Download m L

Orbitals (Ch 5)
Lecture 6
Suggested HW:
5, 7, 8, 9, 10, 17, 20, 43,
44
Bohr’s Theory Thrown Out
• In chapter 4, we used Bohr’s model of the atom to describe
atomic behavior
• Unfortunately, Bohr’s mathematical interpretation fails when
an atom has more than 1 electron (but, it is still a convenient
model).
• Also, Bohr has no explanation of why electrons simply don’t
fall into the positively charged nucleus.
• This failure is due to violation of the Uncertainty principle.
Uncertainty Principle
• The uncertainty principle is a cornerstone of quantum theory.
• It asserts that:
“You can NOT measure accurately both the position and
momentum of an electron simultaneously, and this uncertainty
is a fundamental property of the act of measurement itself”
• In other words, there is a well defined limit to what we can
possibly know about particles as small as electrons.
• This limitation is a direct consequence of the wave-nature of
electrons
Uncertainty Principle
• Consider an electron
• If you wish to locate the electron. To see the electron, we
must use a photon
• When the electron and photon interact, there is a change in
velocity of the electron due to collision with the photon
• Thus, the act of measuring the position results in a change in
its momentum, and therefore its energy
λ’
λ
Why Does the Bohr Model Fail?
• Bohr’s model conflicts with the uncertainty principle because
if the electron is set within a confined orbit, you know both its
momentum and position at a given moment. Therefore, it
violates the Uncertainty Principle and can not hold true.
Contrast Between Bohr’s Theory and Quantum Mechanics
• The primary differences between Bohr’s theory and quantum
mechanics are:
– Bohr restricts the motion of electrons to exact, well-defined
orbits
– In quantum mechanics, the location of the electron is not
known. Instead, we describe the PROBABILTY DENSITY, or
the likelihood that an electron will be found in some region of
volume around the nucleus.
– Both models agree that electrons within different distances of
the nucleus (shells) have different energies.
Probability Density
• This is directly in line with the uncertainty principle.
• We CAN NOT locate an electron accurately
• We CAN calculate a probability of an electron being in a
certain region of space in the atom
• From these calculations, we get ORBITALS
– An orbital is a theoretical, 3-D “map” of the places
where an electron could be.
Quantum Numbers
• An orbital is defined by 4 quantum numbers
n
L
mL
ms
(principle quantum number)
(azimuthal quantum number)
(magnetic quantum number)
(magnetic spin quantum number)
1. The Principle Quantum Number, n
• n = 1, 2, 3…..etc. These numbers correlate to the
distance of an electron from the nucleus. In Bohr’s
model, these corresponded to the “shell” orbiting the
nucleus.
• n determines the energies of the electrons
• n also determines the orbital size. As n increases, the
orbital becomes larger and the electron is more likely to
be found farther from the nucleus
2. The Angular Momentum Quantum Number, L
• L dictates the orbital shape
• L is restricted to values of 0, 1….(n-1)
• Each value of L has a letter designation. This is how we
label orbitals.
Labeling Orbitals from “L”
• Orbitals are labeled by first writing
the principal quantum number, n,
followed by the letter representation
of L
Value of L
0
1
2
3
Orbital type
s
p
d
f
3. The Magnetic Quantum Number, mL
• The 3rd quantum number, mL, relates to the spatial orientation
of an orbital
• mL can assume all integer values between –L and +L
• Number of possible values of mL gives the number of orbitals
of a given type in a specified “shell”
3. The Magnetic Quantum Number, mL
Envisioning Electron Density Distribution Within An Orbital
• Imagine standing in the center of an enclosed volume that
contains an electron
• Now, imagine taking a picture of the electron every ten
seconds for an entire day.
• If you superimposed the photos together, you would have a
statistical representation of how likely the electron is to be
found at each point.
S-orbitals
• When n=1, the wavefunction that
describes this state (Ψ1) only depends on
r, the distance from the nucleus.
• Because the probability of finding an
electron only depends on r and not the
direction, the probability density is
spherically symmetric
• For n=1, only s-orbitals are allowed
• Since L=0, mL can only be 0
• This single value of mL indicates that
each shell contains a single s-orbital
s-orbital
S-orbitals
z
z
1s
3s
2s
y
x
z
y
y
x
x
As n increases, the electron is more likely to be found at distances
further from the nucleus, so the size of the orbital increases.
P-orbitals
• P orbitals exist in all shells where n> 2.
• For a p orbital, L =1.
• Therefore, mL = -1, 0, 1. Three values of mL means there are 3 porbitals in each “shell” n> 2.
• In p-orbitals, electron density is concentrated in lobes around the
nucleus along either the x, y, or z axis (These are labeled as px, py,
and pz respectively)
D-orbitals
• D-orbitals exist in all
shells where n>3
• L= 2, so mL can be any
of the following:
-2,-1,0,1,2
• Thus, there are five dsuborbitals in every
shell where n>3
Examples
• Can a 2d orbital exist?
