Physics 7A – Lecture 6 Winter 2008 Prof. Robin D. Erbacher 343 Phy/Geo Bldg Prof. Robin D. Erbacher 343 Phy/Geo Bldg

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Particle Model of Matter

Normal Matter: Particles Bouncing Around! Understand the particulate nature of matter

Particle Model 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.

Model: Bonded Atoms as Masses on Springs Atom 1 (anchored) Atom 2 (bonded) ~ two atomic size particles interacting via “pair- wise potential” a.k.a. Lennard-Jones Potential

Potential Energy Between Two Atoms “pair-wise potential” a.k.a. Lennard-Jones Potential separation r Distance between the atoms (r) (units of  – atomic diameter) Equilibrium separation r o Potential Energy

“Pairwise” Atom-Atom Potential a.k.a Leonard-Jones Potential separation r Distance between the atoms, r Equilibrium separation r o Potential Energy As the atom-atom separation increases from equilibrium, force from the potential increases. ~ attracting each other when they are a small distance apart

“Pairwise” Atom-Atom Potential a.k.a Leonard-Jones Potential 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 r o Potential Energy

PE KE E tot Separation ( m) Energy ( 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 roro   is the well depth r o is the equilibrium separation  pair-wise  ~ J  ~ m = 1Å Potential Energy between two atoms “pair-wise potential” a.k.a. Lennard-Jones Potential

Clicker: What is breaking a bond? If a bond is “broken” in an atom-atom potential, which of the following must be true: A. E tot  0 B. E tot  0 C. PE  0 D. PE  0 E. KE  0  

Clicker: A Stronger Bond Means: 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  C.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 E bond, Particle Model of E thermal

Molecular Model 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.

Phases Under the Microscope 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.

Particle Model of E bond and E thermal Example: H 2 O What is E bond in terms of KE and PE of individual atom (atom pair)? What is E thermal in terms of KE and PE of individual atom (atom pair)?

Multiple Atom Systems: E bond Typically every pair of atoms interacts Magnitude of E bond for a substance is the amount of energy required to break apart “all” the bonds i.e. we define E bond = 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. E bond = ∑ all pairs (PE pair-wise )

What is the change in bond energy (∆E bond ) by removing the red atom? 2.2 A 4.4 A 6.6 A -8 x J -0.5 x J ~ 0 J 8.8 A 11 A ~ 0 J Bond energy Separation ( m) Atom-atom potential for each atom

What is the bond energy E bond for the entire molecule? -8 x J Separation ( m) Atom-atom potential for each atom -8.5 x J E bond = -42 x Joules

What is the bond energy E bond for the entire molecule? Separation ( m) Atom-atom potential for each atom E bond ≈ -40 x Joules Energy required to break a single pair of atoms apart: +8x J =5 bonds.

Recap: Particle Model of E bond E bond for a substance: amount of energy required to break apart “all” the bonds (magnitude only) i.e. we define E bond = 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. E bond = ∑ all pairs (PE pair-wise ) A useful approximation of the above relation is: E bond ~ - (total number of nearest neighbor pairs) x (  )  E bond of the system is negative, determined by: 1) the depth of the pair-wise potential well   (positive) 2) the number of nearest-neighbors.

Clicker: E total, E bond, E thermal 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 E tot greater? a) Situation A has a greater E tot b) Situation B has a greater E tot c) Both have the same E tot d) Impossible to tell A B

Clicker: E total, E bond, E thermal 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 E bond greater? a) Situation A has a greater E bond b) Situation B has a greater E bond c) Both have the same E bond d) Impossible to tell A B

Clicker: E total, E bond, E thermal 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 E bond greater? a) Situation A has a greater E bond b) Situation B has a greater E bond c) Both have the same E bond d) Impossible to tell A B We did not break a bond - E bond did not change!

