Zumdahl`s Chapter 7
Zumdahl`s Chapter 7
Zumdahl’s Chapter 7
• EM Radiation
• EM Quantization
• H Spectrum
• Quantum Nos.
• Orbital Shapes
• W. Pauli (spin)
• Periodic Table
– Property Trends
• Oscillating E & M fields forever.
• Wavelength, , distance between
• Period, τ, time between peaks.
• Speed, c = / τ =
– = frequency (cycles per second)
– = c / and = c /
• E = h = hc /
• Equipartition Theorem demands kT
worth of thermal energy to all light
and matter modes. Leads to energy.
• Vibration overtones in matter are
truncated by indivisible atoms.
• in light energies overcome by Planck
with QUANTIZED energies.
• h = Planck’s constant = 6.6x10–34 Js
• White light is all colors (all );
diffraction in prisms or
raindrops gives continuous
• Atomic excitation gives instead
discrete colors (few ).
• In H atom, E light = R (n2–2 – n1–2)
• Why so simple?
F centripetal = m r ²
F attraction = – Z e² / r²
Balance: Z e² = m r³ ²
E = K+V = ½ m r² ² – Z e² / r
E = – ½ Z e² / r on substitution
E = – (½ Z e² / r) (Z e² / m r³ ²)
E = – Z² e4 m / (2 m² r4 ²)
E = – Z² e4 m / (2 m² r4 ²)
pθ = m r² is angular momentum
E = – Z² e4 m / (2 pθ²)
pθ = n h / 2 Bohr’s postulate
En = – (2² Z² e4 m / h²) n–²
En = – RH Z² / n² YES!!
– But but but WHY quantize pθ ?!?
• Just as light moves as a wave
but lives and dies as a discrete
energy packet (“photon”),
• Matter too has wave properties
only dominant for light masses:
• = h / p = h / mv = h / (m r² )
• And n = 2 or wave kills itself!
• Waves must S P A N .
• Attempts to narrow the wave,
reduce , increasing p = h /
• So a minimum uncertainty x in
position MUST exist, and
• ( x) ( px) ½ h / 2
– Heisenberg’s Uncertainty Principle
• Bohr’s orbits are infinitesimally
thin trajectories. Being off them
must be infinitely uncertain!
• Need a full 3-D wave, , not 1-D.
• Schrödinger’s Wave Equation
solves H energy in 3-d and finds:
• En = – RH Z² / n² also and more!
l ,ml ,ms)
• Principle Quantum Number, n
– n = 1, 2, 3, 4, 5, … ,
– Governs the number of nodes in
the electron’s matter wave, !
– # of nodes (where =0) is n – 1
– For “hydrogenic” ions (and H itself)
electron energy depends only on n.
– Nodes can be spherical or angular!
Quantum Number, l
= 0, 1, 2, 3, 4, … , (n – 1)
s p d f g …
measures # of nodes that are
angular; so it must stop at n–1.
– Increasing angular nodes squeezes
waves, so E usually depends on l
– Z component of
l is also quantized!
• ml is the component of l along
(up or down) the Z axis in space.
– ml = – l, – l + 1, … , –1, 0, 1, … , l–1,
– Because the component can’t
exceed its vector.
– E only depends upon ml when a
magnetic field is applied.
3-D Shapes of Orbitals
• Governed by n, l, and ml
• l = 0 is spherical
n = 1 means no nodes:
(n – 1) = # of nodes, all spherical.
n = 2 means one spherical node:
Wavefunction, , falls off in
intensity to zero at large distance
Cross-section of 2s
• l = 1 implies one angular node
– Cleave space with an x=0 plane
– But y=0 or z=0 work as well, so
there are three or 2l+1 suborbitals.
– The ml sequence always gives 2l+1
– ml differentiates directions in
space for chemical bonding!
• In the absence of applied
magnetic field, all suborbitals of
a given l have the same energy.
• This identity of energies is
• Even nearly degenerate orbitals
may be mixed to give new ones.
• Dirac applied Einstein’s fixed c
to Schrödinger’s Equation and
found new quantum number, ms.
– ms is electron spin number and
takes on only two values, ½.
• Pauli Principle says only two
electrons can occupy any
orbital, and their ms must differ,
– without which NO CHEMISTRY.
• Beyond H, repulsions BETWEEN
electrons compete with nuclear
attraction & complicate spectra.
– Hund’s Rule: if electrons have the
choice between degenerate
orbitals, they choose NOT to
double occupy them.
• It minimizes electronic repulsion.
• Energies are now a function of l,
the angular quantum number.
• The “filling sequence” shows
the new energy order:
– <5p<6s<4f<5d<6p<7s<5f<6d etc.
– Periodic Table exemplifies it, but a
simple pattern emerges:
And that’s as far
as the known
• Aufbau (filling sequence) follows
– 1s² 2s² 2p6 3s² 3p6 4s² 3d10 4p6 5s²
– 4d10 5p6 6s² 4f14 5d10 6p6 7s² 6d10
– and the latest elements among 7p6
• Irregularities occur where ½
filled suborbitals acquire greater
stability than a predecessor:
• Vanadium: [Ar] 4s² 3d3 suggests
• Chromium: [Ar] 4s² 3d4 is next,
BUT IT ISN’T SO!
• Chromium: [Ar] 4s1 3d5 lowers
its energy by borrowing a 4s to
complete a ½–filled d suborbital.
• Manganese: [Ar] 4s² 3d5 follows.
• Rows are called “periods” on the
– Columns are called “groups.”
• Progression along rows implies
adding new electrons & protons.
– They get added 1:1 for neutrality.
– But new repulsions keep pace with
new attractions. Which wins?
• Protons exert attraction only
toward the atom’s center.
• Electrons exert repulsion from
all over their wavefunctions.
– “Core” electrons are located very
close to the nucleus where they
repel outer electrons as effectively
as their number of protons attract.
Effective Charge =
Protons – Effective e
• So nucleus’s effective charge is
Z – (# of all core electrons) less
the effect of “outer” electrons.
• While core are 100% effective,
“valence” electrons are LESS by
virtue of spanning a greater
fraction of the atom. Only their
inner portion is 100% effective.
• Across a row, added electrons
are valence, not core. So they
repel one another less than the
added protons attract them.
• Effective potential INCREASES
across the row, binding the
subsequent electrons ever more
Trends on Periods
• Increased electron binding
along rows (to the right),
generally results in:
Increasing ionization potentials
Decreasing atomic sizes
Growing electron affinities
• Elements in the same column
have the same number and type
of valence electrons, differing
only by n.
• Because increasing n by 1 puts
one more node in , dimensions
of increase, vaulting electrons
outside their predecessors.
• Dropping down a group (column)
increases efficiency of core and
distance of valence from center.
Both conspire to weaken the
– Atomic size increases.
– Ionization potential decreases.
– Electronegativity decreases.