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Chapter 2 Atomic Structure
I.
Basic Quantum Mechanics
A.
Subatomic Particles
1) The Electron
a) J. J. Thomson credited with the discovery in 1897
i. Previously, atoms were believed to be the smallest particles
ii. William Morgan (1785)—passed current through a vacuum
Produced a glow
iii. Luigi Galvani (1800)—first produced constant current from a battery
Cu (wet salt solution) Sn Ecell = +0.48 V
iv. Michael Faraday (1836)—determines glow comes from the cathode
v.
Eugene Goldstein (1876)—coins the term “cathode ray”
vi. Thomson finds electron is negatively charged with 1/1836 mass of H
b)
Thomson’s Experiment
i)
Repulsion from negative pole of an electric field meant that the cathode
rays must be negatively charged
ii)
The amount of deflection was a function of the mass of the “ray”
iii) Since many different metals all produced the same “cathode rays,” all
atoms must be made up of the same +/- particles
iv) G. F. FitzGerald renames cathode rays as Electrons
2)
The Nucleus
a) Ernest Rutherford credited with discovering the Nucleus in 1911
i. He nuclei (a particle) were deflected as they passed through a Gold foil
ii. Conclusion: Heavy, tiny nucleus and much empty space in an atom
3)
B.
The Proton
a) Rutherford and Moseley quickly discovered the charge of the nucleus
i. Electron beams aimed at an element caused X-rays to be emitted
ii. The square root of the X-ray frequency emitted had a linear relationship
with about half of the atomic mass of the element
iii. This number was clearly Z, the nuclear charge (or the atomic number)
Atomic Spectra
1) Balmer described the emission spectrum of H in 1885
 1
1  nh = integer > 2
a) E
.  RH  2  2  RH = Rydberg constant = 1.097 x 107 m-1
nh 
2
hc
b)
c)
n = principle quantum number = discrete allowed energies for H electrons

 hc
h = Planck’s constant = 6.626 x 10-34 J s
 = frequency of light s-1
c = speed of light = 2.998 x 108 m/s
 = wavelength nm
 = wavenumber cm-1
.E  h 
2)
Bohr’s Quantum Theory of the Atom (1913)
a) Negative electrons move in stable, circular orbits around positive nuclei
b) Electrons absorb or emit light by moving out or moving in to other orbits
c) Bohr replaced Balmer’s equations with better ones
 1
1 

E  RH  2  2 
nh 
 nl
2 2 Z 2e 4
RH 
(4o ) 2 h 2
d)
1
1
1


 me mnucleus
 = reduced mass
E = electron charge
Z = nuclear charge
4o = permittivity of vacuum
Energy levels are far apart at small n, close together at large n
n =  if the nucleus and electron are completely separate
e)
Only worked for H-atom; not a complete description of atomic structure
Hydrogen Atom
Energy Levels
3)
Wave nature of particles: de Broglie (1924)
a) De Broglie Equation relates particles to a property of waves
h = Planck’s constant
m = mass of the particle
v = velocity of the particle
Massive (visible) particles have short wavelengths that can’t be observed; we
call these classical particles that obey classical mechanics
h

mv
b)
c)
d)
4)
Small particles (electrons) have observable wave properties
Electrons around nuclei are like standing waves
Heisenberg’s Uncertainty Principle (1927)
a) Heisenberg’s equation relates position and velocity
h
Dp Dx 
4
p = momentum = mv
b)
The better you know velocity (p), the worse you can know position (x)
Dp , then Dx and vice versa
c)
Can’t exactly describe e- orbits, only orbitals, which are regions of space
where of high probability of finding e- in it
Can’t describe e- as a particle, since we don’t know its location exactly
d)
C.
The Schrödinger Equation
1) Describes wave properties of electrons: position, mass, total E, potential E
2)
Y = wave function = describes an electron wave in space = describes atomic
orbital
3)
The Schrödinger Equation:
HΨ  EΨ
a)
b)
4)
H = the Hamiltonian Operator = calculus operation on Y
E = energy of the electron
When H is carried out on Y , the result is E times Y (E is a constant)
Different orbitals have different Y and consequently, different E’s
The Hamiltonian Operator = H
 h2   2  2  2 
Ze 2
H  2  2  2  2  
8 m  x y z  4o x 2  y 2  z 2
Kinetic Energy part
Potential Energy part
a)
Definitions
i. h = Planck’s constant
ii. m = mass of the particle
iii. E = total energy of the system
iv. e = charge of electron
v. (x2 + y2 + z2)1/2 = r = distance to nucleus
vi. Z = charge of the nucleus
vii. 4o = permittivity of vacuum
b)
Applied to Y:
  h2   2  2  2 

