Quantum model of atom
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Quantum- Model of Atom. Kim Shih Ph.D. The Behavior of the Electrons. Electrons are incredibly small a single speck of dust has more electrons than the number of people who have ever lived on earth Electron behavior determines much of the behavior of atoms

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Quantum model of atom

Quantum- Model of Atom

Kim Shih Ph.D.

Kim Shih


Quantum model of atom

The Behavior of the Electrons

  • Electrons are incredibly small

    • a single speck of dust has more electrons than the number of people who have ever lived on earth

  • Electron behavior determines much of the behavior of atoms

  • Directly observing electrons in the atom is impossible, the electron is so small that observing it changes its behavior

    • even shining a light on the electron would affect it

Kim Shih


Quantum model of atom

A Theory that Explains Electron Behavior

  • The quantum-mechanical model explains the manner in which electrons exist and behave in atoms

  • Understand and predict the properties of atoms that are directly related to the behavior of the electrons

    • why some elements are metals and others are nonmetals

    • why some elements gain one electron when forming an anion, whereas others gain two

    • why some elements are very reactive while others are practically inert

    • and other periodic patterns we see in the properties of the elements

Kim Shih


Quantum model of atom

The Nature of Light:Its Wave Nature

  • Light is a form of electromagnetic radiation

    • composed of perpendicular oscillating waves, one for the electric field and one for the magnetic field

      • an electric field is a region where an electrically charged particle experiences a force

      • a magnetic field is a region where a magnetized particle experiences a force

  • All electromagnetic waves move through space at the same, constant speed

    • 3.00 x 108 m/s in a vacuum = the speed of light, c

Kim Shih


Quantum model of atom

Kim Shih


Quantum model of atom

Characterizing Waves

  • The amplitude is the height of the wave

    • the distance from node to crest

  • The wavelength (l)is a measure of the distance covered by the wave

    • the distance from one crest to the next

  • The frequency (n)is the number of waves that pass a point in a given period of time

    • the number of waves = number of cycles

    • units are hertz (Hz) or cycles/s = s−1

  • The total energy is proportional to the amplitude of the waves and the frequency

    • the larger the amplitude, the more force it has

Kim Shih


Quantum model of atom

Kim Shih


Quantum model of atom

The Relationship Between Wavelength and Frequency

  • For waves traveling at the same speed in the same distance, the shorter the wavelength, the more frequently they pass

  • This means that the wavelength and frequency of electromagnetic waves are inversely proportional

    • because the speed of light is constant, if we know wavelength we can find the frequency, and vice versa

Kim Shih


Quantum model of atom

Calculate the wavelength of red light with a frequency of 4.62 x 1014 s−1

Calculate the wavelength of a radio signal with a frequency of 100.7 MHz. 1MHz= 106 s-1

Kim Shih


Quantum model of atom

What causes different color?

What causes different brightness?

Kim Shih


Quantum model of atom

Color

  • The color of light is determined by its wavelength

    • or frequency

  • White light is a mixture of all the colors of visible light

    • a spectrum

    • RedOrangeYellowGreenBlueViolet

  • When an object absorbs some of the wavelengths of white light and reflects others, it appears colored

    • the observed color is predominantly the colors reflected

Kim Shih


Quantum model of atom

Amplitude & Wavelength

Kim Shih


Quantum model of atom

Electromagnetic Spectrum

Kim Shih


Quantum model of atom

Interference

  • The interaction between waves is called interference

  • When waves interact so that they add to make a larger wave it is called constructive interference

    • waves are in-phase

  • When waves interact so they cancel each other it is called destructive interference

    • waves are out-of-phase

Kim Shih


Quantum model of atom

Interference

Kim Shih


Quantum model of atom

Diffraction

  • When traveling waves encounter an obstacle or opening in a barrier that is about the same size as the wavelength, they bend around it – this is called diffraction

    • traveling particles do not diffract

  • The diffraction of light through two slits separated by a distance comparable to the wavelength results in an interference pattern of the diffracted waves

  • An interference pattern is a characteristic of all light waves

Kim Shih


Quantum model of atom

Diffraction

Kim Shih


Quantum model of atom

2-Slit Interference

Kim Shih


Quantum model of atom

The Photoelectric Effect

Many metals emit electrons when a light shines on their surface

Kim Shih


Quantum model of atom

Photoelectric Effect

Photoelectric Effect: Irradiation of clean metal surface with light causes electrons to be ejected from the metal. Furthermore, the frequency of the light used for the irradiation must be above some threshold value, which is different for every metal.

