In this presentation you will be able to, describe how atomic orbitals arise from the Schrodinger's equation, relate orbital shapes to electron density distribution and interpret the information obtained from a four set of quantum numbers.
2. Explain Heisenberg’s Uncertainty Principle
Describe how atomic orbitals arise from the
Schrodinger equation
Relate orbital shapes to electron density
distribution
Qualitatively sketch the orbital shapes
Interpret the information obtained from a set of
four quantum numbers
Objectives
3. Assign the correct set of quantum numbers
for an electron
Objectives
4. a. Heisenberg’s Uncertainty Principle
b. Schrodinger Equation
c. Wave function
d. Electron probability density
e. Atomic orbital
f. Principal quantum number
g. Angular momentum quantum number
Keywords
6. HEISENBERG’S UNCERTAINTY
PRINCIPLE
formulated by Werner Heisenberg, a German
physicist
“the position of a particle and its momentum
cannot be simultaneously measured with
arbitrarily high precision”
In other words, it is not possible to measure
the exact position and the exact momentum
of a particle at the same time.
7. Mathematically, this is stated as:
where "x is the uncertainty in position, "p is
the uncertainty in momentum, and h is
Planck’s constant
8. The Bohr model violates Heisenberg’s
Uncertainty Principle. Electrons do not go
around the nucleus in well-defined orbits.
Otherwise, we will be able to determine the
exact position and momentum of the electron
in the atom at the same time.
A better model is needed to fully describe the
atom.
9. SCHRODINGER EQUATION
In 1926, Erwin Schrodinger, an Austrian
physicist, formulated a mathematical equation
that describes the behavior and energies of
submicroscopic particles.
The Schrodinger equation incorporates
particle behavior and wave behavior, treating
the electron as a standing wave.
10. SCHRODINGER EQUATION
The solution to the Schrodinger equation is a
wave function called ψ (psi). The wave
functions are also called atomic orbitals (as
distinguished from the Bohr orbits). Aside
from the wave functions, energies are also
obtained from solving the equation.
11. The wave function itself has no physical
meaning. However, the probability of finding
the electron in a particular volume element in
space is proportional to ψ2. In wave theory,
the intensity of light is proportional to the
square of the amplitude of the wave or ψ2.
Similarly, the most likely place to find the
particle is where the value of ψ2 is greatest.
12. The Schrodinger equation began a new field in
physics and chemistry referred to as quantum
mechanics or wave mechanics.
The Schrodinger equation can be solved exactly
for the hydrogen atom but not for atoms with
more than one electron. For many-electron
atoms, approximation methods are used to
solve the Schrodinger equation.
13. THE QUANTUM MECHANICAL
DESCRIPTION OF THE HYDROGEN ATOM
It is not possible to pinpoint the exact location
of the electron in an atom but ψ2 gives the
region where it can most probably be found.
The electron density gives the probability that
the electron will be found in a particular
region of an atom.
14. THE QUANTUM MECHANICAL
DESCRIPTION OF THE HYDROGEN ATOM
representation of the electron density
distribution around the nucleus in the
hydrogen atom
15. THE QUANTUM MECHANICAL
DESCRIPTION OF THE HYDROGEN ATOM
The darker the shade, the higher the
probability of finding the electron in that
region.
the probability distribution is spherical
ψ is the solution to the Schrodinger equation.
It is also referred to as an atomic orbital.
16. THE QUANTUM MECHANICAL
DESCRIPTION OF THE HYDROGEN ATOM
When we say that the electron is in an atomic orbital,
we mean that it is described by a wave function, ψ,
and that the probability of locating the electron is
given by the square of the wave function associated
with that orbital.
The atomic orbital has a characteristic energy as well
as a characteristic electron density distribution. This
electron density distribution in three-dimensions
gives the shape of the atomic orbital.
17. In the mathematical solution of the Schrodinger
equation, three quantum numbers are obtained.
These are the principal quantum number (n), the
angular quantum number, (ℓ) ,and the magnetic
quantum number (ml). They describe the atomic
orbitals. A fourth quantum number, the spin
quantum number (ms) completes the description
of the electrons in the atoms.
The QUANTUM NUMBERS
18. a. Determines the energy of an orbital
b. Determines the orbital size
c. Is related to the average distance of the electron from
the nucleus in a particular orbital; the larger the n
value, the farther the average distance of the electron
from the nucleus
d. Can have the values: n = 1, 2, 3, …
e. Orbitals with the same n are said to be in the same
shell.