 2  2  2  2   V ( x, y, z ) Y ( x, y, z )  EY ( x, y, z )
 8 m  x y z 

c)
V = potential E = electrostatic attraction between electron/nucleus
V
i.
ii.
 Ze 2
4o
 Ze 2

2
2
2
4o r
x y z
Define attractive force = negative energy
Closer to nucleus = large –V; farther from nucleus = small –V
iii. if r = , then V = 0
5)
There are any number of solutions to the Schrödinger equation, each describing an
electron in an atomic orbital: 1s, 2s, 2px, 2py….
6)
The meaning of Y
a) Y has no physical meaning itself, it is just a mathematical expression
b) Y2 = probability of finding the electron at a given point in space
i. If you consider all space, Y2 = 100%
ii. Consider where the electron is 90% of the time, Y2 = atomic orbital (1s)
7)
Physical Reality imposes some conditions on what Y can be
a) Y must have only one value because a given electron only has one energy
b) Y and dY must be continuous because the electron can’t “jump”
Y ----> 0 as r ---->  because the probability must decrease farther away
c)
d)
The total probability of finding the electron somewhere must = 1
i. Normalization = setting values to 1
Y Y
A
8)
*
A
d  1
Y* is used to make imaginary Y real
just use Y2 if not imaginary
dt = x + y +z
All orbitals are orthoganal (perpendicular):
 Y Y d  0
A
B
II.
Applications of the Schrödinger Equation
A.
A Particle in a Box
1) A one-dimensional box is a simple case for using the Schrödinger Equation
a) V(x) = 0 inside the box, between x = 0 and x = a
2)
b)
V(x) =  outside the box
c)
The Particle can never leave the box
Apply the Schrödinger Equation:
 h 2   2 Y ( x) 

  EY( x)
2
2
8 m  x 
3)
Solve the Schrödinger Equation
a) sine/cosine describe waves, so we will make a combination of these functions
A,B,r,s = constants
Y  Asin rx  B cos sx
b)
Substitute into the Schrödinger Equation and solve for r and s
2
r  s  2mE
h
4)
Apply conditions of the particle in a box
a) Y must go to 0 at x = 0 and x = a
b) cos sx = 1 for x = 0
c) Y = 0 only if B = 0
d)
Y = A sin rx
5)
At x = a, Y = 0, so sin ra = 0
a) Only possible if ra = n (n = integer)
b) ra = n
 n
r
a
n0
c)
Keep only the + values (+/- give same results)
d)
Substitute and solve for E
n
2
 2mE
a
h
n2h2
E
8ma 2
r
6)
These are the energy levels predicted by the Particle in a Box Solutions
a) Quantized: n = 1,2,3,4….
2 2
E
b)
nh
8ma2
Substitute r = n/a into the wave equation
 nx 
Y  A sin 

 a 
 YAYA d  1
*
7)
Normalization leads to a more complete solution:
2
A
a
8)
Total solution:
2
nx
Y
sin
a
a
9)
Plotting the wave functions
a) Classical Mechanics: particle has equal probability anywhere in the box
b) Quantum Mechanics: high and low probabilities at different locations
Wave Functions and their
Squares for the Particle in
a Box with n = 1, 2, and 3