Chemistry 431Chemistry 431Lecture 2

Breakdown of classical physicsHeat capacityHeat capacity

Photoelectric effectWave-particle dualityWave-particle duality

Atomic spectraS i l i l h d tSemi-classical hydrogen atom

NC State UniversityNC State University

Breakdown of classical physicsAside from the ultraviolet catastrophe there were a number of experiment observations that did not pagree with classical physics:1. The heat capacity approaches zero as the p y pp

temperature approaches zero2. The “photoelectric effect”. Ionization of a metal p

depends on the frequency, rather the intensity of radiation.

3. Atomic and molecular spectra had discrete lines.

4. The wave-like properties of electrons and other particles.

Heat capacity• The heat capacity is the energy required to raise

the temperature of substance. The definition is:

Cv,m = ∂Um

∂T = ∂< E >∂T

• Solids, liquids and gases all have heat

, ∂T ∂Tq g

capacities.• Um is the molar energy of the substance. This is

also known as the molar internal energy and isthe same as the average energy < E >.

• The heat capacity is given at constant volume.

Internal energy of a solid• Einstein first calculated the internal energy of a

metal by treating it as a collection of oscillators,metal by treating it as a collection of oscillators, which represent the bonds between the atoms

3N hνUm = 3NAhνehν/kT – 1

• This expression assumes that the frequency of the oscillators is hν. The expression has more pthan superficial similarity to the Planck Law. The different is that the Planck Law refers to radiation modes and the Einstein formula refers to vibrational frequencies.

Internal energy of a solid• We can define the Einstein temperature as:

hν

• Using the definition of the Einstein temperature

θE = hνk

• Using the definition of the Einstein temperature we can rewrite the internal energy as:

Um = θE

eθE/T 13R

eθE/T – 1

Limits of the function fof internal energy

• As the temperature approaches 0, the value ofAs the temperature approaches 0, the value of ehν/kT >> 1 so the expression becomes.

U 3NAhν 3N h –hν/kT

• As the temperature becomes large (or

Um ≈ A

ehν/kT = 3NAhνe–hν/kT

As the temperature becomes large (or approaches infinity) we can use the expansion

hν/kT 1 hν

to show that

ehν/kT = 1 – hνkT + ...

to show that Um = 3NAkT = 3RT

Comparison of heat capacities• The classical heat capacity is

C 3R• The classical heat capacity agrees with

Cv,m = 3R• The classical heat capacity agrees with

experiment at room temperature. However, the classical heat capacity fails at low temperature.classical heat capacity fails at low temperature.

• The Einstein heat capacity is

2 2

Cv.m = θE

T

2eθE/2T

eθE/T – 1

2

3RT e E – 1

Comparison of heat capacitiesOne can also write this as follows

C = 3Rfwhere the function.

Cv,m = 3Rf

f θE2

eθE/2T 2

At high temperature f=1 (see page 248) However

f = θE

Te

eθE/T – 1At high temperature f=1 (see page 248). However,at low temperature

f θE2

θ /T

This agrees with experiment As the temperature

f ≈ θE

T e–θE/T

This agrees with experiment. As the temperature goes to zero the heat capacity goes to zero.

Photoelectric Effect• Electrons are ejected

from a metal surface by e-from a metal surface by absorption of a photon.

• Depends on frequency,

hν e

p q ynot on intensity.

• Threshold frequency corresponds to hν = Φ

Metal Surface

KineticEnergycorresponds to hν 0 = Φ

• Φ is the work function. It is essentially equal to

hν

hνΦ Φ

gy

y qthe ionization potential of the metal.

Insufficient energy for photoejection

Photoejection occurs

Photoelectric EffectPhotoelectric Effect• The kinetic energy of the ejected particle is givenThe kinetic energy of the ejected particle is given

by: 1/2 mv2 = hν - Φ

• The threshold energy is Φ, the work function.• This demonstrates the particle-like behavior ofThis demonstrates the particle like behavior of

photons.• A wave-like behavior would be indicated if the a e e be a o ou d be d cated t e

intensity produced the effect.