Kim Shih


Quantum model of atom

Photoelectric Effect:The Problem

  • In experiments it was observed that there was a minimum frequency needed before electrons would be emitted

    • called the threshold frequency

    • regardless of the intensity(amplitude)

  • It was also observed that high-frequency light from a dim source caused electron emission without any lag time

Kim Shih


Quantum model of atom

Einstein’s Explanation

  • Einstein proposed that the light energy was delivered to the atoms in packets, called quanta or photons

  • The energy of a photon of light is directly proportional to its frequency

    • the proportionality constant is called Planck’s Constant, (h)and has the value 6.626 x 10−34 J∙s

Kim Shih


Quantum model of atom

Exercise

Kim Shih


Quantum model of atom

Calculate the number of photons in a laser pulse with wavelength 337 nm and total energy 3.83 mJ

Hint:

Step 1: Convert all the units

Step 2: Find out individual photon energy

Step 3: Total energy /individual energy = total photon

Kim Shih


Quantum model of atom

What is the frequency of radiation required to supply

1.0 x 102 J of energy from 8.5 x 1027 photons?

Kim Shih


Quantum model of atom

Spectra

  • When atoms or molecules absorb energy, that energy is often released as light energy

    • fireworks, neon lights, etc.

  • When that emitted light is passed through a prism, a pattern of particular wavelengths of light is seen that is unique to that type of atom or molecule – the pattern is called an emission spectrum

    • non-continuous

    • can be used to identify the material

      • flame tests

Kim Shih


Quantum model of atom

Emission Spectra

Kim Shih


Quantum model of atom

Na

K

Li

Ba

Identifying Elements with Flame Tests

Kim Shih


Quantum model of atom

Examples of Spectra

Kim Shih


Quantum model of atom

= R∞

= R∞

1

1

1

1

1

1

λ

λ

-

-

m2

22

n2

n2

Electromagnetic Energy and Atomic Line Spectra

Johann Balmer in 1885 discovered a mathematical relationship for the four visible lines in the atomic line spectra for hydrogen.

Johannes Rydberg later modified the equation to fit every line in the entire spectrum of hydrogen.

R (Rydberg Constant) = 1.097 x 10-2 nm-1

Kim Shih


Quantum model of atom

Rutherford’s Nuclear Model

  • The atom contains a tiny dense center called the nucleus

    • the volume is about 1/10 trillionth the volume of the atom

  • The nucleus is essentially the entire mass of the atom

  • The nucleus is positively charged

    • the amount of positive charge balances the negative charge of the electrons

  • The electrons move around in the empty space of the atom surrounding the nucleus

Kim Shih


Quantum model of atom

Problems with Rutherford’s Nuclear Model of the Atom

  • Electrons are moving charged particles

  • According to classical physics, moving charged particles give off energy

  • Therefore electrons should constantly be giving off energy

    • should cause the atom to glow!

  • The electrons should lose energy, crash into the nucleus, and the atom should collapse!!

    • but it doesn’t!

