Quantum theory

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Quantum theory

• Electron Clouds and Probability

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• Bohrs model of the atom is unable to describe
  electron (e-) behavior in an atom
• Problem multiple spectral lines closely spaced
• deBroglie Hypothesis
• Believed e- had a Dual-nature
• Acted as particles with mass and waves of energy
  with no mass, simultaneously
• Combined Einsteins Relativity equation (Emc2)
  with Planks quantum equation (E hv) (Planks
  constant, frequency of a wave)
• mc2 hv substitute c with v ( general
  velocity)
• v(frequency of wave) v/l velocity/wavelength)

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• Final equation l h
• mv
• Enabled de Broglie to predict the wavelength of
  a particle of mass m and velocity v
• Showed that an e- stream acted as a group of
  particles and and in some ways as a ray of light
• Waves can act as particles and particles can act
  as waves
• Wave-particle duality of nature

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• Momentum (p) is the product of mass and velocity
  (speed and direction of motion)
• l h/p
• Wavelength inversely proportional to momentum
• Only worth studying for particles of small mass
• Quantum mechanic small particles traveling near
  speed of light

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• Schrodenger
• Studied e- as waves
• Found amplitude of wave was related to distance
  or point in space an e- was from the nucleus
• Developed an equation using e- energies and
  amplitude along with quantum levels to describe
  wave function
• Max Born found that the square of the amplitude
  gave the probability of finding the e- at that
  point in space for which the equation is solved

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• Since the e- is traveling at the speed of light
  and appearing at all these positions, the e-
  appears to be everywhere
• The area the e- occupies appears to be a region
  of negative charge with a specific shape
• This is referred to as an Electron cloud

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• Heisenberg studied e- as particles
• Noted it was impossible to determine both the
  exact position and exact momentum of an e- at the
  same time
• Due to the fact that you interacted with the
  particle to see it and changed one of the two
  properties
• There is always uncertainty
• Proposed Heisenberg Uncertainty Principle

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• Impossible to know the exact position and
  momentum of an e- at the same time
• Scientists are unable to describe the exact
  structure of an atom due to this
• But it can be determined with probability
• Can determine with high probability where an e-
  is most likely to be found in the energy levels
  of an atom at any one given time

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• Probability
• Is the ratio between the number of times the e-
  is in that current position and the total number
  of times it is at all positions
• The higher the probability, the more likely the
  e- will be found in a given position
• The probability plots give a three dimensional
  shape to a region of space an e- is most likely
  to be found

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How do e- jump to higher energy levels?

• Change amplitude its a wave function not a
  particle function
• Extend into higher energy levels only if unfilled
• Explains gaps or lack of some colors (cannot fit
  into filled levels)
• So what is importance of particle nature?
• Only particles carry charge
• Keep e- from cancelling each other out
• Makes e- occupy specific regions and evenly
  spreads them out

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• So do e- actually jump? Not really. Change the
  amplitude which can extend the reach of the
  energy of the e- into a higher energy state
  (level) Goes from ground state to excited state
  (notice word state).
• e- can extend into lower states or levels, but
  the repulsion of e- in the lower levels force e-
  to maintain their region of space based on the
  average energy of the e- amplitude from its wave
  energy. Wave energy function explains why e-
  seem to go near to nucleus and amplitude also
  explains why e- appear to be in a region and are
  found at probability locations, specific
  distances from nucleus.

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• If a higher region, energy level or state, is
  completely filled (occupied) by e-, an e- that
  gains enough energy to extend its amplitude into
  that region is unable, because there is not
  enough energy for that e- to penetrate that level
  (e- act as a barrier). However, if the e- gains
  enough energy to jump to a higher unfilled
  level, it will pass through the lower energy
  level (now has more energy) and go to the higher
  energy state. This can explain why some colors
  or gaps of energy are skipped when looking at a
  spectrum. But there is still the problem of
  multiple lines of the same color. Jumps between
  energy levels next to each other can explain some
  but not all. There has to be another reason and
  that is the next part of the energy levels that
  needs to be explained.

