The Quantum Nature of the Atom
Silberberg: Ch. 7-8
Structure can be studied by looking at the interaction between electromagnetic
radiation and matter.
Where we're going:
The nature of light
Wave-particle duality
The quantum nature of the atom:
the Schrödinger model of the Hydrogen atom
- atomic orbitals
- quantum numbers
- quantized energies
- how do we know that the
sub-microscopic world is quantized?
From one-electron atoms to
many-electron atoms: what's different?
Electron configuration and periodicity
- originally, a form of energy thought of as waves traveling through space
- travels (in a vacuum) at the speed of light (c = 3.00 x 108 m/s)
Amplitude
1. wavelength (λ, lambda):
t
Amplitude
Wave characteristics:
t
2. frequency (ν, nu):
νλ=c
νλ=c
3. amplitude (A): height of wave
- an interference
of waves:
Amplitude
diffraction
the double slit
experiment
waves interfere:
constructive
destructive
3. amplitude (A): classically related to energy (intensity) of light
- light of greater amplitude appears brighter
Light – has many wave properties but does not
behave as classical wave
1. To eject an electron from the metal,
a minimum frequency of light is necessary.
e- with
kinetic energy
2. Above this minimum frequency, the kinetic energy
of the ejected electron increases with light frequency.
light
3. Above this minimum frequency, increased light intensity increases
the number of ejected electrons but not their kinetic energy.
metal
A green laser pointer emits lights having a wavelength of 532 nm. A red laser pointer emits light having a
wavelength of 656 nm.
1.
Which light has a greater frequency? Green or Red
2.
Which light has a greater energy? Green or Red
3. A laser pointer emits 4.0 mW of light at 532 nm. How many photons of light are emitted in one minute?
(1.00 W is 1.00 J/s)
Warm Up Questions
Of the two waves shown below, which most accurately depicts electromagnetic
radiation of higher energy?
Wave A
Wave B
Originally, matter: - discrete particles with determinate mass, position, etc.
The double-slit experiment revisited:
particles
waves
electrons
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
electrons diffract!
beam of electrons
Passage of electricity through gas of atoms causes atoms to emit light
H-atoms
Passage of electricity through gas of atoms causes atoms to emit light
ACTUALLY observed only four frequencies
in visible part of spectrum:
H-atoms
These observed frequencies are
different for excited atoms of
different elements.
experiments
Before 1900:
light: wave character – e.g. wavelength, frequency, diffraction
matter: particulate character – e.g. mass, position
light:
matter:
photoelectric effect
diffraction by electrons
atomic line spectra
Wave-particle duality:
Matter has wave characteristics as well as particle characteristics
Schrödinger model of the atom:
HY = EY
wave equation
of ical
t
se mat ns
solve
e tio
h
t ra
a
m pe
o
- has many solutions Y
Y = "wavefunction" or "orbital"
= a mathematical function
describing the shape of a wave
HY = EY
wave equation – has many solutions Y
Y = "wavefunction" or "orbital"
1. principal quantum number (n)
®
2. angular momentum quantum number ( ! )
®
3. magnetic quantum number (m! or m)
®
4. electron spin quantum number (ms or s)
ms =
Size of the orbital
1. principal quantum number (n)
n = 1, 2, 3, ...
Shape of the orbital
2. angular momentum quantum number ( ! )
! = 0, 1, … , n-1
The hydrogen atom is predicted to have:
an s orbital for n = 1;
s and p orbitals for n = 2;
s, p, and d orbitals for n = 3;
and so on…
Orientation of the orbital
3. magnetic quantum number (m! or m)
m = -!, -!+1, … , !-1, !
n
1
2
2
3
3
3
!
Notation
m!
Number of orbitals
Among these five sets of quantum numbers:
I.
II.
III.
IV.
V.
n = 3, l = 2, ml = -2
n = 3, l = 1, ml = 0
n = 3, l = 0, ml = -1
n = 3, l = 2, ml = 0
n = 3, l = 3, ml = -2
which does not describe a state for an electron in
an atom?
A.
B.
C.
D.
II and IV
III and V
I and III
IV and V
3
2
3d
-2, -1, 0, +1, +2
Warm Up Question
How many orbitals are there in the subshell designated by n = 3, l = 2?
A.
B.
C.
D.
E.
2
3
5
6
10
Solving H Y = E Y for H-atom,
1s orbital: n = 1, ! = 0, m! = 0
2s orbital:
n = 2, ! = 0, m! = 0
"wave-like"!
As n increases, the
size of the orbital
and number of
nodes increase.
2p orbitals:
3d orbitals:
From H Y = E Y,
many solutions
Y = orbital
E = energy of
each orbital
−0.242 x 10−18 J
−0.545 x 10−18 J
For H-atom,
1
En = -2.18x10- 18 Jæç 2 ö÷
èn ø
For every orbital,
- doesn’t depend on !, m
® subshells have the same energy
(i.e. they are degenerate)
Energy of the atom is quantized!
