Physics 7A – Lecture 6

Winter 2008

Physics 7A – Lecture 6

Winter 2008

Prof. Robin D. Erbacher343 Phy/Geo Bldg

erbacher@physics.ucdavis.edu

Prof. Robin D. Erbacher343 Phy/Geo Bldg

erbacher@physics.ucdavis.edu

• Quiz 4 is today, focused on material from DLMs 9-12.

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Particle Model of Matter

Particle Model of Matter

Understand the particulate nature of matter

We are modeling real atoms of liquids and solids as oscillating masses and springs. r

Goal : To understand macroscopic phenomena (e.g. melting, vaporizing) and macroscopic properties of matter such as phases, temperature, heat capacities, in terms of microscopic constituents and its behavior.

Atom 1(anchored)

Atom 2(bonded)

~ two atomic size particles interacting via “pair-wise potential” a.k.a. Lennard-Jones Potential

separation

r

Distance between the atoms (r) (units of – atomic diameter)

Equilibrium separation

ro

Po

ten

tial

En

erg

y

separation

r

Distance between the atoms, r

Equilibrium separation

ro

Po

ten

tial

En

erg

y

As the atom-atom separation increases from equilibrium, force from the potential increases. ~ attracting each other when

they are a small distance apart

separation

Flattening: atoms have negligible forces at large separation.

r

Distance between the atoms, r

Repulsive: Atoms push apart as they get too close.

Equilibrium separation

ro

Po

ten

tial

En

erg

y

PE

KE

Etot

Separation (10-10 m)

En

erg

y (1

0-21 J

)

This is what is meant by a “bond” - the particles cannot escape from one another

The bond is an abstraction: Atoms that don’t have enough energy cannot escape the potential (force), so we treat them as bound until we add enough energy to free them.

Energy

r (atomic diameters)

r

is the atomic diameter

ro

is the well depth

ro is the equilibrium separation

pair-wise

~ 10-21 J

~ 10-10m = 1Å

Potential Energy between two atomsPotential Energy between two atoms“pair-wise potential” a.k.a. Lennard-Jones “pair-wise potential” a.k.a. Lennard-Jones

PotentialPotential

Potential Energy between two atomsPotential Energy between two atoms“pair-wise potential” a.k.a. Lennard-Jones “pair-wise potential” a.k.a. Lennard-Jones

PotentialPotential

If a bond is “broken” in an atom-atom potential, which of the following must be true:

A. Etot 0

B. Etot 0

C. PE 0

D. PE 0

E. KE 0

If a bond is considered stronger, it means that:

A. A greater well depth and a greater

B. A smaller well depth and a greater

A. A smaller well depth and

no constraint on

D. A smaller well depth and a smaller

E. A greater well depth and no constraint on

Multiple- Atom Systems:

Particle Model of Ebond ,Particle Model of Ethermal

Multiple- Atom Systems:

Particle Model of Ebond ,Particle Model of Ethermal

• If the atoms in the molecule do not move too far, the forces between them can be modeled as if there were springs between the atoms.

• The potential energy acts similar to that of a simple oscillator.

Liquid: Molecules can move around, but are loosely held together by molecular bonds. Nearly incompressible.

Gas: Molecules move freely through space. Compressible.

Solid: Rigid, definite shape. Nearly incompressible.

Example: H2O

What is Ebond in terms of KE and PE of individual atom (atom pair)?

What is Ethermal in terms of KE and PE of individual atom (atom pair)?

• Typically every pair of atoms interacts

• Magnitude of Ebond for a substance is the amountof energy required to break apart “all” the bonds i.e. we define Ebond = 0 when all the atoms are separated

• We treat bonds as “broken” or “formed”. Bond energy (per bond) exists as long as the bond exists.

• The bond energy of a large substance comes from adding all the potential energies of particles at their equilibrium positions.

