Download CHAPTER 7: The Quantum-Mechanical Model of the Atom Energy

CHAPTER 7: The Quantum-Mechanical Model of the Atom
2. Frequency (v – Read as ‘nu’): Number of waves per second that pass a
-1
given point in space (unit: 1/s or s or Hz - hertz).
8
Energy
8
-1
3. Speed (c): Speed of light is 2.9979 X 10 m/s (3.0 X 10 ms )
Energy is the capacity of a physical system to perform work.
It exists in several forms such as heat, light, electrical, mechanical energy,
etc.
Radiation
Radiation is energy that comes from a source (eg. atom or molecule) and
travels through some material or through space.
Electromagnetic Radiation
Electromagnetic radiation is the emission and transmission of energy in the
form of electromagnetic waves.
It consists of electric and magnetic field.
Wavelength and frequency can be interconverted
c = vλ
λ or v = c/λ
Exercise:
Calculate the frequency of red light of wavelength 6.50 x 102 nm.
8
Speed of light (c) = 2.9979 x 10 m/s
2
9
= 6.50 x 10 nm (convert it to meter, 1 meter = 10 nm)
2
9
= 6.50 x10 nm x (1m/10 nm)
= 6.50 x 10-7 m
v = c/λ
λ
8
-7
Therefore, v = (2.9979 x 10 m/s) / (6.50 x 10 m)
14
= 4.61 x 10
Electromagnetic radiation travels through space at the speed of light in a
vacuum.
Electromagnetic Spectrum
Electromagnetic Waves
s
-1
Eg. Light from the sun, Energy used to cook food in microwave oven, X-ray,
etc.
The electromagnetic spectrum is the range of all possible wavelengths (or
frequencies) of electromagnetic radiation.
Electromagnetic Waves have 3 primary characteristics:
1. Wavelength (λ): Distance between two peaks in a wave (unit: meter).
General Chemistry Chapter 7 (Page:
21)
•
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.
The "electromagnetic spectrum" of an object is the characteristic distribution
of electromagnetic radiation emitted or absorbed by that particular object.
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.
In te r fe re n c e
The Photoelectric Effect: It was observed that many metals
emit electrons when a light shines on their surface.
This is called the photoelectric effect
effect.
•
According to classic wave theory, if the wavelength of light is made
shorter, or the light wave’s intensity made stronger, more electrons
should be ejected.
General Chemistry Chapter 7 (Page:
T ro , P ri n ci p le s o f C h e m i st ry: A M o l e c u la r A p p ro a c h
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22)
In experiments with the photoelectric effect, it was observed
that there was a maximum wavelength for electrons to be emitted.
called the threshold frequency
Atomic Spectrum of Hydrogen
Characteristic distribution of electromagnetic radiation emitted or absorbed
by hydrogen atom.
When a sample of hydrogen gas receives a high-energy
ergy spark, the H2
molecules absorb energy, and some of the H-H
H bonds are broken.
The resulting Hydrogen atoms contain some excess energy (they are called
excited hydrogen atoms).
Excited atoms release the energy by emitting light (in the form of photons) of
various wavelengths (hydrogen emission spectrum).
High energy
spark
The wavelength of light is depends on the energy loss from the excited
hydrogen atom.
H2
Excited ‘H’ atom
(Has more energy)
H
Light (photons)
Changes in energy between discrete energy levels in hydrogen will produce
only certain wavelengths of emitted light.
To understand the hydrogen emission spectrum, we need to know about
continuous and line (discrete) spectrum.
Continuous spectrum
It contains all the wavelengths of visible light.
It is produced when white light is passed through a prism (It looks like a
rainbow) (see figure ‘a’).
Line (discrete) spectrum
It contains only some of the (discrete) wavelengths of light.
It is produced when hydrogen emission spectrum is passed through a prism
(see figure ‘b’).
General Chemistry Chapter 7 (Page:
23)
In summary, when excited hydrogen atom fall back to ground state (that is
from the high to lower energy level), it release energy in the form of light
(photons).
The wavelength of the emitted light can be calculated by Planck’s Equation.
-18
18
RH (Rydberg constant) = 2.18 x 10 J
Z = Atomic number (number of protons)
Exercise
What is the energy levels of electron in hydrogen atom when it is in orbital n
= 1.
Atomic number (Z) = 1; Number of orbital (n)) = 1
∆E = change in energy, in J
h = Planck’s constant, 6.626 X 10
v = frequency, in s
-34
Js
-1
E =
=
=
-18
2
2
-2.18 X 10 J (Z /n )
-18
2 2
-2.18 X 10 J (1 /1 )
-18
-2.18 X 10 J
λ = wavelength, in m
c = Speed of light
When electron moving from one orbit to another orbit, the energy difference
(∆E) can be calculated using the following formula.
