1. Chapter 7 Quantum Theory of Atom Bushra Javed

2. Contents The Wave Nature of Light Quantum Effects and Photons
The Bohr Theory of the Hydrogen Atom
Quantum Mechanics
5. Quantum Numbers and Atomic Orbitals
6. Quantum Numbers and Atomic Orbitals

3. A Theory that Explains Electron Behavior
The quantum-mechanical model explains the manner in which electrons exist and behave in atoms
It helps us understand and predict the properties of atoms that are directly related to the behavior of the electrons

4. 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

5. Electromagnetic Radiation

6. Electromagnetic Spectrum
The range of frequencies and wavelengths of electromagnetic radiation is called the electromagnetic spectrum.

7. Characterizing Waves The amplitude is the height of the wave
the distance from node to crest
or node to trough
the amplitude is a measure of how intense the light is – the larger the amplitude, the brighter the light
The wavelength (l) is a measure of the distance covered by the wave
the distance from one crest to the next

8. Wave Characteristics
Tro: Chemistry: A Molecular Approach, 2/e

9. Characterizing Waves
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
1 Hz = 1 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

10. Characterizing Waves
The longer the wavelength of light, the lower the frequency.
The shorter the wavelength of light, the higher the frequency.

11. 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

12. Wavelength, Frequency & Speed of Light
Wavelength and Frequency are related as:
λ α 1/ v
c = λ v
Where c is the speed of light
Light waves always move through empty space at the same speed.
Therefore speed of light is a fundamental constant.
c = 3.00 x 108 m/s

13. The Wave Nature of Light
Example 1
Which of the following statements is incorrect?
a. The product of wavelength and frequency of electromagnetic radiation is a constant.
b. As the energy of a photon increases, its frequency decreases.
c. As the wavelength of a photon increases, its frequency decreases.
d. As the frequency of a photon increases, its wavelength decreases.

14. Units of Wavelength & Frequency
nanometers, micrometers, meters and Angstrom
1 nm = 10-9m
1A° = 10-10m
Units of Frequency:
1/second (per second) or 1s-1
Hertz = Hz, MHz and GHz
1-1s = 1 Hz

15. Wavelength & Frequency
Example 2 What is the wavelength of blue light with a frequency of 6.4  1014/s?
c = nl so
l = c/n
n = 6.4  1014/s
c = 3.00  108 m/s
l = 4.7  10−7 m

16. Wavelength & Frequency
Example 3 What is the frequency of light having a wavelength of 681 nm?
c = nl so
n = c/l
l = 681 nm = 6.81  10−7 m
c = 3.00  108 m/s
l = 4.41  1014 /s

17. Electromagnetic Spectrum
Example 4 Which type of electromagnetic radiation has the highest energy? a. radio waves b. gamma rays c. blue light d. red light

18. Wave Theory of light
One property of waves is that they can be diffracted—that is, they spread out when they encounter an obstacle about the size of their wavelength. In 1801, Thomas Young, a British physicist, showed that light could be diffracted. By the early 1900s, the wave theory of light was well established.

19. Diffraction

20. The Wave/Particle Nature of Light
Max Planck worked on the light of various frequencies emitted by the hot solids at different temperatures.
In 1900, Max Planck proposed that radiant energy is not continuous, but is emitted in small bundles.
Planck theorized that the atoms of an heated object vibrate with a definite frequency.
An individual unit of light energy is a photon.

21. Dual Nature of Light Einstein expanded on Planck’s work.
Through Photoelectric Effect ,he showed that when light of particular frequency shines on a metal surface it knocks out electrons.
This knocking out of electrons is dependent only on the frequency of the shining light.

22. The Photoelectric Effect

23. 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
inversely proportional to its wavelength
the proportionality constant is called Planck’s Constant, (h) and has the value x 10−34 J∙s

24. Einstein’s Explanation
Light, therefore, has properties of both waves and matter. Neither understanding is sufficient alone. This is called the particle–wave duality of light.

25. Kinetic Energy = Ephoton – Ebinding
Ejected Electrons
One photon at the threshold frequency gives the electron just enough energy for it to escape the atom
binding energy, f
When irradiated with a shorter wavelength photon, the electron absorbs more energy than is necessary to escape
This excess energy becomes kinetic energy of the ejected electron
Kinetic Energy = Ephoton – Ebinding
KE = hn − f

26. The Photoelectric Effect
Example 5:
Suppose a metal will eject electrons from its surface when struck by yellow light. What will happen if the surface is struck with ultraviolet light?
a. No electrons would be ejected.
Electrons would be ejected, and they would have the same kinetic energy as those ejected by yellow light.
c. Electrons would be ejected, and they would have
greater kinetic energy than those ejected by yellow
light.
d. Electrons would be ejected, and they would have lower
kinetic energy than those ejected by yellow light.