Kim Shih


Quantum model of atom

The Bohr Model of the Atom

Neils Bohr (1885–1962)

  • Bohr’s major idea was that the energy of the atom was quantized, and that the amount of energy in the atom was related to the electron’s position in the atom

    • quantized means that the atom could only have very specific amounts of energy

  • The electrons travel in orbits that are at a fixed distance from the nucleus

    • stationary states

    • therefore the energy of the electron was proportional to the distance the orbit was from the nucleus

  • Electrons emit radiation when they “jump” from an orbit with higher energy down to an orbit with lower energy

    • the emitted radiation was a photon of light

    • the distance between the orbits determined the energy of the photon of light produced

Kim Shih


Quantum model of atom

Kim Shih


Quantum model of atom

Bohr Model of H Atoms

Kim Shih


Quantum model of atom

Wave Behavior of Electrons

Louis de Broglie (1892–1987)

  • de Broglie proposed that particles could have wave-like character

  • de Broglie predicted that the wavelength of a particle was inversely proportional to its momentum

  • Because it is so small, the wave character of electrons is significant

Kim Shih


Quantum model of atom

Exercise

Kim Shih


Quantum model of atom

Calculate the wavelength of an electron traveling at 2.65 x 106 m/s

Hint: v = 2.65 x 106 m/s, m = 9.11 x 10−31 kg

h= 6.626 x 10−34 J∙s, J = kg m2 /s2

l=h/mv

Determine the wavelength of a neutron traveling at 1.00 x 102 m/s (Massneutron = 1.675 x 10−24 g)

Kim Shih


Quantum model of atom

Complementary Properties

  • When you try to observe the wave nature of the electron, you cannot observe its particle nature – and vice-versa

    • wave nature = interference pattern

    • particle nature = position, which slit it is passing through

  • The wave and particle nature of the electron are complementary properties

    • as you know more about one you know less about the other

Kim Shih


Quantum model of atom

Quantum Mechanics and the Heisenberg Uncertainty Principle

In 1926 Erwin Schrödinger proposed the quantum mechanical model of the atom which focuses on the wavelike properties of the electron.

In 1927 Werner Heisenberg stated that it is impossible to know precisely where an electron is and what path it follows—a statement called the Heisenberg uncertainty principle.

Kim Shih


Quantum model of atom

Uncertainty Principle

  • Heisenberg stated that the product of the uncertainties in both the position and speed of a particle was inversely proportional to its mass

    • x = position, Dx = uncertainty in position

    • v = velocity, Dv = uncertainty in velocity

    • m = mass

  • This means that the more accurately you know the position of a small particle, such as an electron, the less you know about its speed

    • and vice-versa

Kim Shih


Quantum model of atom

Heisenberg Uncertainty Principle

Based on de Broglie equation, electron velocity

related to λ.

Electron has both wave and particle properties called

Duality

It means we can not measure its position and velocity

at the same time.

Kim Shih


Quantum model of atom

You are wrong, because

your brain was hit by an apple

I know gravity

Kim Shih


Quantum model of atom

Determinacy vs. Indeterminacy

  • According to classical physics(Newton’s Law), particles move in a path determined by the particle’s velocity, position, and forces acting on it

    • determinacy = definite, predictable future

  • Because we cannot know both the position and velocity of an electron, we cannot predict the path it will follow

    • indeterminacy = indefinite future, can only predict probability

  • The best we can do is to describe the probability an electron will be found in a particular region using statistical functions

Kim Shih


Quantum model of atom

Trajectory vs. Probability

Kim Shih


Quantum model of atom

Schrödinger’s Equation

  • Schödinger’s Equation allows us to calculate the probability of finding an electron with a particular amount of energy at a particular location in the atom

  • Solutions to Schödinger’s Equation produce many wave functions, Y

    • A plot of distance vs. Y2 represents an orbital, a probability distribution map of a region where the electron is likely to be found

Kim Shih


Quantum model of atom

Solutions to the Wave Function, Y

  • Calculations show that the size, shape, and orientation in space of an orbital are determined to be three integer terms in the wave function

    • added to quantize the energy of the electron

  • These integers are called quantum numbers

    • principal quantum number, n

    • angular momentum quantum number, l

    • magnetic quantum number, ml

Kim Shih


Quantum model of atom

Probability of finding

electron in a region

of space (Y2)

solve

Wave

equation

Wave function

or orbital (Y)

Wave Functions and Quantum Numbers

A wave function is characterized by three parameters called quantum numbers, n, l, ml.

