Download Ch. 1: Atoms: The Quantum World

Ch. 1: Atoms: The Quantum World
CHEM 4A: General Chemistry with
Quantitative Analysis
Fall 2009
Instructor: Dr. Orlando E. Raola
Santa Rosa Junior College
Overview
1.1The nuclear atom
1.2 Characteristics of electromagnetic radiation
1.3 Atomic spectra
1.4 Radiation, quanta, photons
1.5 Wave-particle duality
1.6 Uncertainty principle
Spherical polar coordinates
colatitude
azimuth
radius
General formula of wavefunctions for the
hydrogen atom
ψ (r,θ ,ϕ ) = R(r )Y(θ ,ϕ )
For n = 1
ψ (r,θ ,ϕ ) =
2e
−
r
a0
3
2
0
a
×
1
2π
1
2
=
e
−
r
a0
1
3 2
0
(π a )
a0 =
4πε 0  2
me e 2
General formula of wavefunctions for the
hydrogen atom
ψ (r,θ ,ϕ ) = R(r )Y(θ ,ϕ )
1
For n = 2 and E 2 = − hℜ
4
ψ (r,θ ,ϕ ) =
1
1
2 6
5
2
0
a
re
−
r
2a0
1
2
r
−
⎛
⎞
⎛ 3 ⎞
1
1
2a0
×⎜
sin
θ
cos
φ
=
r
e
sin θ cos φ
⎜
5⎟
⎟
4 ⎝ 2π a0 ⎠
⎝ 4π ⎠
Quantum numbers
n: principal quantum number
determines the energy
indicates the size of the orbital
 : angular momentum quantum number,
relates to the shape of the orbital
m : magnetic quantum number, possible
orientations of the angular momentum
around an arbitrary axis.
magnetic
quantum number
principal
quantum number
orbital angular momentum
quantum number
Electron probability in the
ground-state H atom.
Radial probability distribution
Allowable Combinations of Quantum Numbers
l = 0, 1, …, (n – 1)
ml = l, (l – 1), ..., -l
No two electrons in the same atom have the
same four quantum numbers.
Higher probability
of finding an
electron
Lower probability
of finding an
electron
most probable radii
The most probable
radius increases as n
increases.
boundary surface
•  90% likelihood of finding
electron within
radial
nodes
Wavefunction (Ψ) is nonzero
at the nucleus (r = 0).
For an s-orbital, there is a nonzero
probability density (Ψ2) at the nucleus.
radial nodes
n=1
l=0
no radial nodes
n=2
l=0
1 radial node
n=3
l=0
2 radial nodes
2p-orbital
n=2
l = 1, 0, or -1
no radial nodes
1 nodal plane
Plot of
wavefunction is
for yellow lobe
along blue arrow
axis.
The three p-orbitals
nodal planes
The labels “x”, “y”, and “z” do not correspond
directly to ml values (-1, 0, 1).
The five d-orbitals
n = 3, 4, …
dark orange (+)
l = 2, 1, 0, -1, -2
light orange (–)
nodal planes
The seven f-orbitals
n = 4, 5, …
dark purple (+)
l = 3, 2, 1, 0, -1, -2, -3
light purple (–)
Allowed orbitals
Allowed subshells
2 electrons
per orbital
Maximum of
32 electrons
for n = 4 shell
Stern and Gerlach Experiment: Electron Spin
Atoms with
one type of
electron spin
Atoms with
other type of
electron spin
Silver atoms
(with one unpaired electron)
Spin States of an Electron
Spin magnetic quantum number (ms) has two possible values:
Relative Energies of Orbitals in a Multi-electron Atom
Z is the atomic number.
After Z = 20, 4s orbitals have
higher energies than 3d orbitals.
Probability maxmima
for orbitals within a
given shell are close
together.
A 3s-electron has a
greater probability of
being found near the
nucleus than 3p- and
3d-electrons due to
contribution of peaks
located closer to the
nucleus.
Paired spins
Lower energy
Parallel spins
Higher energy
Electron Configurations: H and He
1s electron (n, l, ml, ms)
•  1, 0, 0, (+½ or –½)
1s electrons (n, l, ml, ms)
•  1, 0, 0, +½
•  1, 0, 0, –½)
Electron Configurations: Li and Be
1s electrons (n, l, ml, ms)
•  1, 0, 0, +½
•  1, 0, 0, –½
1s electrons (n, l, ml, ms)
•  1, 0, 0, +½
•  1, 0, 0, –½
2s electron*
•  2, 0, 0, +½
2s electrons
•  2, 0, 0, +½
•  2, 0, 0, –½
* one possible assignment
Electron Configurations: B and C
1s electrons (n, l, ml, ms)
•  1, 0, 0, +½
•  1, 0, 0, –½
1s electrons (n, l, ml, ms)
•  1, 0, 0, +½
•  1, 0, 0, –½
2s electrons
•  2, 0, 0, +½
•  2, 0, 0, –½
2s electrons
•  2, 0, 0, +½
•  2, 0, 0, –½
2p electron*
•  2, 1, +1, +½
2p electrons*
•  2, 1, +1, +½
•  2, 1, 0, +½
* one possible assignment
* one possible assignment
Filling order for orbitals
subshell being filled
maximum number of electrons in subshell
The Hydrogen atom: atomic orbitals
The potential in a hydrogen atom can be
expressed as
2
e
V(x) = −
4πε 0r
Schrödinger (1927) found that the exact
solutions for his equation give expression for
the energy as
hℜ
E=− 2
n
ℜ=
me e
4
8h ε
3 2
0
n = 1,2,3....