Ebond = ∑all pairs(PEpair-wise)

What is the change in bond energy (∆Ebond) by removing the red atom?

2.2 A

4.4 A

6.6 A

-8 x 10-21J

-0.5 x 10-21J

~ 0 J8.8 A

11 A~ 0 J

~ 0 J

Bond energy

Separation (10-10m)

What is the bond energy Ebond for the entire molecule?

-8 x 10-21J

Separation (10-10m)

-8.5 x 10-21J

-8.5 x 10-21J -8.5 x 10-21J

-8.5 x 10-21J

Ebond = -42 x 10-21 Joules

What is the bond energy Ebond for the entire molecule?Separation (10-10m)

Ebond ≈ -40 x 10-21 Joules

Energy required to break a single pair of atoms apart: +8x10-21 J

=5 bonds.

• Ebond for a substance: amount of energy requiredto break apart “all” the bonds (magnitude only) i.e. we define Ebond = 0 when all the atoms are separated

• The bond energy of a substance comes from adding all the potential energies of particles at their equilibrium positions. Ebond = ∑all pairs(PEpair-wise)

A useful approximation of the above relation is:Ebond~ - (total number of nearest neighbor pairs) x ()

Ebond of the system is negative, determined by:1) the depth of the pair-wise potential well (positive) 2) the number of nearest-neighbors.

A pair of atoms are interacting via a atom-atom potential. Only these two atoms are around. In two different situations, the pair have a different amount of total energy. In which situation is Etot greater?

a) Situation A has a greater Etot

b) Situation B has a greater Etot

c) Both have the same Etot

d) Impossible to tell

A

B

A pair of atoms are interacting via a atom-atom potential. Only these two atoms are around. In two different situations, the pair have a different amount of total energy. In which situation is Ebond greater?a) Situation A has a greater Ebond

b) Situation B has a greater Ebond

c) Both have the same Ebond

d) Impossible to tell

A

B

A pair of atoms are interacting via a atom-atom potential. Only these two atoms are around. In two different situations, the pair have a different amount of total energy. In which situation is Ebond greater?a) Situation A has a greater Ebond

b) Situation B has a greater Ebond

c) Both have the same Ebond

d) Impossible to tell

A

B

We did not break a bond - Ebond did not change!

A pair of atoms are interacting via a atom-atom potential. Only these two atoms are around. In two different situations, the pair have a different amount of total energy. In which situation is Ethermal greater?a) Situation A has a greater Ethermal

b) Situation B has a greater Ethermal

c) Both have the same Ethermal

d) Impossible to tell

A

B

A

BKE

A pair of atoms are interacting via a atom-atom potential. Only these two atoms are around. In two different situations, the pair have a different amount of total energy. In which situation is Ethermal greater?a) Situation A has a greater Ethermal

b) Situation B has a greater Ethermal

c) Both have the same Ethermal

d) Impossible to tell

KE

We increased Ethermal by putting more energy into the system

Initial

Final

Okay. Now let us look at the same problem from the perspective of the KE and PE of individual atoms.

Which situation is correct in going from initial to final states?

Initial

Final

Okay. Now let us look at the same problem from the perspective of the KE and PE of individual atoms.

Which situation is correct in going from initial to final states?

Initial

Final

Okay. Now let us look at the same problem from the perspective of the KE and PE of individual atoms.

Which situation is correct in going from initial to final states?

• Ethermal is the energy associated with the random microscopic motions and vibrations of the particles.

• We increased Ethermal by putting more energy into the system.

• KE and PE keep changing into one another as the atoms vibrate, just like in the mass-spring system, so we cannot make meaningful statements about instantaneous KE and PE.

• We can make statements about average KE and PE.

• Increasing Ethermal increases both KEaverage and PEaverage .

QuestionQuestion

What is TemperatureWhat is Temperature

in terms of Ein terms of Ethermalthermal??

??

Answer: Answer:

Temperature IS Thermal Energy!Temperature IS Thermal Energy!