27. Photoelectric Effect Example 6
The blue–green line of the hydrogen atom spectrum has a wavelength of 486 nm. What is the energy of a photon of this light?
E = hn and
c = nl so
E = hc/l
l = 486 nm = 4.86  10−7 m
c = 3.00  108 m/s
h = 6.63  10−34 J  s
E = 4.09  10−19 J

28. 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
The electrons should lose energy, crash into the nucleus, and the atom should collapse!!
but it doesn’t

29. Continuous Vs Line Spectra
In addition, this understanding could not explain the observation of line spectra of atoms. A continuous spectrum contains all wavelengths of light. A line spectrum shows only certain colors or specific wavelengths of light. When atoms are heated, they emit light. This process produces a line spectrum that is specific to that atom.

30. 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

31. Exciting Gas Atoms to Emit Light with Electrical Energy

32. Identifying Elements with Flame TestsNa K Li Ba

33. Emission Spectra

34. Emission Line Spectra
When an electrical voltage is passed across a gas in a sealed tube, a series of narrow lines is seen.
These lines are the emission line spectrum. The emission line spectrum for hydrogen gas shows three lines: 434 nm, 486 nm, and 656 nm.

36. The Bohr Model of the Atom
The nuclear model of the atom does not explain what structural changes occur when the atom gains or loses energy
Bohr developed a model of the atom to explain how the structure of the atom changes when it undergoes energy transitions
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

37. Bohr’s Model
The electrons travel in orbits that are at a fixed distance from the nucleus
Energy Levels
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

38. Atomic Line Spectra

39. Bohr’s Model of Atom
Example 7 Which of the following energy level changes in a hydrogen atom produces a visible spectral line? a. 3 → 1 b. 4 → 2 c. 5 → 3 d. all of the above

40. Rydberg’s Spectrum Analysis
Rydberg analyzed the spectrum of hydrogen and found that it could be described with an equation that involved an inverse square of integers

41. Bohr’s Model of Atom
Energy-Level Postulate An electron can have only certain energy values, called energy levels. Energy levels are quantized. For an electron in a hydrogen atom, the energy is given by the following equation: RH =  10−18 J n = principal quantum number

42. Bohr’s Model of Atom
The energy of the emitted or absorbed photon is related to DE: We can now combine these two equations:

43. Bohr’s Model of Atom
Light is absorbed by an atom when the electron transition is from lower n to higher n (nf > ni). In this case, DE will be positive. Light is emitted from an atom when the electron transition is from higher n to lower n (nf < ni). In this case, DE will be negative. An electron is ejected when nf = ∞.

44. Electron transitions for an electron in the hydrogen atom.

45. Example 8.
What is the wavelength of the light emitted when the electron in a hydrogen atom undergoes a transition from n = 6 to n = 3?
ni = 6
nf = 3
RH =  10−18 J
= −1.816  10−19 J
1.094  10−6 m

46. Bohr’s Model of Atom Example 9
What is the frequency (in kHz ) of light emitted when
the electron in a hydrogen atom undergoes a transition
from level n = 6 to level n = 2?
h = 6.63 × J . s), RH = × J)
a × 105 kHz
b × 1011 kHz
c × 1014 kHz
d × 10–28 kHz

47. Wave properties of Matter
In 1923, Louis de Broglie, a French physicist, reasoned that particles (matter) might also have wave properties. The wavelength of a particle of mass, m (kg), and velocity, v (m/s), is given by the de Broglie relation:

48. Wave properties of Matter
Example 10
Compare the wavelengths of (a) an electron traveling at a speed that is one-hundredth the speed of light and (b) a baseball of mass kg having a speed of 26.8 m/s (60 mph).
Baseball
m = kg
v = 26.8 m/s
Electron
me = 9.11  10−31 kg
v = 3.00  106 m/s

49. Electron
me = 9.11  10−31 kg
v = 3.00  106 m/s
2.43  10−10 m
Baseball
m = kg
v = 26.8 m/s
1.71  10−34 m

50. Quantum (or Wave) mechanics
Building on de Broglie’s work, Erwin Schrodinger devised a theory that could be used to explain the wave properties of electrons in atoms and molecules. The branch of physics that mathematically describes the wave properties of submicroscopic particles is called or wave mechanics.