Kim Shih


Quantum model of atom

Principal Quantum Number (n)

  • Describes the size andenergylevel of the orbital

  • Commonly called shell

  • Positive integer (n = 1, 2, 3, 4, …)

  • As the value of n increases:

    • The energy of the electron increases

    • The average distance of the electron from the nucleus increases

    • an electron would have E = 0 when it just escapes the atom

Kim Shih


Quantum model of atom

Principal Energy Levels in Hydrogen

Kim Shih


Quantum model of atom

Angular-Momentum Quantum Number (l)

  • Defines the three-dimensional shape of the orbital

  • Commonly called subshell

  • There are n different shapes for orbitals

    • If n = 1 then l = 0

    • If n = 2 then l = 0 or 1

    • If n = 3 then l = 0, 1, or 2

    • etc.

  • Commonly referred to by letter (subshell notation)

    • l = 0s (sharp) (spherical shape)

    • l = 1p (principal) (two balloons tied at the knots shape)

    • l = 2d (diffuse) (four balloons tied at the knot)

    • l = 3f (fundamental) (eight balloons tied at the knot)

    • etc.

Kim Shih


Quantum model of atom

Magnetic Quantum Number (ml )

  • Defines the spatial orientation of the orbital

  • There are 2l + 1 values of ml and they can have any integral value from -l to +l

    • If l = 0 then ml = 0

    • If l = 1 then ml = -1, 0, or 1

    • If l = 2 then ml = -2, -1, 0, 1, or 2

    • etc.

Kim Shih


Quantum model of atom

Energy Shells and Subshells

Kim Shih


Quantum model of atom

Wave Functions and Quantum Numbers

Kim Shih


Quantum model of atom

Number of Electrons in Electron Shells and Subshells

Orbitals

Available

(2l + 1)

Number of Electrons

Possible in Subshell

[2(2l+1)]

Maximun Electrons

Possible for nth

Shell (2n2)

Subshells

Available

Shell

1

s 1 2 2

s 1 2 8

2

p 3 6

s 1 2 18

3

p 3 6

d 5 10

s 1 2 32

4

p 3 6

d 5 10

f 7 14

Kim Shih


Quantum model of atom

Exercise

Kim Shih


Quantum model of atom

What are the quantum numbers and names (for example, 2s, 2p) of the orbitals in then = 4principal level? How many orbitals exist?

n = 4

 l = 0, 1, 2, 3

  • n = 4, l = 1 (p)

  • ml : −1,0,+1

  • 3 orbitals

  • 4p

  • n = 4, l = 0 (s)

  • ml : 0

  • 1 orbital

  • 4s

  • n = 4, l = 3 (f)

  • ml : −3,−2,−1,0,+1,+2,+3

  • 7 orbitals

  • 4f

  • n = 4, l = 2 (d)

  • ml : −2,−1,0,+1,+2

  • 5 orbitals

  • 4d

total of 16 orbitals: 1 + 3 + 5 + 7 = 42 = n2

Kim Shih


Quantum model of atom

1. Identify the shell and subshell when n = 3 l = 1 ml = 1

2. What are the possible quantum numbers for 4p?

3. Which of the following is not correct?

a. n = 3 l = 0 ml = 0

b. n = 2 l = 1 ml = -1

c. n = 1 l = 0 ml = 0

d. n = 4 l = 1 ml = -2

Kim Shih


Quantum model of atom

Quantum Mechanical Explanation of Atomic Spectra

  • Each wavelength in the spectrum of an atom corresponds to an electron transition between orbitals

  • When an electron is excited, it transitions from an orbital in a lower energy level to an orbital in a higher energy level

  • When an electron relaxes, it transitions from an orbital in a higher energy level to an orbital in a lower energy level

  • When an electron relaxes, a photon of light is released whose energy equals the energy difference between the orbitals

Kim Shih


Quantum model of atom

Electron Transitions

  • To transition to a higher energy state, the electron must gain the correct amount of energy corresponding to the difference in energy between the final and initial states

