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Chapter 7 Quantum Theory and Atomic Structure

• Key concepts

• Correlate the frequency, wavelength, and energy

• Apply Bohr model to identify the atomic spectrum

• Identify the quantum numbers of the electrons in atoms

• Derive the electron capabilities of energy level

• Determine the electron configuration

• Draw the orbital diagram for atoms or ions

• Identify the periodical trend (radii, ionization energy, electronegativity)

Firework : Different color created by the atomic spectra of various elements

Firework displays are fascinating to watch.

Neon light transforms the skyline with brilliant color.

Sodium vapor lamps transforms the skyline with brilliant color.

1. Light is electromagnetic radiation. Light travels through space as a wave which is made of successive crests and troughs.

2. The elementary particle that defines light is photon (wave and particle properties; it exhibits wavewave--particle dualityparticle duality).

3. The three basic dimensions of light are:Wave length (λ) / Frequency (γ) -- perceived by humans as the color of the light.

Intensity / amplitude (ψ ) -- related to the human perception of brightness the light. Polarization / angle of vibration – described direction of their transverse electric field.

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Frequency and Wavelength

c = l n

The Wave Nature Of LightThe Wave Nature Of Light

Frequency (γ) – the number of wave cycles passing a given point in unit time. Unit Hz (a cycle per second)

Wavelength (λ) – the distance between two consecutive crests most often measured in meter (m) or nanometer (nm)

Crest

Trough

The speed of light can be determined by multiplying the length (λ) of wave cycle by the number of cycles (γ).

Speed of light in vacuum : 3.00 × 10 8 m/s

amplitude

Regions of the electromagnetic spectrum (EM)

• The EM of an object is the range of electromagnetic radiation that it emits, reflects, or transmits.

• The EM extends from the frequencies at the long-wavelength end (modern radio) to gamma radiation (at the short-wavelength end).

• The EM covers wavelengths from thousands of kilometers down to fractions of the size of an atom.

• The visible light to the eyes is only a small portion of the entire of EM spectrum.• Ranging from 400 nm to 700 nm. (barbecue -charcoal - red light given off)

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SOLUTION:PLAN:

Interconverting Wavelength and Frequency

wavelength in units given

wavelength in m

frequency (s-1 or Hz)

γ = c/λ

10-2 m1 cm

10-9 m1 nm

= 1.00x10-10 m

= 325x10-2 m

= 473x10-9 m

γ =3x108 m/s

1.00x10-10 m= 3x1018 s-1

γ =

γ =

3x108 m/s325x10-2 m

= 9.23x107 s-1

3x108 m/s473x10-9 m

= 6.34x1014 s-1

PROBLEM: A dental hygienist uses x-rays (λ = 1.00 A) to take a series of dental radiographs while the patient listens to a radio station (λ = 325 cm) and looks out the window at the blue sky( λ= 473 nm). What is the frequency (in s-1) of the electromagnetic radiation from each source? (Assume that the radiation travels at the speed of light, 3.00x108 m/s.)

o

1 A = 10-10 m1 cm = 10-2 m1 nm = 10-9 m

o

325 cm

473nm

1.00Ao 10-10 m

1AoUse equation c = λ γ

Blackbody radiation –A hot and glowing object can emit or absorb a certain quantitiesof energy. The atoms have only certain quantities of energy (energy is quantized).

When a solid object is heated to ca. 1000K, it begins to emit light.

E = n h ν

The difference in atomic energy state:

∆E = ∆ nhγ

h is plank constant6.626 ×10-34 J∙s

smoldering coal1000 K – soft red

electric heating element1500 K – brighter , yellow

light bulb filament2000 K – brighter and whiter

The particle nature of lightwave-particle duality

Black-body radiation and the quantization of energy

The photoelectric effect and the photon theory

Atomic spectra

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The line spectra of several elements.Take-home message

The Bohr explanation of the three series of spectral lines.Take-home message

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Emission and absorption spectra of sodium atoms.

Flame tests.

strontium 38Sr copper 29Cu

Take-home message

Quantum Numbers and Atomic Orbitals

An atomic orbital is specified by three quantum numbers.

n the principal quantum number - a positive integer

l the angular momentum quantum number - an integer from 0 to n-1

ml the magnetic moment quantum number - an integer from -l to +l

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The Hierarchy of Quantum Numbers for Atomic Orbitals

Name, Symbol(Property) Allowed Values Quantum Numbers

Principal, n(size, energy)

Angular momentum, l

(shape)

Magnetic, ml(orientation)

Positive integer(1, 2, 3, ...)