Clicker: E total, E bond, E thermal 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 E thermal greater? a) Situation A has a greater E thermal b) Situation B has a greater E thermal c) Both have the same E thermal d) Impossible to tell A B

A B KE Clicker: E total, E bond, E thermal 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 E thermal greater? a) Situation A has a greater E thermal b) Situation B has a greater E thermal c) Both have the same E thermal d) Impossible to tell KE We increased E thermal by putting more energy into the system

Initial Final Clicker: Individual Atoms 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 Clicker: Individual Atoms 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 Clicker: Individual Atoms 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?

Particle Model of E thermal E thermal is the energy associated with the random microscopic motions and vibrations of the particles. We increased E thermal 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 E thermal increases both KE average and PE average.

Question What is Temperature in terms of E thermal ? in terms of E thermal ? ? ? Answer: Temperature IS Thermal Energy!

Particle Model of E bond and E thermal The energy associated with the random motion of particles is split between PE oscillation and KE.

Mass on a Spring As we increase E tot we increase PE avg and KE avg PE avg = KE avg = E tot /2 Energ y position E tot PE KE

Particle Model of E bond and E thermal The energy associated with the random motion of particles is split between PE oscillation, KE. For particles in liquids and solids, let’s not forget the part that corresponds to E bond of the system. E bond of the system is determined by both the depth of the pair-wise potential well and the number of nearest neighbors (# NN).

Particle Model of E bond and E thermal E bond of the system is determined by both the depth of the pair-wise potential well and the # NN. Solids/Liquids: KE all atoms = (1/2)E thermal PE all atoms = PE bond + PE oscillation = E bond (PE bond )+ (1/2)E thermal (PE oscillation )  KE all atoms + PE all atoms = E thermal + E bond Gases (monoatomic): The gas phase has no springs: no PE oscillation or PE bond

Equipartition of Energy

Intro to 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.

Intro to Equipartition of Energy 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”.

Question What is Temperature in terms of E thermal ? in terms of E thermal ? ? ? Answer: Temperature IS Thermal Energy!

But Wait a Minute… Answer revised: Temperature is proportional to Thermal Energy E thermal. The constant of proportionality is k B : Boltzman’s Constant [Energy] = [Joule][Temperature] = [Kelvin] k B = 1.38  Joule per degree Kelvin

To be precise, energy associated with the component of motions/vibrations (“modes”) in any particular direction is (1/2)k B T : E thermal per mode = (1/2) k B T a.k.a. Equipartition of Energy Gas Liquids and Solids Equipartition of Energy

ModesModes Modes : Ways each particle has of storing energy. Ex. Mass-spring has one KE mode and one PE mode.

Equipartition of Energy Restated In thermal equilibrium, E thermal is shared equally among all the “active” modes available to the particle. In other words,each “active” mode has the same amount of energy given by : E thermal per mode = (1/2) k B T Gas Liquids and Solids Equipartition of Energy: Restated

Modes of an atom in solid/liquid 3 KE translational modes Every atom can move in three directions

Modes of an atom in solid/liquid 3 KE translational modes Every 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 E thermal = 6  (1/2)k B T

Modes of an atom in a Monoatomic Gas 3 KE translational modes Every atom can move in three directions 0 PE modes (no bonds) Gas: No bonds, i.e. no springs Total number of modes is 3KE = 3: E thermal = 3  (1/2)k B T

Modes of a Molecule in a Diatomic Gas 3 KE translational modes 2 KE rotational modes

Modes of a Molecule in a Diatomic Gas 3 KE translational modes 2 KE rotational modes 2 vibrational modes (1 KE, 1PE) (associated with atom-atom interaction within the molecule)

Modes of a Molecule in a Diatomic Gas 3 KE translational modes 2 KE rotational modes 2 vibrational modes (1 KE, 1PE) (associated with atom-atom interaction within the molecule) Total number of modes is 6KE + 1PE = 7 E thermal = 7  (1/2)k B T Sometimes (at lower temperatures), however, not all the modes are “active”. (Freezing out of modes)

Equipartition of Energy: E thermal per Mode = ½ k B T KE mode PE mode Total Solids336 Liquids336 Monatomic gases 303 Diatomic gases

Next Time: From Molecular Models to Macroscopic Properties