Chemistry 431 Chemistry 431 Lecture 2 Breakdown of classical - - PowerPoint PPT Presentation

Chemistry 431 Chemistry 431

Lecture 2 Breakdown of classical physics Heat capacity Heat capacity Photoelectric effect Wave-particle duality Wave-particle duality Atomic spectra S i l i l h d t Semi-classical hydrogen atom NC State University NC State University

Breakdown of classical physics

Aside from the ultraviolet catastrophe there were a number of experiment observations that did not p agree with classical physics:

• 1. The heat capacity approaches zero as the

p y pp temperature approaches zero

• 2. The “photoelectric effect”. Ionization of a metal

p depends on the frequency, rather the intensity

• f 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 ∂T

q g capacities.

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

also known as the molar internal energy and is the 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 hν Um = 3NAhν ehν/kT – 1

• This expression assumes that the frequency of

the oscillators is hν. The expression has more p than 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 = hν k

• 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 f

• f internal energy
• As the temperature approaches 0, the value of

As 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 – hν kT + ...

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

2

eθE/2T eθE/T – 1

2

3R T e

E

– 1

Comparison of heat capacities

One can also write this as follows

C = 3Rf

where the function.

Cv,m = 3Rf

f θE

2

eθE/2T

2

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

f = θE T e eθE/T – 1

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

f θE

2 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 y not on intensity.

• Threshold frequency

corresponds to hν = Φ

Metal Surface Kinetic Energy

corresponds to hν 0 = Φ

• Φ is the work function.

It is essentially equal to

hν hν Φ Φ

gy

y q the ionization potential

• f the metal.

Insufficient energy for photoejection Photoejection occurs

Photoelectric Effect Photoelectric Effect

• The kinetic energy of the ejected particle is given

The 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 of

This demonstrates the particle like behavior of photons.

• A wave-like behavior would be indicated if the

a e e be a o

• u d be

d cated t e intensity produced the effect.

The Wa e Particle D alit The Wave-Particle Duality

Th 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 l hydrogen atom is a phenomenal success.

• Other wave-like behavior of particles

includes electron diffraction.

• Particle-like behavior of waves is shown

in the photoelectric effect p

De Broglie Relation

• The wave-like properties of particles

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

h

• The practical importance of this

λ = h p

• The practical importance of this

expression is realized in electron microscopy By tuning the accelerating

• microscopy. By tuning the accelerating

voltage in an electron microscope we can alter the momentum and therefore can 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 t hydrogen atom

• Hydrogen atom emission is “quantized”. It

y g q

• ccurs 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 spectra Atomic 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 ) ( 1 1 1 ~

1 2 2 2

n n R > − = =

⎟ ⎟ ⎟ ⎟ ⎟ ⎞ ⎜ ⎜ ⎜ ⎜ ⎜ ⎛

λ ν

1 2 2 2 2 1

n n

⎟ ⎟ ⎟ ⎠ ⎜ ⎜ ⎜ ⎝

λ where R is the Rydberg constant and n and n where R is the Rydberg constant, and n1 and n2 are integers.

The hydrogen atom: semi-classical approach

• Why should the hydrogen atom care

Why 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 Centrifugal Balance of forces

f =

e 4 π ε 0r 2 = m v r e- 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 quantization

This is a condition for quantization

• f angular momentum

Example of self-interference p

• According to the Bohr picture the

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

• The quantization of angular

momentum implies quantization momentum implies quantization

• f the radius:

2

r = 4πε0n 2h

2

me 2

The significance of quantized orbits g q

• The Bohr model is consistent with quantized

bit f th l t d th l

• rbits of the electron around the nucleus.
• This implies a relationship between quantized

angular momentum and the wavelength angular 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 l all particles.

The Bohr Model Predicts The 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 = 1 2 m v 2 – e 2 4 = e 2 8

Substituting in for r gives

2 4 π ε 0r 8 π ε 0r

E n = – m e 4 8 ε 0

2h 2 1

n 2 8 ε 0h