Quantum Numbers and Atomic Orbitals
An atomic orbital is specified by three
quantum numbers.
n the principal quantum number
- a positive integer
ℓ the angular momentum quantum number
- an integer from 0 to n-1
mℓ the magnetic moment quantum number
- an integer from -ℓ to +ℓ
Quantum Numbers
1.Principal (n = 1, 2, 3, . . .) - related to size and
energy of the orbital.
2.Angular Momentum (ℓ = 0 to n  1) - relates to
shape of the orbital.
3.Magnetic (mℓ = ℓ to ℓ) - relates to orientation
of the orbital in space relative to other
orbitals.
4.Electron Spin (ms = +1/2, 1/2) - relates to the
spin states of the electrons.
Table 7.2 The Hierarchy of Quantum Numbers for Atomic Orbitals
Name, Symbol
Allowed Values
(Property)
Principal, n
Positive integer
(size, energy)
(1, 2, 3, ...)
Quantum Numbers
1
2
3
Angular
momentum, ℓ
(shape)
0 to n-1
Magnetic, mℓ
(orientation) -ℓ,…,0,…,+ℓ
0
0
0
0
1
0
1
2
0
-1
0 +1
-1
-2
0 +1
-1
0
+1 +2
Sample Problem 7.5
Determining Quantum Numbers for an Energy Level
PROBLEM: What values of the angular momentum (ℓ) and magnetic (m )
ℓ
quantum numbers are allowed for a principal quantum number (n) of
3? How many orbitals are allowed for n = 3?
PLAN: Follow the rules for allowable quantum numbers found in the text.
l values can be integers from 0 to n-1; mℓ can be integers from -ℓ
through 0 to + ℓ.
SOLUTION: For n = 3, ℓ = 0, 1, 2
For ℓ = 0 mℓ = 0
For ℓ = 1 mℓ = -1, 0, or +1
For ℓ= 2 mℓ = -2, -1, 0, +1, or +2
There are 9 mℓ values and therefore 9 orbitals with n = 3.
Sample Problem 7.6
Determining Sublevel Names and Orbital Quantum
Numbers
PROBLEM: Give the name, magnetic quantum numbers, and number of orbitals
for each sublevel with the following quantum numbers:
(a) n = 3, ℓ = 2
(b) n = 2 ℓ= 0
(c) n = 5, ℓ = 1 (d) n = 4, ℓ = 3
PLAN: Combine the n value and ℓ designation to name the sublevel.
Knowing ℓ, we can find mℓ and the number of orbitals.
SOLUTION:
n
ℓ
sublevel name possible mℓ values # of orbitals
(a)
3
2
3d
-2, -1, 0, 1, 2
5
(b)
2
0
2s
0
1
(c)
5
1
5p
-1, 0, 1
3
(d)
4
3
4f
-3, -2, -1, 0, 1, 2, 3
7
1s
2s
3s
The 2p orbitals.
Representation of the 1s, 2s and 3s orbitals in
the hydrogen atom
Representation of the 2p orbitals of the
hydrogen atom
Representation of the 3d orbitals
Representation of the 4f orbitals
Pauli Exclusion Principle
In a given atom, no two electrons can have
the same set of four quantum numbers (n, ℓ,
mℓ, ms).
Therefore, an orbital can hold only two
electrons, and they must have opposite spins.
Types of
Atomic
Orbitals
Levels and sublevels
s orbital are spherical
Dot picture of
electron cloud in
1s orbital.
Surface
density
4πr2y versus
distance
Surface of 90%
probability
sphere 1s orbital
2s orbitals
3s orbital
p orbitals
When n = 2, then ℓ = 0 and 1
Therefore, in n = 2 levell there are
2 types of orbitals — 2 sublevels
For ℓ = 0
mℓ = 0
this is a s sublevel
For ℓ = 1 mℓ = -1, 0, +1
this is a p sublevel with 3 orbitals
p Orbitals
The three p orbitals lie 90o apart in space
2px Orbital
3px Orbital
d Orbitals
When n = 3, what are the values of ℓ?
ℓ = 0, 1, 2 and so there are 3 sublevels in level n=3.
