Download - Lecture2(Quantum Chem) Ks
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Dr. K. Sumithra
CHEMISTRY I (CHEM C141)
Lecture 2: 6/8/2010
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Summary of the last Lecture:
Failure of Classical MechanicsProblems that led to Quantum Theory
Black Body Radiation – FeaturesWien’s Displacement LawStephan-Boltzmann LawClassical Theory : Rayleigh-Jeans TheoryUV Catastrophe
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Major experimental observations
Not all wavelengths of light are emitted equally
At any temperature, the intensity of emitted light → 0 as the wavelength → 0
It increases to some maximum intensity Imax at some wavelength
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Black body radiation- Features1. Wien’s Displacement Law
maxT = 2.9 mm K (Constant)
max
T
T
Common observation with heated bodies; Red blue
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Stephan-Boltzman Law ; Emittance
Emittance : Area under the curve
Rapid increase with increasing temperature
M3000K = 81 x M1000K
M = Power/Area = aT4
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Black body radiation : Rayleigh-Jeans formula
Energy density d - energy per unit volume associated with radiation of wavelength from to +d
Rayleigh-Jeans formula : d =k
d
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Rayleigh-Jeans Law
kB = 1.38065 x 10-23 J K-1
No quantization of energy, the oscillators could emit any energy
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Consequences
Works at long wavelengths (low frequencies) but fails badly at short wavelengths( high frequencies) As λ decreases, ρ increases without going through maximum
Oscillations of short wavelength areOscillations of short wavelength arestrongly excited at room temperaturestrongly excited at room temperature
k
dd =
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The function rises without bound as decreases
•Even cold objects would emit UV and visible!
Rayleigh-Jeans Formula: UV Catastrophe
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Rayleigh-Jeans Theory
Expt
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h = 6.626 x 10-34 J s, Planck constant
Classically :
1.Radiation from a blackbody is the result of electrons oscillating with frequency .
( It is like electrons in antenna, emitting radio waves!)
2. The electrons can oscillate (& radiate) equally well at any frequency.
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Planck Formula (1900)Crucial assumption
An oscillator of frequency cannot be excited to any arbitrary energy, but only to integral multiples of a fundamental unit or quantum of energy h
h = 6.626 x 10-34 J s, the Planck constant
E = nh, n = 0,1,2,….
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Planck’s Formula (1900)
k
hk
ehkB
T _ 1
hc
ehckB
T _ 1
h = 6.626 x 10-34 J s, Planck constant
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Success of Planck’s formula:
M=aT4
Stefan Boltzman Law is obtained
Integrate over to get total power radiated
Take derivative of w-r-t to get peak
maxT = constant Wien’s displacement Law is obtained
hc
ehckB
T _ 1
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max =hc
4.956kT
x=4.956
maxT = hc/4.9k = constant
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Success of Planck’s formula:
•ehc/kT faster than 5
(Exponential is large) 0 as 0•Energy density 0 as 0•UV Catastrophe avoided
hc
ehckB
T _ 1
Case1 : small
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Success of Planck’s formula:
Case2 : large values of
Reduces to Rayleigh-Jeans formula
hc
ehckB
T _ 1
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h = 6.626 x 10-34 J s, Planck constant
Quantum Ideas1.The energy of the oscillator α ν2. E = nh, n = 0,1,2,….
hν : Quantum of energy
Quantum Mechanics
Restriction on the value of energy
The energy of oscillators is proportional to the frequency of the oscillators.
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Quantization of energy
• Energies in atoms are quantized, not continuous.
• Quantized means only certain energies allowed. continuum discrete
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Quantization of energy!
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Quantization
Planck expression reproduces the experimental distribution with h = 6.63 x 10–34 J s
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Planck's hypothesis: An oscillator cannot be excited unless it receives an
energy of at least hν (as this the minimum amount of energy an oscillator of frequency ν may possess above zero).
For high frequency oscillators (large ν), the amount of energy hν is too large to be supplied by the thermal motion of the atoms in the walls, and so they are not excited.
Catastrophe avoided
Success of Planck’s formula
hc
ehckT _ 1
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Basic Idea behind Planck’s formula
Quantum Ideas1.The energy of the oscillator ν2. E = nh, n = 0,1,2,….
hν : Quantum of energy
Are electromagnetic radiations that simple as we think?
A new view of light?
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Photoelectric Effect
Emission of electrons from metals when exposed to (ultraviolet) radiation.
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Observations1. No emission of electrons below a threshold value
characteristic of the metal – Work function
2. Kinetic energy varies linearly with the frequency
3. Above the threshold value, emission of electrons is instantaneous.
Emission - Independent of light intensity.
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Explanation (EINSTEIN 1905)1. Light : collection of particles, called photons,
each of energy h.
2. If h < , no emission of electrons occurs.
3. Threshold frequency 0 , = h0
4. For > 0, the kinetic energy of the emitted electron Ek = h = h( 0).
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ExampleThe work function of rubidium is 2.09 eV (1 eV = 1.602 x 10-19 J). Can blue (470 nm) light eject electrons from the metal? Need to find out energy of radiation, convert 470 nm to eV.
hν = hc/λ = (6.626 x 10-34 J s) x (3.00 x 108
m/s) / (470 x 10-9 m) = 4.23 X 10-19 J = 2.63 eV
2.63 eV > 2.09 eV
Photoelectrons will be ejected
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Line Spectra
Molecules Dissociate to atoms
Excited Atoms emit radiations of discrete wavelengths.
A spectrum of discrete lines!
Electric discharge
Most compelling evidence for QUANTIZATION
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Line SpectraHot gas emits photons with the characteristic wavelengths corresponding to the transitions between different energy levels of the atoms or molecules in the gas. This leads to bright lines in the spectrum.
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Transitions between quantized energy levels of atom or molecule, with absorption or emission of photon accounts for line spectra.
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Line Spectrum of Hydrogen atom
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The frequencies (in wave numbers) at which the lines occur in the spectrum of hydrogen :
= 1/ = RH(1/n12 1/n2
2)
where RH = 109677 cm-1 , is the Rydberg constant
n1 and n2 > n1 are positive integers
n1 n2 Region
Lyman 1 2,3,4,…. Ultraviolet
Balmer 2 3,4,5,…. Visible
Paschen 3 4,5,6,…. Near IR
Bracket 4 5,6,7,…. IR
Pfund 5 6,7,8,…. Far IR
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Bohr atom model
•Coulombic force and the centripetal force balance
Electron of mass m, in circular orbit of radius r, about stationary nucleus of mass mN, charge Ze
mv2/r = Ze2/40r2
Atomic ModelsRutherford’s Planetary Model
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Bohr Model
1. Specific orbits, discrete quantized energies.
2. The electrons do not continuously lose energy – gain or lose by jumping from one orbit to another
3. quantization of angular momentum
L = mvr = nh/2 = nħ, n = 1,2,3,….
Success
Could explain Rydberg’s formula
Theoretical background for Line Spectra
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Bohr model – Inadequacies
Primitive Model
Semi-classical
•The spectra of larger atoms.
•The relative intensities of spectral lines
•The existence of fine and hyperfine structure in spectral lines.
•The Zeeman effect - changes in spectral lines due to external magnetic fields
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Waves and Particles
Main experiment showing light as particles is the Photoelectric effect
Two properties of waves are:InterferenceDiffraction
The ability for something to behave as a wave and a particle at the same time is known as wave-particle duality.
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Wave-Particle Duality
Double-slit ExperimentInterference: Superposition of two or more waves to generate new patterns
Constructive; destructive
Wave-Particle duality shows:Light can act like a wave and like a particle.Particles can act as waves