Lecture #6 Physics 7A Cassandra Paul Summer Session II 2008

Today… Finish: Intro to Particle Model of Energy Particle Model of Bond Energy Particle Model of Thermal Energy Solid and liquid relations Modes and equipartition

Last time: PE pair- wise System: Two Particles, one bond Initial: v=0, r=1.12σ Final: v~0 r=3σ Wait! We don’t have an equation for PE pair-wise! It’s ok, we have something better… a graph! Work ΔPE = Work PE f – PE i = Work 0ε – (-1ε) = WorkWork = 1ε Energy Added i f

Is the energy added in our example the same as the ±ΔEbond needed for a phase change? A.Yes, and this is always the case B.No never, this energy added is equal to the total energy of the system. C.Yes, but only when the system consists of only two particles and one bond. Energy Added i f i f Energy Removed

What is ΔE bond at the microscopic level? A.The total amount of energy it takes to break (or form) one bond in a system. B.The total amount of energy released when one bond in a system breaks (or forms). C.The average amount of energy it takes to break (or form) all of the bonds in a system D.The total amount of energy it takes to break (or form) all of the bonds in a system. E.The total amount of energy released when all bonds in a system are broken (or formed).

So what happens when we have more particles? We will get there, but first…

Particle Model of Bond Energy A tool for exploring the energy associated with breaking bonds at the microscopic level

Lets go back to our definition of ΔE bond at the microscopic level: In the Particle Model of Bond Energy: ΔE bond is equal to the total amount of energy it takes to break (or form) all of the bonds in a system. Is there such a thing as instantaneous E bond ? E bond + E thermal = E tot ???? Yes! We need a definition for instantaneous Eb…

The particle is at rest at equilibrium, what is it’s total energy? What form(s) is it in? r0r0 E tot = -1ε =E bond = E bond + E thermal E bond is equal to the potential energy of the system when the particle is at rest at equilibrium. KE + PE = E tot ε = E tot = -1ε + 0 Note: don’t think is proof Ebond = PE we will talk about this soon...

Now back to the question of Ebond when you have more than two particles… Let’s start a little easier than a 18 particle system.

What is the total bond energy of this system? A.~1ε B.~2ε C.~-1ε D.~-2ε E.~-3ε r0r0 r0r0

Are these two particles bonded? We need to draw to scale to answer: 1σ How far apart are the two on the outside? ~2σ (2.24σ to be exact) 1.21σ

1σ 2.24σ The Ebond of the system is: -1ε + -1ε +.03ε=2.03ε

So what is our definition of instantaneous E bond ? Ebond is the total amount of potential energy a system of particles possesses when the particles are at rest. E bond = Σ all pairs (PE pair-wise )  EXACT DEFINITION But what about when we have too many particles to count?

1σ 1.21σ We don’t want to spend all day counting, so we need to develop an approximation

Closest Atomic Packing The red particle has 6 nearest-neighbors in the same plane, three more on top and then three more on the bottom for a total of 12 nearest-neighbors. If you add any more to the system, they are no longer nearest- neighbors. (They are NEXT-nearest-neighbors.) Here’s what it looks like when there are all packed together.

Developing an approximation: How many nearest neighbors does every particle have? Condtions for our approximation: 1.We only want to consider nearest neighbor bonds E bond = Σ n-n bonds (-ε) 2.We don’t want to have to count E bond = (tot # n-n bonds) (-ε) 12 bonds associated with every particle (for close packing, other packings have different #’s) But we know there are two particles associated with every bond So we must divide by 2 in order to get the total number of NN bonds Start with: E bond = Σ all pairs (PE pair-wise ) E bond = n-n/2 (tot # of particles) (-ε)  E bond = 6*(tot # of particles)(-ε)

What if you were given this 2-D packing? How many nearest neighbors does each atom have? A. 9 nearest neighbors B. 8 nearest neighbors C. 2 nearest neighbors D. 4 nearest neighbors

What if you were given this 2-D packing? How many nearest neighbors does each atom have? D. 4 nearest neighbors

If we had 1 mole of this substance what would be the value of E bond ? E bond = n-n/2 (tot # of particles) (-ε) A x10 24 ε B x10 24 ε C x10 24 ε D x10 24 ε E x10 24 ε Hint, how many particles are in a mole?

