal

to infinity. As the electron gets closer to the nucleus (as n decreases), En

becomes larger in absolute value, but also more negative.

The most negative value is reached when n=1, corresponds to the most

stable energy state called the ground state, or the ground level, which

refers to the lowest energy state of the atom

The stability of the electron diminishes for n= 2, 3, . . . . Each of these

levels is called an excited state, or excit ed level, which is higher in

energy than the ground state.

E.g. A hydrogen electron for which n is greater than 1 is in an excited

state. The radius of each circular orbit in Bohrs model depends on n2.Thus, as n increases from 1 to 2 to 3, the orbit radius increases very

rapidly. The higher the excited state, the farther away the electron is

from the nucleus (and the less tightly it is held by the nucleus)

Bohrs theory explain the line spectrum of the hydrogen atom


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E = hn

E = hn

7.3

Figure 7.10A mechanical analogy for the

emission processes. The ball can rest onany step but not between steps

Bohr s theory explain the line spectrum of the hydrogen atom.

- Radiant energy absorbed by the atom causes the electron to move

from a lower-energy state (characterized by a smaller n value) to a

higher-energy state (characterized by a larger n value).

- Conversely, radiant energy (in the form of a photon) is emittedwhen the electron moves from a higher-energy state to a lower-

energy state.

- The quantized movement of the electron from one energy state to

another is analogous to the movement of a tennis ball either up or

down a set of stairs ( Figure 7.10 ).

- The ball can be on any of several steps but never between steps.

The journey from a lower step to a higher one is an energy-

requiring process, whereas movement from a higher step to a lower

step is an energy-releasing process.

- The quantity of energy involved in either type of change is

determined by the distance between the beginning and ending

steps.

- Similarly, the amount of energy needed to move an electron in the

Bohr atom depends on the difference in energy levels between the

initial and final states.


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7.3

To apply Equation (7.5) to the emission process in a hydrogen atom,

- Suppose that the electron is initially in an excited state characterized by the

principal quantum number ni. During emission, the electron drops to a lower

energy state characterized by the principal quantum number nf(subscripts i and

f denote the initial and final states, respectively).- This lower energy state may be either a less excited state or the ground state.

- The difference between the energies of the initial and final states is


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- When a photon is emitted, ni>nf.

- Consequently the term in parentheses is negative and E is negative (energy islost to the surroundings).

- When energy is absorbed,ni, n

fand the term in parentheses is positive, so E is

positive.

- Each spectral line in the emission spectrum corresponds to a particular

transition in a hydrogen atom.

- When we study a large number of hydrogen atoms, we observe all possibletransitions and hence the corresponding spectral lines.

- The brightness of a spectral line depends on how many photons of the same

wavelength are emitted.

- The emission spectrum of hydrogen includes a wide range of wavelengths fromthe infrared to the ultraviolet.


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- Table 7.1 lists the series of transitions in the hydrogen spectrum; they are

named after their discoverers.

- The Balmer series was easy to study because a number of its lines fall in the

visible range.

- Figure 7.9 shows a single transition.

- Better to express transitions as shown in Figure 7.11 (next slide)

- Each horizontal line represents an allowed energy. level for the electron in a

hydrogen atom. The energy levels are labeled with their principal quantum

+


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+

Ephoton= DE = Ef- Ei

Ef= -RH( )1

n2f

Ei= -RH( )

1

n2i

i f

DE = RH( )1

n2

1

n2

nf = 1

ni = 2

nf = 1

ni = 3

nf

= 2

ni = 3

7.3

Figure 7.11 The energy levels in the hydrogen atom and the various emission

series. Each energy level corresponds to the energy associated with an allowed

energy state for an orbit, as postulated by Bohr and shownin Figure 7.9 . The emission lines are labeled based on the scheme in Table 7.1 .


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+


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+Limitation of the Bohr Model

What are the definite paths that electrons take when movingaround the atom?

Cannot explain splitting of spectral lines in a magnetic field

Answered by the quantum mechanical model!