The Wa e Particle D alitThe Wave-Particle DualityTh f t th t th D B li l th• The fact that the DeBroglie wavelength explains the quantization of the h d t i h lhydrogen atom is a phenomenal success.

• Other wave-like behavior of particles includes electron diffraction.

• Particle-like behavior of waves is shown in the photoelectric effectp

De Broglie Relation• The wave-like properties of particles

can be described very simply in the y p yrelationship of wavelength and momentum:

h

• The practical importance of thisλ = h

p• The practical importance of this

expression is realized in electron microscopy By tuning the acceleratingmicroscopy. By tuning the accelerating voltage in an electron microscope we can alter the momentum and thereforecan alter the momentum and therefore the wavelength of the electron.

The definition of a photon• The wave-particle duality goes both

ways.• If a particle can act like a wave, then a

wave can act like a particle.• Light particles are called photons. The g p p

absorption of photons can explain how atoms and molecules can absorb discrete amounts of energy.

• The energy of a photon is:The energy of a photon is:E = hν

Experimental observation of h d thydrogen atom

• Hydrogen atom emission is “quantized”. It y g qoccurs at discrete wavelengths (and therefore at discrete energies).

• The Balmer series results from four visible lines at 410 nm, 434 nm, 496 nm and 656 nm.

• The relationship between these lines was shown to follow the Rydberg relation.

Atomic spectraAtomic spectra• Atomic spectra consist of series of narrow lines.• Empirically it has been shown that the

wavenumber of the spectral lines can be fit by

)(111~1222 nnR >−==

⎟⎟⎟⎟⎟⎞

⎜⎜⎜⎜⎜⎛

λν 122

22

1 nn ⎟⎟⎟

⎠⎜⎜⎜

⎝λ

where R is the Rydberg constant and n and nwhere R is the Rydberg constant, and n1 and n2are integers.

The hydrogen atom: semi-classical approach

• Why should the hydrogen atom careWhy should the hydrogen atom care about integers?

• What determines the value of the Rydberg constant R=109 677 cm-1?Rydberg constant R=109,677 cm 1?

• Bohr model for the hydrogen atom.

f e 2 m v 2+

Coulomb CentrifugalBalance of forces

f = e4π ε 0r 2 = m v

re-r

• Balance of forces.• Assume electron travels in a radius r.• There must be an integral number of g

wavelengths in the circumference.2πr = nλ n = 1,2,3….

The electron must not interfere with itself

• The condition for a stable orbit is: 2πr = nλ, n=1,2,3..

• The Bohr orbital shown has n = 16.• The DeBroglie wavelength

λ h/ λ h/λ = h/p or λ = h/mv gives: mvr = nh/2π n=1,2,3…

• This is a condition for quantizationThis is a condition for quantization of angular momentum

Example of self-interferencep• According to the Bohr picture the

diti h ill l d tcondition shown will lead to cancellation of the wave and is not a stable orbitnot a stable orbit.

• The quantization of angular momentum implies quantizationmomentum implies quantization of the radius:

2

r = 4πε0n2h2

me2

The significance of quantized orbitsg q

• The Bohr model is consistent with quantized bit f th l t d th lorbits of the electron around the nucleus.

• This implies a relationship between quantized angular momentum and the wavelengthangular momentum and the wavelength.

• Einstein argued (based on relativity) that λ = h/p, where the wavelength of light is λ, and the g g ,momentum of a photon is p.

• DeBroglie argued that the same should hold for ll ti lall particles.

The Bohr Model PredictsThe Bohr Model Predicts Quantized Energies

• The radii of the orbits are quantized and therefore the energies are quantized.

• According to classical electrostatics:

E = T + V = 12

m v 2 – e 2

4 = e 2

8

Substituting in for r gives

2 4π ε 0r 8π ε 0r

E n = – m e 4

8 ε 02h 2

1n 28 ε 0h