Therefore, wavelength λ = hc / ∆E
∆E = -2.18 X 10
-18
2
2
2
2
J X (Z /n
nfinal – Z /ninitial )
Where, Z = nuclear charge
The Bohr model (Bohr model of atom)
Bohr developed a quantum model for the hydrogen atom.
ninitial = Orbit of electron in initial stage
nfinal = Orbit of electron after receiving or loosing energy (final stage).
Electrons in hydrogen atom moves around the nucleus only in certain
allowed circular orbits.
Exercise:
Light is emitted as e- moves from one energy level to a lower energy level.
What is the energy change if a excited hydrogen atom fall back from the orbit
n = 6 to ground state (n=1)
Where, Z for hydrogen atom (number of protons) = 1
ninitial = 6
nfinal = 1
Energy change (∆E) = -2.18 X 10
2
J X (1/1 – 1/36)
-18
J X (1 – 0.028)
= -2.18 X 10
2
2
= -2.18 X 10-18 J X (0.972)
-18
= - 2.119 X 10
Where, n (principal quantum number) = 1,2,3,B
2
J X (1 /1 – 1 /6 )
-18
= -2.18 X 10
Energy of an electron in an orbit can be calculated by the following formula:
-18
J
When the electron is fall from higher energy stage to lower energy stage, it
emits light (in the form of photons).
General Chemistry Chapter 7 (Page:
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The energy of photons (Ephoton)= Energy change (∆E)
The wavelength of the photon (light) can be calculated using the following
formula.
Ephoton = h x (c / λ)
Here Ephoton = ∆E
Quantum numbers
Therefore,
Each of the orbitals is characterized by a series of numbers called quantum
numbers.
∆E = h x (c / λ)
1. Principal quantum number (n) is related to size and energy of the
orbital (n = 1, 2, 3, . . .).
When the value of ‘n’ is increase, the energy and size of the orbital
also increase.
λ = h x (c / ∆E)
Where h = Planck’s constant, 6.626 X 10
-34
J.s
8
c = Speed (speed of light = 2.9979 x 10 m/s)
λ = wavelength, in m
Exercise:
Calculate the wavelength of light emitted from the excited hydrogen atom
when it is fall back from orbit n=6 to ground state (n=1).
Wavelength (λ) = h x (c / ∆E)
-34
= 6.626 X 10
8
-18
J.s X (2.9979 x 10 m/s) / (- 2.119 X 10
(-34+8+18)
= 9.374 X 10
J)
m
-8
= 9.374 X 10 m
Quantum mechanical model of the atom
It is known as quantum mechanics or wave mechanics. It is based on the
wave properties of the atom.
“The electron bound to the nucleus seemed similar to a standing wave”.
2. Angular Momentum quantum number (l) is relates to the shape of
the orbital (l = 0 to ‘n – 1’).
Eg.
If n = 3, l = 0, 1, 2
The value of ‘l’ (called subshell) for a particular orbital is commonly
assigned letter designation.
Eg.
l=0 is called ‘s’
l=1 is called ‘p’
l = 2 is called ‘d’
l =3 is called ‘f’
l=4 is called ‘g’
3. Magnetic quantum number (ml) is relates to orientation of the
orbital in space relative to other orbitals (ml = l to -l).
Eg.
If n = 3
l = 0, 1, 2
Ml = -2, -1, 0, -1, -2
A specific wave function is often called an orbital.
Eg. The hydrogen electron visualized as a standing wave around the nucleus
General Chemistry Chapter 7 (Page:
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Representation of 1s, 2s, and 3s orbitals
Representation of 2p orbitals
Exercise
Q:
A:
For the quantum level n=5, determine the number of allowed
subshells (different values of l) and give the designation of each.
n=5
l = 0 to ‘n-1’ = 0, 1, 2, 3, 4
Subshell (l) = p = 1
ml = -1, 0, +1
Number of orbitals = 3
Maximum number of electrons (2 electrons per 1 orbital) = 2 X 3 = 6
Designation: 5s, 5p, 5d, 5f, and 5g (where 5 is the indication of ‘n’)
Exercise
Q:
How many orbitals in the quantum level n=5?
A:
n=5
l = 0 to ‘n-1’ = 0, 1, 2, 3, 4
ml = l to –l = -4, -3, -2, -1, 0, 1, 2, 3, 4
Number of orbitals = 9
Note: d, f and g subshells are little complicated to understand from the
picture. Just know about the total number of orbitals (and maximum
electron occupancy) in d, f and g subshells (see table 7.2).
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