Louis de Broglie (1892-1987)

Erwin Schrodinger (1887-

1961) Werner Heisenberg

(1879-1976)

+ Louis De Broglie


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+ Louis De Broglie

Electron is both a particle and a wave.

Erwin Schrdinger Wave functions

Schrdinger Equation: = E

where = wave function

= Hamiltonian- a set of mathematicalinstructions, called an operator

E = total energy of the atom

Werner Heisenberg Heisenberg Uncertainty Principle

It is impossible to know precisely where anelectron is and what path it follows.

+


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+

The energy of the electron is quantized.

The electron moves in 3-D space around the nucleus butnot in an orbit of definite radius.

The position of the electron cannot be defined exactly,only the probability.

Features of the Wave Mechanical Model

Electron probability for a 1s orbital

The size of the Hydrogen 1s orbital is thesphere that encloses 90% of the total

e- probability (90% of the time, the e-

is found inside this sphere)

+ 57Schrodinger Wave Equation


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In 1926 Schrodinger wrote an equation that

described both the particle and wave nature of the e-

Wave function (y) describes:

1. energy of e-with a given y

2. probability of finding e-in a volume of space

Schrodingers equation can only be

solved exactly for the hydrogen atom.

Must approximate its solution for

multi-electron systems.



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+ 58g q

yis a function of four numbers called

quantum numbers(n, l, ml, ms)

principal quantum number n

n= 1, 2, 3, 4, .

n=1 n=2 n=3

distance of e-

from the nucleus

59Schrodinger Wave Equation


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59

quantum numbers: (n, l, ml, ms)

angular momentum quantum number l

for a given value of n, l= 0, 1, 2, 3, n-1

n= 1, l = 0

n= 2, l= 0 or1

n= 3, l= 0, 1, or2

Shape of the volume of space that the e-occupies

l= 0 sorbital

l= 1 porbital

l= 2 dorbital

l= 3 forbital

Schrodinger Wave Equation



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60


magnetic quantum number ml

for a given value of lml= -l, ., 0, . +l

orientation of the orbital in space

if l= 1 (p orbital), ml= -1, 0, or1

if l= 2 (d orbital), ml= -2, -1, 0, 1, or2




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61

(n, l, ml, ms)

spin quantum number ms

ms= + or-


ms= -ms= +

62


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62

63


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Where 90% of the

e-density is found

for the 1s orbital

64l = 0 (s orbitals)


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l 0 (sorbitals)

l= 1 (porbitals)

65


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l= 2 (dorbitals)

7.7


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List the values of n, , and mfor orbitals in the 4d subshell.

7.7


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StrategyWhat are the relationships among n, , and m?

What do 4 and d represent in 4d?

Solut ionAs we saw earlier, the number given in thedesignation of the subshell is the principal quantum number, so

in this case n = 4. The letter designates the type of orbital.Because we are dealing with d orbitals, = 2. The values of mcan vary from to . Therefore, mcan be 2, 1, 0, 1, or 2.

CheckThe values of n and are fixed for 4d, but mcan haveany one of the five values, which correspond to the five dorbitals.

68m = 1 0 or 1 3 orientations is space


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ml= -1, 0, or1 3 orientations is space

69m = 2 1 0 1 or 2 5 i t ti i


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ml= -2, -1, 0, 1, or2 5 orientations is space

7.8


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What is the total number of orbitals associated with the principal

quantum number n = 3?

7.8


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StrategyTo calculate the total number of orbitals for a given n value,

we need to first write the possible values of . We then determinehow many m

values are associated with each value of . The total

number of orbitals is equal to the sum of all the m values.

Solut ionFor n = 3, the possible values of are 0, 1, and 2. Thus,

there is one 3s orbital (n = 3, = 0, and m= 0); there are three 3porbitals (n = 3, = 1, and m= 1, 0, 1); there are five 3d orbitals (n =3, = 2, and m= 2, 1, 0, 1, 2). The total number of orbitals is 1 +3 + 5 = 9.

CheckThe total number of orbitals for a given value of n is n2. So

here we have 32= 9. Can you prove the validity of this relationship?