For ℓ = 0, mℓ = 0  s sublevel with single orbital
For ℓ = 1, mℓ = -1, 0, +1  p sublevel with 3 orbitals
For ℓ = 2, mℓ = -2, -1, 0, +1, +2
d
sublevel with 5 orbitals
s orbitals have no planar node
(ℓ = 0) and so are spherical.
p orbitals have ℓ = 1, and
have 1 planar node,
and so are “dumbbell”
shaped.
This means d orbitals
(with ℓ = 2) have 2 planar
nodes
One of 7 possible f
orbitals.
All have 3 planar
surfaces.
Can you find the 3
surfaces here?
2 s orbital
Summary of Quantum Numbers of Electrons in Atoms
Name
Symbol
Permitted Values
Property
principal
n
positive integers(1,2,3,…) orbital energy (size)
angular
momentum
ℓ
integers from 0 to n-1
magnetic
mℓ
integers from -ℓ to 0 to +ℓ
orbital shape (The ℓ values
0, 1, 2, and 3 correspond to
s, p, d, and f orbitals,
respectively.)
orbital orientation
spin
ms
+1/2 or -1/2
direction of e- spin
Experimental observation of the spin of
the electron (Stern and Gerlach, 1920)
A comparison of the radial probability
distributions of the 2s and 2p orbitals
The radial probability distribution for an electron in a 3s orbital. The radial
probability distribution for the 3s, 3p, and 3d orbitals.
The 3d orbitals
One of the seven
possible 4f orbitals.
Schematic representation of the
energy levels of the hydrogen atom
CHEM 4A:
General Chemistry
with Quantitative Analysis
Dr. Orlando E. Raola
FALL 2008
1 / 25
Ch.1: Atoms: The Quam World
2 / 25
Expression forms for the wavefunction ψ
1. The wavefunction ψ describes the movement of the particle.
2. The probability density ψ 2 describes the probability of finding the
electron in a region of space .
3. The radial distribution function P(r ) = r 2 R 2 (r ) is the probability
that the electron would be found at a certain distance r from the
nucleus.
3 / 25
Allowable Combinations of Quantum Numbers
` = 0, 1, 2, ...,n-1; m` = -`,...,0,...`
4 / 25
Stern and Gerlach Experiment (1925)
5 / 25
Stern and Gerlach Experiment (1925)
6 / 25
Stern and Gerlach Experiment (1925)
7 / 25
Electron spin
Electrons (and many other subatomic particles) have intrinsic
magnetic moment. This is a purely quantum mechanical effect
without parallel in classical physics.
Spin is characterized by the spin magnetic quantum number ms , that
can have values + 21 and − 21
To help our understanding of this phenomenon, we resort to a rather
crude physical model:
8 / 25
EPR: electron paramagnetic resonance
9 / 25
Energy of hydrogen electronic orbitals
10 / 25
Many-electron atoms: The form of the potential for
the helium atom
V =−
2e2
2e2
e2
−
+
2
4π0 r12 4π0 r22 4π0 r12
11 / 25
Orbital energies in many-electron atoms
12 / 25
Shieding: Effective nuclear charge
As a result of electron-electron repulsion, in many-electron atoms,
the pull of the nucleus in an electron is decreased, as if the charge of
the nucleus were smaller than it actually is.
En = −
2 h <
Zeff
n2
13 / 25
Penetration: how close to the nucleus
Because s electrons are in
average closer to the nucleus,
they experience less shielding and
“see” a stronger effective nuclear
charge than p electrons.
Penetration effects can explain
why the 4s orbital has such a low
energy than even the 3d.
14 / 25
Pauli exclusion principle, 1925
In an atomic system, no more than two electrons can occupy any
given orbital. When two electrons occupy one orbital, their spins must
be antiparallel.
No two electrons in an atom can have the same set of four quantum
numbers.
15 / 25
Aufbau principle
To predict the electronic configuration of a neutral atom of an
element:
1. Electrons are added, one by one, to the orbitals in the order of
their increasing energies. No more than two electrons per orbital
(Pauli exclusion priniciple).
2. If more than one orbital in a subshell is available, electrons are
added to the orbitals in that subshell with parallel spins until the
subshell is half-full. Then electrons are start to be paired with
antiparallel spins (Hund’s rule of maximum multiplicity).
16 / 25
Valence shell and the Periodic Table
Interactive periodic table
17 / 25
Periodic properties: Effective nuclear charge
18 / 25
Atomic radius
Three types of atomic radii:
1. metallic radius
2. covalent radius
3. van der Waals radius
19 / 25
Periodic trends: atomic radius
20 / 25
Ionic radius
Ionic radii are determined on the
distance between neighboring
ions in an ionic solid, based on the
asumption that the radius of the
oxide ion (O 2− ) is 140. pm
21 / 25
Periodic trends: ionic radius
22 / 25
Ionization energy
Ionization energy is the enery required to remove one electron from
an atom in the gas phase.
Mg(g) → Mg + (g) + e− (g)
I = E(X + ) − E(x)
23 / 25
Succesive ionization energies
24 / 25
Electron affinity
25 / 25