51. 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

52. Uncertainty Principle
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

53. Schrodinger’s Equation
Schodinger’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, A plot of distance vs. Y2 represents an orbital, a probability distribution map of a region where the electron is likely to be found

54. 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

55. Principal Quantum Number, n
Characterizes the energy of the electron in a particular orbital
corresponds to Bohr’s energy level
n can be any integer  1
The larger the value of n, the more energy the orbital has
Energies are defined as being negative
an electron would have E = 0 when it just escapes the atom
The larger the value of n, the larger the orbital
As n gets larger, the amount of energy between orbitals gets smaller

56. Angular Momentum Quantum Number, l
The angular momentum quantum number determines the shape of the orbital
l can have integer values from 0 to (n – 1)
Each value of l is called by a particular letter that designates the shape of the orbital

57. Angular Momentum Quantum Number,
s orbitals are spherical
p orbitals are like two balloons tied at the knots
d orbitals are mainly like four balloons tied at the
Knot
f orbitals are mainly like eight balloons tied at the
knot

58. Magnetic Quantum Number, ml
The magnetic quantum number is an integer that specifies the orientation of the orbital
the direction in space the orbital is aligned relative to the other orbitals
Values are integers from −l to +l
including zero
gives the number of orbitals of a particular shape
when l = 2, the values of ml are
−2, −1, 0, +1,+2
which means there are five orbitals with l = 2

59. Describing an Orbital Each set of n, l, and ml describes one orbital
Orbitals with the same value of n are in the same principal energy level
aka principal shell
Orbitals with the same values of n and l are said to be in the same sublevel

60. Energy Shells and Subshells

61. Quantum Numbers Spin Quantum Number, ms
This quantum number refers to the two possible orientations of the spin axis of an electron.
It may have a value of either +1/2 or -1/2.

62. The Quantum Numbers
Example 11 Give the notation (using letter designations for 1) of subshells for the following quantum numbers. n. = 3, l = 1 n. = 4, l = 0 n. = 5,1 = 3 n. = 6,1 = 2 n. = 2,1 = 1

63. The Quantum Numbers Example 12
Determine how many valid sets of quantum numbers
exist for a 3p orbital ; Give two examples.
Strategy:
Write the the number of possible values for each
quantum number.
Quantum number n 1 m ms
Possible values 3 1 1, 0, /2, -1/2
# of possible values
Possible sets of values for a 3p electron: (1)(1)(3)(2) = 6

64. The Quantum Numbers
There are six sets of quantum numbers that describe
a 3p electron.
Examples:
n = 3 1=1 m1= ms = +1/2
n = 3 1=1 m1 = ms = +1/2

65. The Quantum Numbers
Example 13 Which of the following sets of quantum numbers (n, l, ml, ms) is not permissible? a ½ b ½ c ½ d ½

66. The Quantum Numbers
Example 14 Determine how many valid sets of quantum numbers exist for 4d orbitals ;Give three examples.

67. The Quantum Numbers Example 15
Determine how many valid sets of quantum
numbers exist for 5f orbitals ;Give three
examples.

68. The Shapes of Orbitals s Orbitls : 1. All s orbitals are spherical
2. The probability of finding an s electron depends only on distance from the nucleus, not on direction
3. There is only one orientation of a sphere in space
Therefore S orbital has ml = 0
The value of ψ2 for an s-orbital is greatest near the nucleus
The size of the s orbital increases as the number of shell increases i.e. 3s > 2s>1s

69. S Orbitals

70. Probability Density Function
The probability density function represents the total probability of finding an electron at a particular point in space
Tro: Chemistry: A Molecular Approach, 2/e

71. p Orbitals p orbitlas are dumbbell –shaped.
2. The electron distribution is concentrated in identical lobes on either side of the nucleus and separated by a planar node cutting through the nucleus
3. Probability of finding a p electron near the nucleus is zero.
4. The two lobes of a p orbital have different phases of a wave (different algebraic signs)
5. The size of p orbitals increases in the order 4p>3p>2p

73. d Orbitals
1. The third and higher shells contain each contain five d orbitals 2. d orbitals have two shapes 3. Four of the five d orbitals are cloverleaf –shaped and have four lobes of maximum electron probability 4. The fifth d orbital is similar in shape to a pz orbital but has an additional donut –shaped region of electron probability centered in the xy plane 5. All five d orbitals have the same energy 6. Alternating lobes of the d orbitals have different phases

74. d Orbitals

75. f orbitals
Tro: Chemistry: A Molecular Approach, 2/e