The energy associated with the random motion of particles is split

between PEoscillation and KE .

As we increase Etot we increase PEavg and KEavg

PEavg = KEavg = Etot/2

En

erg

y

position

Etot

PE

KE

• The energy associated with the randommotion of particles is split between PEoscillation , KE.

• For particles in liquids and solids, let’s not forgetthe part that corresponds to Ebond of the system.

• Ebond of the system is determined by both the depth of the pair-wise potential well and the number of nearest neighbors (# NN) .

Equipartition of Energy

Equipartition of Energy

• If the atoms in the molecule do not move too far, the forces between them can be modeled as if there were springs between the atoms.

• Each particle in a solid or liquid oscillates in 3 dimensions about its equilibrium positions as determined by its single-particle potential.

• Another way of stating: Each particle has six “ways” to store the energy associated with its random thermal motion.

• We call this “way” for a system to have thermal energy a “mode”.

QuestionQuestion

What is TemperatureWhat is Temperature

in terms of Ein terms of Ethermalthermal??

??

Answer: Answer:

Temperature IS Thermal Energy!Temperature IS Thermal Energy!

But Wait a Minute…But Wait a Minute…

Answer revised: Answer revised:

Temperature is Temperature is proportionalproportional to to

Thermal Energy EThermal Energy Ethermalthermal..

The constant of proportionality is The constant of proportionality is kkBB: Boltzman’s Constant: Boltzman’s Constant

[Energy] = [Joule] [Temperature] = [Kelvin]

kB = 1.38 10-23 Joule per degree Kelvin

To be precise, energy associated with the To be precise, energy associated with the component of motions/vibrations component of motions/vibrations (“modes”) in any particular direction is (“modes”) in any particular direction is (1/2)k(1/2)kBBT :T :

EEthermal per modethermal per mode = (1/2) k = (1/2) kBBTT

a.k.a. Equipartition of Energya.k.a. Equipartition of Energy

GasLiquids and Solids

Modes : Ways each particle has of storing energy.

Ex. Mass-spring has one KE mode

and one PE mode.

Equipartition of Energy RestatedEquipartition of Energy Restated

In thermal equilibrium, EIn thermal equilibrium, Ethermal thermal is shared equally is shared equally among all the “active” modes available to the among all the “active” modes available to the particle. In other words,each “active” mode has particle. In other words,each “active” mode has the same amount of energy given by :the same amount of energy given by :

EEthermal per modethermal per mode = (1/2) k = (1/2) kBBTT

Gas

Liquids and Solids

3 KEtranslational modesEvery atom can move in three directions

3 KEtranslational modesEvery atom can move in three directions

Plus 3 potential energy modes along three

directions

3 PE modes

Total number of modes is 3PE + 3KE = 6

Ethermal = 6(1/2)kBT

3 KEtranslational modesEvery atom can move in three directions

0 PE modes(no bonds)

Gas:

No bonds, i.e. no springsTotal number of

modes is 3KE = 3:Ethermal = 3(1/2)kBT

3 KEtranslational modes

2 KErotational modes

3 KEtranslational modes

2 KErotational modes

2 vibrational modes (1 KE, 1PE)(associated with atom-atom interaction

within the molecule)

3 KEtranslational modes

2 KErotational modes

2 vibrational modes (1 KE, 1PE)(associated with atom-atom interaction

within the molecule)

Total number of modes is 6KE + 1PE = 7Ethermal = 7(1/2)kBT

•Sometimes (at lower temperatures), however, not all the modes are

“active”. (Freezing out of modes)

KE KE modemode

PE PE modemode

TotalTotal

Solids 3 3 6Liquids 3 3 6

Monatomic gases

3 0 3

Diatomic gases

3+2+1 1 7

Next Time: From Molecular

Models to Macroscopic Properties

Next Time: From Molecular

Models to Macroscopic Properties