0 to n-1

-l,…,0,…,+l

1

0

0

2

0 1

0

3

0 1 2

00-1 +1 -1 0 +1

0 +1 +2-1-2

Summary of Quantum Numbers of Electrons in Atoms

Name Symbol Permitted Values Property

principal n positive integers(1,2,3,…)

orbital energy (size)

angular momentum

l integers from 0 to n-1 orbital shape (The l values 0, 1, 2, and 3 correspond to s, p, d, and f orbitals, respectively.)

magnetic ml integers from -l to 0 to +l

orbital orientation

spin ms +1/2 or -1/2 direction of e- spin

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Sample Problem

SOLUTION:

PLAN:

Determining Quantum Numbers for an Energy Level

PROBLEM: What values of the angular momentum (l) and magnetic (ml) quantum numbers are allowed for a principal quantum number (n) of 3? How many orbitals are allowed for n = 3?

Follow the rules for allowable quantum numbers found in the text.l values can be integers from 0 to n-1; ml can be integers from -l through 0 to + l.

For n = 3, l = 0, 1, 2

For l = 0 ml = 0

For l = 1 ml = -1, 0, or +1

For l = 2 ml = -2, -1, 0, +1, or +2

There are 9 ml values and therefore 9 orbitals with n = 3.

Orbital orentatin = n2

Sample Problem

SOLUTION:

PLAN:

Determining Sublevel Names and Orbital Quantum Numbers

PROBLEM: Give the name, magnetic quantum numbers, and number of orbitals for each sublevel with the following quantum numbers:

(a) n = 3, l = 2 (b) n = 2, l = 0 (c) n = 5, l = 1 (d) n = 4, l = 3

Combine the n value and l designation to name the sublevel. Knowing l, we can find ml and the number of orbitals.

n l sublevel name possible ml values # of orbitals

(a)

(b)

(c)

(d)

3

2

5

4

2

0

1

3

3d

2s

5p

4f

-2, -1, 0, 1, 2

0

-1, 0, 1

-3, -2, -1, 0, 1, 2, 3

5

1

3

7

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Atomic orbitals – shape and size

1s 2s 3s

s orbital• s sublevels are spherical. • The s orbital can take 2 electrons in total• They differ from another in size.• As n increases, the radius of orbital become larger.

The 2p orbitals.

The p orbitals

P orbital consists two lobes along an axis (x,y,z).

The p orbital can take 6 electrons in total.

There is zero probability of finding electrons at the origin (the nucleus of the atom).

Three orbitals are oriented at the right angles to one another and designated as px, py, pz.

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The 3d orbitals.

The d orbitalsThere are five orbitals in total. The d orbital can take 10 electrons in total. The d orbitals are very influential when it comes to the transition metals.

Take-home message

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One of the seven possible 4f orbitals.

Take 14 electrons in total.

Take-home message

Different behaviors of waves and particles.

Take-home message

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The Schrödinger Equation

HΨ = EΨ

d2Ψdy2

d2Ψdx2

d2Ψdz2

+ +8π2mΘ

h2 (E-V(x,y,z)Ψ(x,y,z) = 0+

how ψ changes in space

mass of electron

total quantized energy of the atomic system

potential energy at x,y,zwave function

Take-home message

The photoelectric effect.

The presence of a threshold frequency • a beam of light consists of an enormous numbers of photons.• light intensity is related to the # of photons striking the surface per unit time.• the energy of the photon is absorbed by the electron and, if sufficient, the electron can

escape from the material with a finite kinetic energy. • a single photon can only eject a single electron, as the energy of one photon may only be

absorbed by one electron.

Absence of a time lag• The electron can not save energy from several photons.• One electron is ejected at the moment while it absorbs sufficient energy.

Incoming EM radiation striking substance.

Electrons flying off from a substance.

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The hydrogen atom• The H containing one electron.• H play important role in developing

modern electronic structure.

• The Bohr assumed hydrogen consists of a central proton and an electron moves in a circular orbit.

• The Bohr model depicts the atom as a small, positively charged nucleus surrounded by waves of electrons orbit with electrostatic forces providing attraction, the waves spread over entire orbit

• The model's key success was in explaining the spectral emission lines of atomic hydrogen; • The Bohr model is a primitive model of the hydrogen atom that cannot explain the fine

structure of the hydrogen atom nor any of the heavier atoms.

The Bohr Model

• Three points of Bohr model• Bohr designated 0 energy of an atom if the electron is completed removed from the nucleus.• Ordinary H has its lowest energy state, referred to as the ground state, n=1.• When an excited electron gave off energy as a photon of light, it drops back to its ground

state.