Lecture #6 Physics 7A Cassandra Paul Summer Session II 2008.
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Transcript of Lecture #6 Physics 7A Cassandra Paul Summer Session II 2008.
Lecture #6
Physics 7ACassandra 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:
PEpair-wise
System: Two Particles, one bondInitial: v=0, r=1.12σFinal: 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
PEf – PEi = Work
0ε – (-1ε) = Work Work = 1ε
EnergyAdded
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.
EnergyAdded
i
fi
f
EnergyRemoved
What is ΔEbond 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 ΔEbond
at the microscopic level:
In the Particle Model of Bond Energy:ΔEbond 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 Ebond?
Ebond + Ethermal = Etot ????
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?
r0
Etot = -1ε =Ebond
= Ebond + Ethermal
Ebond is equal to the potential energy of the system when the particle is at rest at equilibrium.
KE + PE = Etot
0 + -1ε = Etot = -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. ~1εB. ~2εC. ~-1εD. ~-2εE. ~-3ε
r0r0
Are these two particles bonded?
We need to draw to scale to answer:
1σ 1σ 1σ
How far apart are the two on the outside?
~2σ (2.24σ to be exact)
1.21σ 1.21σ
1σ 1σ 1σ
2.24σ
The Ebond of the system is:
-1ε + -1ε +.03ε=2.03ε
So what is our definition of instantaneous Ebond?
• Ebond is the total amount of potential energy a system of particles possesses when the particles are at rest.
• Ebond = Σall pairs (PEpair-wise) EXACT DEFINITION
But what about when we have too many particles to count?
1σ 1σ 1σ
1.21σ 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 thenthree 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 areall 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
Ebond = Σn-n bonds (-ε)
2.We don’t want to have to count
Ebond = (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 bondSo we must divide by 2 in order to get the total number of NN bonds
Start with: Ebond = Σall pairs (PEpair-wise)
Ebond = n-n/2 (tot # of particles) (-ε) Ebond = 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 neighborsB. 8 nearest neighborsC. 2 nearest neighborsD. 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 Ebond?
Ebond = n-n/2 (tot # of particles) (-ε)
A. -1.204x1024εB. -2.408x1024εC. 1.204x1024εD. 2.408x1024εE. -4.816x1024ε
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?
= Ebond + Ethermal KE + PE = Etot
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 = Ebond and KE = Ethermal?
NO!!!
In DL You should have derived:
• For Solids and Liquids:PE = Ebond + ½ Ethermal
KE = ½ Ethermal
Why is Ethermal 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 = Ebond + ½ Ethermal
KE = ½ Ethermal
Do these equations hold for gases?Lets look at monatomic gases…
AtomAtom
Hmmm, no spring.
MONOTOMIC gases
What MUST be equal to zero?
A.PEB.KEC.Ebond
D.Ethermal
E.PE and Ebond
AtomAtom
Hmmm, no spring.
MONOTOMIC gases
What MUST be equal to zero?
D. PE and Ebond
AtomAtom
Hmmm, no spring.
= Ebond + Ethermal KE + PE = Etot
KE = Ethermal = EtotFor 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 kb T) of energywhere kb is Boltzmann’s constant: kb = 1.38x10-23 J/K,and T = Temperature in Kelvin
But more on this value later……
3 KEtranslational modes
Modes of an atom in monoatomic gasModes of an atom in monoatomic gas
Every atom can move in three directions
0 PE modes
GasNo bonds, i.e. no springs
3 KEtranslational modes
Modes of an atom in solid/liquidModes 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!?
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 EnergyEquipartition of EnergyIn thermal equilibrium, EIn thermal equilibrium, Ethermal thermal is shared equally among all the is shared equally among all the “active” modes available to the particle. In other words, each “active” modes available to the particle. In other words, each “active” mode has the same amount of energy given by :“active” mode has the same amount of energy given by :
EEthermal per modethermal per mode = (1/2) k = (1/2) kBBTT
Liquids and Solids Gas
Let’s calculate the Thermal Energy of a mole of monatomic gas at 300K….
• EEthermal per modethermal per mode = (1/2) k = (1/2) kBBTT
• = ½ (1.38x10-23J/K)(300K) • = ½ (1.38x10-23J/K)(300K)(3)(6.02x1023)• = 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 Ethermal of a monatomic gas?
A. KE>EthermalB. KE<EthermalC. KE=EthermalD. Depends on the SubstanceE. Impossible to tell
Monatomic gases (only)
Etot = KE +PE =Ebond +Ethermal
Etot = KE =Ethermal
Same question as before!
Quiz Monday
I will send out an email saying what you should know, no later than Thursday afternoon.
Have a good weekend!
DL sections
• Swapno: 11:00AM Everson Section 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 Introduction to the Particle Model Potential Energy between two atomsPotential Energy between two atoms
separation
Flattening: atoms have negligible forcesat large separation.
Repulsive: Atoms push apart as they get too close
r
PE
Distance between the atoms