• Can a 1p orbital exist?
• Can a 4s orbital exist?
4. The Magnetic Spin Number, ms
Stern- Gerlach experiment
• A beam of Ag atoms was
passed through an uneven
magnetic field. Some of the
atoms were pulled toward
the curved pole, others
were repelled.
• All of the atoms are the
same, and have the
same charge. Why
does this happen?
4. The Magnetic Spin Number, ms
• Spinning electrons have magnetic fields. The direction of
spin changes the direction of the field. If the field of the
electron does not align with the magnetic field, it is
repelled.
• Thus, because the beam splits two ways, electrons must
spin in TWO opposite directions with equal probability.
We label these “spin-up” and “spin-down”
1
2
1
2
Two possible orientations: ms = + (𝑠𝑝𝑖𝑛 𝑢𝑝), − (𝑠𝑝𝑖𝑛 𝑑𝑜𝑤𝑛)
Pauli Exclusion Principle
NO TWO ELECTRONS IN THE SAME ATOM CAN HAVE
THE SAME 4 QUANTUM NUMBERS!!!
1s
Quantum numbers:
1, 0 , 0, + ½
1, 0, 0, – ½
* Allowed
1s
Quantum numbers:
1, 0, 0, + ½
1, 0, 0 , + ½
* Forbidden !!
Example: What Are The Allowed Sets of Quantum Numbers
For An Electron In A 2p Orbital? (n, L, mL, ms)
1
1
1
1
2
1
n
l
0
1
1
-1
ml
1
2
n
l
ml
ms
2
1
1
+½
2
2
1
1
-½
2
2
1
0
+½
2
1
0
-½
2
1
-1
+½
2
1
-1
-½
2
2
2
ms
Spin up:
+½
Spin down: - ½
-1
0
1
Representation of the three 2p-orbitals
Example: List ALL Possible Sets of Quantum Numbers In the
n=2
• List all possible sets of quantum numbers in the n=2 shell?
• n=2
S
• L = 0, 1
• mL = 0
= -1, 0, 1
• ms = +/- ½
(L=0)
(L=1)
P
Electron Configurations
• As previously stated, the energy
of an electron depends on n.
• Orbitals having the same n, but
different L (like 3s, 3p, 3d) have
different energies.
• When we write the electron
configuration of an atom, we list
the orbitals in order of energy
according to the diagram shown
on the left (the periodic table is
aligned to depict this).
REMEMBER: S-orbitals can hold no more than TWO
electrons. P- orbitals can hold no more than SIX, and Dorbitals can hold no more than TEN electrons.
Example
• Write the electron configurations of N, Cl, and Ca
𝑁 (7 𝑒𝑙𝑒𝑐𝑡𝑟𝑜𝑛𝑠): 1𝑠 2 2𝑠 2 2𝑝3
𝐶𝑙 (17 𝑒𝑙𝑒𝑐𝑡𝑟𝑜𝑛𝑠): 1𝑠 2 2𝑠 2 2𝑝6 3𝑠 2 3𝑝5
𝐶𝑎 (20 𝑒𝑙𝑒𝑐𝑡𝑟𝑜𝑛𝑠): 1𝑠 2 2𝑠 2 2𝑝6 3𝑠 2 3𝑝6 4𝑠 2
Energy
Example, contd.
• If we drew the orbital representations of N based on the
configuration in the previous slide, we would obtain:
Energy
2p
2s
1s
For any set of orbitals of
the same energy, fill the
orbitals one electron at a
time with parallel spins.
(Hund’s Rule)
Noble Gas Configurations
• In chapter 4, we learned how to write Lewis dot configurations.
Now that we can assign orbitals to electrons, we can write
proper valence electron configurations.
Group Examples
• Give the noble gas configurations of:
• K
• K+
• Cl• Zn
• Sr
Excited States
• When we fill orbitals in order, we obtain the ground state
(lowest energy) configuration of an atom.
• What happens to the electron configuration when we
excite an electron?
• Absorbing light with enough
energy will bump a valence
electron into an excited state.
• The electron will move up to
the next available orbital.
This is the 1st excited state.
Example
• Ground state Li:
1s2 2s1
• 1st excited state Li: 1s2 2s0 2p1
• 2nd excited state Li: 1s2 2s0 2p0 3s1
Energy
3s
2nd excited state
2p
1st excited state
2s
Ground state
1s
ns2
np6
ns1
1
ns2 ns2 ns2 ns2 ns2
np1 np2 np3 np4 np5
ns2
2
3
4
5
6
7
ns2 (n-1)dx
Transition Metals
• As you know, the d-orbitals hold a max of 10 electrons
• These d-orbitals, when possible, will assume a half-filled, or
fully-filled configuration by taking an electron from the ns orbital
• This occurs when a transition metal has 4 or 9 valence d
electrons
Example: Cr  [Ar] 4s2 3d4
[Ar] 4s1 3d5
3d
3d
Unfavorable
4s
Favorable
4s