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• Now lets put this into the Bohr Model
• Electrons are assumed to have a circular path and
  to always be found at a specific distance from
  the nucleus dependent on their P.E.(ground state)
• But there is the probability of any e- at
  trillions if not more points in space
• Many of these points have high probability
• Connect all these points together and you obtain
  a 3D shape
• The most probable place to find the e- is on the
  surface of this shape

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• Now lets put this into the Bohr Model
• Electrons are assumed to have a circular path and
  to always be found at a specific distance from
  the nucleus dependent on their P.E.(ground state)
• But there is the probability of any e- at
  trillions if not more points in space
• Many of these points have high probability
• Connect all these points together and you obtain
  a 3D shape
• The most probable place to find the e- is on the
  surface of this shape

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• Remember that e- move near speed of light
• The e- random movement causes it to appear as a
  cloud
• The e- occupies all the volume of this cloud
• Does not normally go beyond the outer volume
  area(ground state)

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• Now in order to describe an e- behavior we need
  to represent different energy states
• Do this by use of quantum numbers
• The differences correspond to the lines observed
  in the spectrum of atoms
• Easily described using H
• When an e- moves from ground to excited state,
  energy emitted as a form of light
• Represented by a line in the H spectrum

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• Atoms with more than one e- have problems
• Interactions of the other e- cause problems
• as well as the increased nuclear pull
• It is assumed that the various e- in a multi
  electron atom occupy the same energy states
  without affecting each other
• To describe an e- in an atom, four quantum
  numbers are required
• Quantum numbers are IDd by the letters n, l, m,
  s
• Each e- has its own unique set of these four
  numbers

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• Any one e- can occupy only a specific energy
  level based on its total and P.E.
• These energy levels are represented by whole
  integers starting with 1
• The number of the energy level, represented by
  the letter n, is called the Principle Quantum
  Number 1,2,3.n
• Electrons can be found in each energy level of an
  atom
• Greatest number of e- in a level is 2n2

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• Second Quantum number is l Azimuthal quantum
  number
• Represents the energy sublevels and orbitals
• Each energy level is a group of energy states
• Represented by the number of spectral lines we
  saw of the same color
• These are closely grouped together
• States called sublevels
• sublevels in an energy level is equal to the
  principle quantum number

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• Principle quantum level 1 has 1 sublevel
• Principle quantum level 2 has 2 sublevels
• Principle quantum level 3 has 3 sublevels
• And so on
• The lowest sublevel of energy in any principle
  energy is always designated by the letter s
• 2nd sublevel is p so 2nd level has an s and p
• 3rd sublevel is d so 3rd has a s,p,d
• 4th sublevel is f so 4th has s,p,d,f

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• Each sublevel holds a maximum number of e-
• Every s can contain 2e- (one pair)
• Every p can contain 6e- (three pair)
• Every d can contain 10e- (five pair)
• Every f can contain 14e- (seven pair)

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• Each pair in a sublevel has a different place in
  space
• Due to the interactions of the e- within a
  sublevel on each other
• Try to be as far from each other as possible due
  to the fact they are all same charge
• The space occupied by one pair of e- is called an
  orbital
• Designated by quantum number ml magnetic quantum
  number
• Defines each orbital by indicating its direction
  in space

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• Ex. Sublevel s is simply spherical in nature
• Sublevel p with three orbitals (6e-, 3 pairs) has
  e- in 3D along the x,y,z axis
• Orbitals of the same sublevel are alike in size
  and shape and have same energy

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• Electrons in the same orbital must coexist
  together
• How if they are repulsive
• Fourth quantum number is spin ms spin magnetic
  quantum number
• Electrons in the same orbital spin in opposite
  directions
• Sets up opposite magnetic fields, so e- become
  slightly attractive to each other
• Up and down arrows E used to show spin direction
• Pauli Exclusion Principle no two e- in the same
  atom can have the exact same four quantum values

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• So lets see how e- would start to occupy
  positions in an atom
• Aufbau principle states that e- will always
  occupy lowest available energy levels first
• So lets see how this might look

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• As more e- are added to the atom and occupy
  higher energy levels, the interactions become
  greater between e- of different energy levels and
  sublevels
• Also remember that the nucleus is also gaining
  protons and its overall charge is increasing
  causing it to pull harder
• All these interactions force sublevels to begin
  to overlap each other
• It changes the filling pattern of e- in atoms

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• Note starting with energy level three, 4s fills
  before 3d
• Thus have a new filling pattern for e-
• Can use an ARROW DIAGRAM to determine the pattern
• ECN Electron Configuration Notation
• Short cut

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• One more e- filling fact
• Hunds Rule
• The most stable atoms are those which have the
  most parallel (same direction) spinning e-
• Designated by using boxes and the arrows for e-
  spin
• EON Electron Orbital Notation
• EDD Electron Dot Diagrams

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Quantum Numbers

• n,l,m,s
• O
• Na
• Cu