The Hydrogen Atom and the Periodic Table

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The Hydrogen Atom and the Periodic Table

• Chapter 4 of Solymar

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The Hydrogen Atom

• The simplest atom
• A proton and an electron
• Questions
• What is the probability of finding the electron
  at a distance from the proton?
• What are the allowed energy levels?
• Time-independent Schrodingers equation

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Schrodingers Equation

• Proton sits in the center (origin)
• Electron orbits proton in a spherical fashion
• The potential energy of the electron given by
  electrostatics
• Schrodingers equation becomes
• Its tedious and boring to solve it, so lets
  limit to the simplest case, the spherically
  symmetric solution

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Spherical Coordinate System

• The spherical coordinate system best suited for
  the spherical solution (Fig. 4.1)
• r, q, and f
• For a spherically symmetric case, y depends
  neither on f nor on q, but only on r

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More Derivation

• Similar for y and z
• Since

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More Derivation

• In spherical coordinate system

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Schrodingers Equation Again

• Schrodingers equation in spherical coordinate
  system
• A spherically symmetric solution is

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Solution

• When put back into the equation
• For any r
• This agrees with the lowest energy level of
  hydrogen

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Probability of Finding Electron

• Probability of finding the electron
• Highest probability at origin ( r 0 )
• What is the probability of finding the electron
  in the spherical shell between r and r dr?
• The maximum is (Fig. 4.2)
• This is Bohr radius of the first orbit

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Other Solutions

• Still consider spherically symmetric cases
• Ln(r) is a polynomial
• Energy level
• n 1, 2, 3,
• n 1 gives the lowest energy level, which is
  called the ground state
• Probability distributions of excited states (Fig.
  4.3, n 2, 3, )
• The maximum gets further away from the proton

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Quantum Numbers

• The general solution
• n, l, ml are quantum numbers
• n 1, 2, 3,
• l 0, 1, 2, , n-1
• ml 0, ?1, ?2, , ?l
• For n 1, l 0, ml 0, the solution is
• For n 2, 3, l 0, ml 0, the solutions are in
  Fig. 4.3
• These are all spherically symmetric

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Cylindrically Symmetric Solution

• For n 2, l 1, ml 0, the solution
• Its no longer spherically symmetric, but
  cylindrically symmetric
• Dumbbell shaped (Fig. 4.4)
• A plot in the x-z plane
• Maximum probability at x 0 and z ?2/Co (or q
  0 and p, r 2/Co)
• For n 2, l 1, ml ?1, the dumbbell is in the
  x or y direction

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Notation

• The notation for quantum numbers
• For example, in 3 p
• 3 is quantum number n
• p is quantum number l
• l 0 1 2 3 4 5 6 7
• s p d f g h i k

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Electron Spin

• There is one more quantum number
• Spin quantum number s
• It takes values ?1/2
• There are in total four quantum numbers
• n, l, ml, s
• Any permissible combination of these quantum
  numbers gives a state
• Once the four quantum numbers are given, the
  wavefunction is defined, and the electrons
  energy is determined

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Paulis Exclusion Principle

• Question
• How many electrons can occupy the same state
  defined by a permissible combination of the 4
  quantum numbers?
• Answer is ONE
• There can be no more than one electron in any
  given state Paulis exclusion principle
• It can be derived from a relativistic quantum
  theory

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The Helium Atom

• The helium atom has two protons and two electrons
• Two differential operators for two electrons
• Potential energy for electron 1
  , for electron 2
• For interaction between 1 and 2
• Schrodingers equationelectrons
• It can not be solved without many assumptions

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No Electron Interaction

• One assumption we can make is
• There are no interactions between electrons
• When there are Z protons in the nucleus
• More energy is needed to liberate an electron
  when Z 1
• This allows us to build the periodic table

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Building the Periodic Table

• We have all the knowledge required to build the
  periodic table
• Lets begin with the lowest energy level n 1, l
  0, ml 0, s ?1/2
• The state can take up to two electrons
• One electron gives hydrogen
• Two electrons give helium
• Now the n 1 shell is fully occupied
• It doesnt want to give up any of its electrons
• It cant take another electron
• Helium is chemically inert

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Periodic Table n 2

• Now n has to go to 2 (n 2, Table 4.1)
• Lithium has two electrons with n 1 and l 0
  one electron with n 2 and l 0 1s22s1
• It can give up the 2s electron easily, so its a
  metal
• Beryllium 1s22s2
• Boron has two electrons with n 1 and l 0 two
  electrons with n 2 and l 0 one electron with
  n 2, l 1 1s22s22p1
• Carbon 1s22s22p2
• Nitrogen 1s22s22p3
• Oxygen 1s22s22p4

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Periodic Table n 2

• n 2
• Fluorine 1s22s22p5
• For n 2 and l 1, ml can be 0, 1, -1 and s
  can be ½ and 1/2, so it can take up to six
  electrons
• Fluorine has five 2p electrons and wants to take
  one more electron
• When Li meets F, Li gives up one electron and F
  takes it. Both now have closed n 2 shell
• They form a compound or chemical bond due to
  electrostatic attraction
• Neon is chemically inert 1s22s22p6

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Periodic Table n 3

• Now n has to go to 3 (n 3, Table 4.1)
• Sodium behaves like Li 1s22s22p63s1
• Magnesium 1s22s22p63s2
• Aluminum 1s22s22p63s23p1
• Silicon 1s22s22p63s23p2
• Phosphorous 1s22s22p63s23p3
• Sulfur 1s22s22p63s23p4
• Chlorine behaves like F 1s22s22p63s23p5
• Argon is chemically inert 1s22s22p63s23p6

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Periodic Table n 4

• Potassium 1s22s22p63s23p64s1
• 4s instead of 3d, because 4s states have lower
  energy than 3d states
• Due to electron interactions
• 4p states have higher energy than 3d states
• Transition metals
• Scandium 1s22s22p63s23p63d14s2
• Chromium 1s22s22p63s23p63d54s1
• Copper 1s22s22p63s23p63d104s1
• Krypton inert 1s22s22p63s23p63d104s24p6

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Periodic Table n 5

• More irregularities
• Rare-earth metals
• Eleven electrons occupy the n 5 shell while 4f
  states are being filled
• They are chemically similar

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Periodic Table Summary

• Schrodingers equation gives a set of discrete
  states
• Paulis exclusion principle prohibits more than
  one electron per state
• Electrons are filled into these states
• There is a slow variation of energy within a
  shell
• There is a large energy difference between shells
• Alkali elements starts new shells with one
  electron
• They can easily lose the electron
• Halogen electrons have one less electron in the
  out shell
• They want to get an extra electron to close the
  shell

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HW Assignment

• 4.2, 4.3, 4.5, 4.7