Ok ready to start another model? We’ve talked about everything except Eth at the microscopic level… so guess what we’re going to cover next? = E bond + E thermal KE + PE = E tot

Particle Model of Thermal Energy A tool for exploring the energy associated with the movement and potential movement of particles at the microscopic level

What is thermal energy at the particle level? Bond Energy is that which is associated with the PE of the particles when they are at rest.

What is thermal energy at the particle level? Bond Energy is that which is associated with the PE of the particles when they are at rest. Thermal Energy is that which is associated with the oscillations (or translational motion) of the particle.

So can we say that PE = E bond and KE = E thermal ? NO!! !

In DL You should have derived: For Solids and Liquids: PE = E bond + ½ E thermal KE = ½ E thermal Why is E thermal split between PE and KE? Think about a mass spring, in order to make the spring oscillate faster through equilibrium, we must stretch the spring further from equilibrium, thus increasing the PE as well.

In DL You should have derived: For Solids and Liquids: PE = E bond + ½ E thermal KE = ½ E thermal Do these equations hold for gases? Lets look at monatomic gases… Atom Hmmm, no spring.

MONOTOMIC gases What MUST be equal to zero? A.PE B.KE C.E bond D.E thermal E.PE and E bond Atom Hmmm, no spring.

MONOTOMIC gases What MUST be equal to zero? D. PE and E bond Atom Hmmm, no spring. = E bond + E thermal KE + PE = E tot KE = E thermal = E tot For gases:

This brings us to MODES Mode: A ‘way’ for a particle to store energy. Gases have different ‘ways’ to have energy than liquids and gases! Each mode contains (½ k b T) of energy where k b is Boltzmann’s constant: k b = 1.38x J/K, and T = Temperature in Kelvin But more on this value later……

3 KE translational modes Modes of an atom in monoatomic gas Every atom can move in three directions 0 PE modes Gas No bonds, i.e. no springs

3 KE translational modes Modes of an atom in solid/liquid Every atom can move in three directions Plus 3 potential energy along three directions 3 PE modes So solids and Liquids have 6 modes total!

Cassandra don’t solids and each have liquids have 12 nearest neighbors and thus 12 springs, and so if each spring has a KE and PE mode, aren’t there 24 modes total!? This is tricky! Yes they each have 12 BONDS but they can only move in 3 DIMENTIONS. (We live in 3-D not 12-D) So the while the particle can move diagonally, this is really only a combination of say to the right, up, and out therefore, the number of modes are DIFFERENT than the number of bonds.

In DL you will figure out how to count modes for diatomic gases too… But there is one more part about the Particle model of bond energy that we have not talked about yet…

Equipartition of Energy 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 Liquids and Solids Gas

Let’s calculate the Thermal Energy of a mole of monatomic gas at 300K…. E thermal per mode = (1/2) k B T E thermal per mode = (1/2) k B T = ½ (1.38x J/K)(300K) = ½ (1.38x J/K)(300K)(3)(6.02x10 23 ) = 3.74 kJ (# of modes per particle)(# of particles) What about the Total KE for a monatomic gas?

How does the KE compare to the E thermal of a monatomic gas? A.KE>Ethermal B.KE<Ethermal C.KE=Ethermal D.Depends on the Substance E.Impossible to tell

Monatomic gases (only) Etot = KE +PE =E bond +E thermal Etot = KE =E thermal Same question as before!

Quiz Monday I will send out an saying what you should know, no later than Thursday afternoon. Have a good weekend!

DL sections Swapno: 11:00AM EversonSection 1 Amandeep: 11:00AM Roesller Section 2 Yi: 1:40PM Everson Section 3 Chun-Yen: 1:40PM Roesller Section 4

Introduction to the Particle Model Potential Energy between two atoms separation Flattening: atoms have negligible forces at large separation. Repulsive: Atoms push apart as they get too close r PE Distance between the atoms