  • Electrons in high energy states are unstable and tend to lose energy and transition to lower energy states

  • Each line in the emission spectrum corresponds to the difference in energy between two energy states

Kim Shih


Quantum model of atom

Quantum Leaps

Kim Shih


Quantum model of atom

Predicting the Spectrum of Hydrogen

  • The wavelengths of lines in the emission spectrum of hydrogen can be predicted by calculating the difference in energy between any two states

  • For an electron in energy state n, there are (n – 1) energy states it can transition to, therefore (n – 1) lines it can generate

  • Both the Bohr and Quantum Mechanical Models can predict these lines very accurately for a 1-electron system

Kim Shih


Quantum model of atom

Hydrogen Energy Transitions

Kim Shih


Quantum model of atom

Kim Shih


Quantum model of atom

Exercise

Kim Shih


Quantum model of atom

= R∞

= R∞

1

1

1

1

λ

1

1

λ

-

-

m2

n2

22

n2

Please identify all 4 lines(wavelength) in hydrogen

spectrum.

Rydberg Equation:

R (Rydberg Constant) = 1.097 x 10-2 nm-1

Johann Balmer Equation:

Kim Shih


Quantum model of atom

1

1

1

λ

-

m2

n2

Calculate the wavelength of light emitted when the hydrogen electron transitions from n = 6 to n = 5, E = hc/l

Rydberg Equation:

E

= R∞

=

hc

R (Rydberg Constant) = 1.097 x 10-2 nm-1

h (Planck’s constant) = 6.626 x 10-34 J s

Kim Shih


Quantum model of atom

The Shapes of Orbitals

Kim Shih


Quantum model of atom

Phases

Kim Shih


Quantum model of atom

Probability Density Function

The probability density function represents the total probability of finding an electron at a particular point in space

Kim Shih


Quantum model of atom

2s and 3s

2s

n = 2,

l = 0

3s

n = 3,

l = 0

Kim Shih


Quantum model of atom

The Shapes of Orbitals

Node: A surface of zero probability for finding the electron.

Number of nodes = (n – 1)

Kim Shih


Quantum model of atom

l = 1, p orbitals

  • Each principal energy state above n = 1 has three p orbitals

    • ml = −1, 0, +1

  • Each of the three orbitals points along a different axis

    • px, py, pz

  • 2nd lowest energy orbitals in a principal energy state

  • Two-lobed

  • One node at the nucleus, total of n nodes

Kim Shih


Quantum model of atom

The Shapes of Orbitals

Kim Shih


Quantum model of atom

l = 2, d orbitals

  • Each principal energy state above n = 2 has five d orbitals

    • ml = −2, − 1, 0, +1, +2

  • Four of the five orbitals are aligned in a different plane

    • the fifth is aligned with the z axis, dz squared

    • dxy, dyz, dxz, dx squared – y squared

  • 3rd lowest energy orbitals in a principal energy level

  • Mainly four-lobed

    • one is two-lobed with a toroid

  • Planar nodes

    • higher principal levels also have spherical nodes

Kim Shih


Quantum model of atom

d orbitals

Kim Shih


Quantum model of atom

l = 3, f orbitals

  • Each principal energy state above n = 3 has seven d orbitals

    • ml = −3, −2, −1, 0, +1, +2, +3

  • 4th lowest energy orbitals in a principal energy state

  • Mainly eight-lobed

    • some two-lobed with a toroid

  • Planar nodes

    • higher principal levels also have spherical nodes

Kim Shih


Quantum model of atom

f orbitals

Kim Shih


Quantum model of atom

The Phase of an Orbital

  • Orbitals are determined from mathematical wave functions

  • A wave function can have positive or negative values

    • as well as nodes where the wave function = 0

  • The sign of the wave function is called its phase

  • When orbitals interact, their wave functions may be in-phase (same sign) or out-of-phase (opposite signs)

    • this is important in bonding

Kim Shih


Quantum model of atom

ORBITALS

s orbital

p orbitals

d orbitals

f orbitals

Kim Shih


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