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Existence (and energy) of electron in atom is described

by its un iquewave function y.

Paul i exclusion princ iple- no two electrons in an atom

can have the same four quantum numbers.



Each seat is uniquely identified (E, R12, S8).Each seat can hold only one individual at a

time.



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Shellelectrons with the same value of n

Subshellelectrons with the same values of nandl

Orbitalelectrons with the same values of n, l, andml

74Energy of orbitals in a s ingleelectron atom


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Energy only depends on principal quantum number n

En= -RH( )1

n2

n=1

n=2

n=3

75Energy of orbitals in a mul t i-electron atom


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Energy depends on nand l

n=1 l= 0

n=2 l= 0n=2 l= 1

n=3 l= 0 n=3 l= 1

n=3 l= 2

76Fill up electrons in lowest energy orbitals (Aufbau p r inc ip le)


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77The most stable arrangement of electrons in

b h ll i th ith th t t b f


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subshells is the one with the greatest number of

parallel spins (Hunds rule).

+ 78Order of orbitals (filling) in multi-electron atom


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+

1s < 2s < 2p < 3s < 3p < 4s < 3d < 4p < 5s < 4d < 5p < 6s

7.9


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Write the four quantum numbers for an electron in a 3p orbital.

7.9


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Strategy

What do the 3 and p designate in 3p?

How many orbitals (values of m) are there in a 3p subshell?

What are the possible values of electron spin quantum number?

Solut ionTo start with, we know that the principal quantum number nis 3 and the angular momentum quantum number must be 1(because we are dealing with ap orbital). For = 1, there are three

values of mgiven by 1, 0, and 1. Because the electron spinquantum number mscan be either + or , we conclude that thereare six possible ways to designate the electron using the (n, , m,ms) notation.

7.9


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These are:

Check In these six designations we see that the values of n

and are constant, but the values of mand mscan vary.

82Electron con f igurat ion is how the electrons are


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Electron con f igurat ionis how the electrons are

distributed among the various atomic orbitals in an

atom.

1s1

principal quantum

number n

angular momentum

quantum number l

number of electronsin the orbital or subshell

Orbi tal d iagram

H

1s1

83


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Paramagnetic

unpaired electrons

2p

Diamagnetic

all electrons paired

2p

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What is the maximum number of electrons that can be present

in the principal level for which n = 3?

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StrategyWe are given the principal quantum number (n) so

we can determine all the possible values of the angular

momentum quantum number (). The preceding rule showsthat the number of orbitals for each value of is (2 + 1). Thus,we can determine the total number of orbitals. How many

electrons can each orbital accommodate?

Solut ionWhen n = 3, = 0, 1, and 2. The number of orbitalsfor each value of is given by

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The total number of orbitals is nine. Because each orbital can

accommodate two electrons, the maximum number of electrons

that can reside in the orbitals is 2 9, or 18.

CheckIf we use the formula (n2) in Example 7.8, we find that

the total number of orbitals is 32and the total number of

electrons is 2(32

) or 18. In general, the number of

electrons in a given principal energy level n is 2n2.

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An oxygen atom has a total of eight electrons. Write the four

quantum numbers for each of the eight electrons in the ground

state.

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Strategy

We start with n = 1 and proceed to fill orbitals in the order

shown in Figure 7.24.

For each value of n we determine the possible values of .

For each value of , we assign the possible values of m.

We can place electrons in the orbitals according to the Pauli

exclusion principle and Hunds rule.

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Solut ion

We start with n = 1, so = 0, a subshell corresponding to the 1sorbital. This orbital can accommodate a total of two electrons.

Next, n = 2, and / may be either 0 or 1. The = 0 subshellcontains one 2s orbital, which can accommodate two electrons.

The remaining four electrons are placed in the = 1 subshell,

which contains three 2p orbitals. The orbital diagram is

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The results are summarized in the following table:

Of course, the placement of the eighth electron in the orbital labeled m= 1 iscompletely arbitrary. It would be equally correct to assign it to m= 0 or m= 1.

chemistry: quantum theory and electronic structure of electrons

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