Dr. K. Sumithra

CHEMISTRY I (CHEM C141)

Lecture 2: 6/8/2010

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

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

Black body radiation- Features1. Wien’s Displacement Law

maxT = 2.9 mm K (Constant)

max

T

T

Common observation with heated bodies; Red blue

Stephan-Boltzman Law ; Emittance

Emittance : Area under the curve

Rapid increase with increasing temperature

M3000K = 81 x M1000K

M = Power/Area = aT4

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

Rayleigh-Jeans Law

kB = 1.38065 x 10-23 J K-1

No quantization of energy, the oscillators could emit any energy

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 =

The function rises without bound as decreases

•Even cold objects would emit UV and visible!

Rayleigh-Jeans Formula: UV Catastrophe

Rayleigh-Jeans Theory

Expt

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.

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,….

Planck’s Formula (1900)

k

hk

ehkB

T _ 1

hc

ehckB

T _ 1

h = 6.626 x 10-34 J s, Planck constant

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

max =hc

4.956kT

x=4.956

maxT = hc/4.9k = constant

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

Success of Planck’s formula:

Case2 : large values of

Reduces to Rayleigh-Jeans formula

hc

ehckB

T _ 1

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.

Quantization of energy

• Energies in atoms are quantized, not continuous.

• Quantized means only certain energies allowed. continuum discrete

Quantization of energy!

Quantization

Planck expression reproduces the experimental distribution with h = 6.63 x 10–34 J s

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

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?

Photoelectric Effect

Emission of electrons from metals when exposed to (ultraviolet) radiation.

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.

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).

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

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

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.

Transitions between quantized energy levels of atom or molecule, with absorption or emission of photon accounts for line spectra.

Line Spectrum of Hydrogen atom

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

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

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

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

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.

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