Principal Quantum Number (n)
19. a. Describes the “shape” of the orbitals
b. Can have the following values: ℓ = 0, 1, 2, up to n-1.
Examples
n value ℓ value
1 0
2 0, 1
3 0, 1, 2
Angular Momentum Quantum Number
(ℓ)
20. c. Orbitals with the same n and ℓ values belong
to the same subshell.
d. It is usually designated by letters s, p, d, f, …
which have a historical origin from spectral
lines. The designations are as follows
Angular Momentum Quantum
Number (ℓ)
21. The s, p, d, f designations of the orbitals refer
to sharp, principal, diffuse, and fundamental
lines in emission spectra.
22. a. Describes the orientation of the orbital in
space
b. Can have the values: - ℓ, (-ℓ + 1), … 0, … (+ ℓ -
1), + ℓ
Magnetic Quantum Number
(ml)
23. a. The first three quantum numbers describe
the energy, shape and orientation of
orbitals. The 4th quantum number refers to
two different spin orientations of electrons
in a specified orbital.
Electron Spin Quantum
Number (ms)
24. b. When lines of the hydrogen spectrum are
examined at very high resolution, they are
found to be closely spaced doublets and
called as the Zeeman effect. This splitting is
called fine structure, and was one of the first
experimental evidences for electron spin. The
direct observation of the electron's intrinsic
angular momentum was achieved in the
Stern–Gerlach experiment.
Electron Spin Quantum
Number (ms)
25. Electron Spin Quantum
Number (ms)
c. Uhlenbeck, Goudsmit, and Kronig (1925)
introduced the idea of the self-rotation of
the electron. The spin orientations are called
"spin-up" or "spin-down" and is assigned
the number ms = ½ ms = -½, respectively.
26. d. The spin property of an electron would give
rise to magnetic moment, which was a
requisite for the fourth quantum number.
The electrons are paired such that one spins
upward and one downward, neutralizing the
effect of their spin on the action of the atom
as a whole. But in the valence shell of atoms
where there is a single electron whose spin
remains unbalanced, the unbalanced spin
Electron Spin Quantum
Number (ms)
27. creates spin magnetic moment, making the
electron act like a very small magnet.
The four quantum numbers compose the
numbers that describe the electron in an atom.
The quantum numbers shall be in the order:
energy level (n), sub-level or orbital type (ℓ), the
orientation of the orbital specified in ℓ (mℓ), and
the orientation of the spin of the electron (ms). It
is written in the order (n, ℓ, mℓ, ms).
Electron Spin Quantum
Number (ms)
28. 1. An electron is found in the first energy level.
What is the allowed set of quantum numbers
for this electron?
2. What is the total number of orbitals
associated with the principal quantum
number n=1?
3. What is the total number of orbitals
associated with the principal quantum
number n=2?
Example
29. 4. What is the total number of orbitals
associated with the principal quantum
number n=3?
5. List the values of n, ℓ , mℓ for an orbital in the
4d subshell.
30. What are the shapes of the atomic orbitals?
• Strictly speaking, an orbital does not have a
definite shape because the wave function
extends to infinity. However, while the electron
can be found anywhere, there are regions
where the probability of finding
it is much higher.
The Representations of the
Shapes of Atomic Orbitals
31. The Representations of the
Shapes of Atomic Orbitals
The electron density distribution of a 1s electron
around the nucleus.
32. The p orbitals starts when n =2 for which ℓ has a
value of 1 and mℓ has values -1, 0, +1. There are
three 2p orbitals: 2px, 2py, 2pz indicating the axes
along which they are oriented. For the p
orbitals, the electron probability density is not
spherically symmetric but has a double teardrop
shape, or a dumbbell shape. The greatest
probability of finding the electron is within the
two lobes of the dumbbell region;
The Representations of the
Shapes of Atomic Orbitals
33. it has zero probability along the nodal planes found
in the axes. All three 2p orbitals are identical in
shape and energy but differ in orientation.
The Representations of the
Shapes of Atomic Orbitals
34. The d orbitals occur for the first time when n = 3.
The angular function in these cases possesses
two angular (or planar) nodes. Four of the
orbitals have the same basic shapes except for
the orientation with respect to the axes. The
wave functions exhibit positive and negative
lobes along the axes and shows zero probability
of finding the electron at the origin.
The Representations of the
Shapes of Atomic Orbitals
35. The fifth wave function,
dx2 , has a similar
shape with that
of the p-orbital
with a donut-shape
region along the x-axis.