−2.18 x 10−18 J
Y describes relative
probability of finding
electron around nucleus
E tells us how stable it is
having that probability
Passage of electricity through gas of atoms causes atoms to emit light
ACTUALLY observed only four frequencies
in visible part of spectrum:
H-atoms
410 nm
434 nm
656 nm
486 nm
These observed frequencies are
different for excited atoms of
different elements.
n
∞
6
5
4
For the H-atom:
−0.242 x 10−18 J
3
−0.545 x 10−18 J
2
Infrared
(>656 nm)
656 nm
486 nm Visible
434 nm
410 nm
Ultraviolet
(<400 nm)
2.18 ´10 -18 J
En = n2
Detecting screen
En
Final state: n = 2
Discrete emission lines
can only be explained
by quantization of the
energy levels of atoms
and molecules.
−2.18 x 10−18 J
Initial state: n = 3
1
Absorptions
Emissions
H-atoms
The figure below represents the emission spectrum of a one-electron ion in the gas phase. All the
lines shown result from electronic transitions from excited states to the n = 3 state.
A
B
Wavelength
A) A: n = 3 → n = 1; B: n = 2 → n = 1
What transitions correspond to A and B?
B) A: n = 1 → n = 3; B: n = 1 → n = 2
C) A: n = 6 → n = 3; B: n = 5 → n = 3
D) A: n = 5 → n = 3 ; B: n = 6 → n = 3
E) A: n = 3 → n = 1; B: n = 2 → n = 1
1. no two electrons in an atom can have the same values
of all 4 quantum numbers: Pauli exclusion principle
1. principal q. n. (n)
n = 1, 2, 3, ...
3. magnetic q. n. (m!)
m = −!, −!+1, … , !−1, !
2. angular momentum q. n. ( ! )
! = 0, 1, … , n−1
4. electron spin q. n. (ms)
ms = +1/2, −1/2
If, for an electron, n = 2, ! = 1, m = 0, ms = −1/2
Then, for any other electron, n = 2, ! = 1, m = 0, ms = −1/2
↓
N
2. repulsions between the electrons within the atom
Effective nuclear charge, Zeff : the nuclear charge actually felt by an electron
↑
Two factors determine an atom's energy:
2. electron-electron repulsions
electrons further from
nucleus are shielded
from full +ve nuclear
charge by electrons
closer to the nucleus
E.g. comparing a 2s electron to
a 2p electron:
1s electron cloud
1. electron-nucleus attractions;
many-electron atom: E2s < E2p
2s experience higher Zeff than 2p
Generally, for any given value of n:
Ens < Enp < End < Enf
Outcome: for a many-electron atom, subshells are not degenerate:
H-atom
Ens < Enp < End < Enf
many-electron
atom
For (almost) all elements of the periodic table, we can now predict which orbitals are occupied by electrons
= the ground state (i.e. most stable) electron configuration:
1. Electrons will fill orbitals of lowest energy ('Aufbau')
2. 2 electrons per orbital with opposing spin (­ & ¯)
3. If two or more degenerate orbitals are available, one electron goes
into each until they are all half-full. All electrons in half-filled
orbitals have the same spin quantum number.
'Hund's rule'
Order of orbital
filling with electrons:
6s 6p 6d …
5s 5p 5d 5f 5g
4s 4p 4d 4f
3s 3p 3d
2s 2p
1s
many-electron
atom
E.g. ground state electron configuration of P atom (Z = 15):
Ne
P
Note: some atoms do not follow these simple
rules. For our purposes, we will not consider
these anomalous electron configurations.
Structure of the atom from Schrödinger is evident in the structure of the periodic table
pm
pm
atomic size decreases
atomic size decreases
1. The 1s22s22px2 state of a carbon atom is
A. a ground state
B. an excited state
2. The electronic configuration of element X is 1s22s22p63s23p3. The formula of
the most likely compound that X will form with Ca is
A. CaX2
B. Ca2X3
C. Ca2X
D. Ca3X2
Light has both wave and particle properties = photons. Each photon has energy E = hν
Matter has both wave and particle properties
HY = EY ® cannot pinpoint electron’s position but can describe a PROBABILITY of finding the
electron around an atom
Y = orbital; Y2 = probability of finding the electron in a region of space about the atom.
Orbitals are regions where the electron spends most of its time; each uniquely specified
by its set of three quantum numbers (+ one quantum number specifying electron spin)
Energy levels of atoms and molecules are quantized (i.e. discrete). Transition between
energy levels occurs with either absorption (↑ in energy) or emission (↓ in energy) of a
photon of radiation of the corresponding energy (and therefore wavelength/frequency).
The absorption/emission spectrum of each element and compound is unique.
Quantum mechanics gives us the electronic configuration of atoms which helps us understand
their properties.