Transcript Slide 1

Lecture #6
Physics 7A
Cassandra Paul
Summer Session II 2008
Today…
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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:
System: Two Particles, one bond
Initial: v=0, r=1.12σ
Final: v~0 r=3σ
Work
f
PEpairwise
ΔPE = Work
Energy
Added
Wait! We don’t have an equation for PE pair-wise!
It’s ok, we have something better… a graph!
PEf – PEi = Work
0ε – (-1ε) = Work
Work = 1ε
i
Is the energy added in our example the
same as the ±ΔEbond needed for a phase
change?
if
Energy
Added
Energy
Removed
f
i
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.
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
KE + PE = Etot
= Ebond + Ethermal
0 + -1ε = Etot = -1ε + 0
Note: don’t think is proof Ebond = PE we will talk about this soon...
Etot = -1ε =Ebond
Ebond is equal to the potential energy of the
system when the particle is at rest
at equilibrium.
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.
B.
C.
D.
E.
~1ε
~2ε
~-1ε
~-2ε
~-3ε
r0
r0
Are these two particles bonded?
We need to draw to scale to answer:
How far apart are the two on the outside?
1σ
1σ
1.21σ
1σ
1.21σ
~2σ (2.24σ to be exact)
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?
We don’t want to spend all day counting,
so we need to develop an approximation
1σ
1σ
1.21σ
1σ
1.21σ
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 nearestneighbors. (They are NEXT-nearest-neighbors.)
Here’s what it looks like when there are
all packed together.
Developing an approximation:
Start with: Ebond = Σall pairs (PEpair-wise)
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)
How many nearest neighbors does every particle have?
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
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.
B.
C.
D.
9 nearest neighbors
8 nearest neighbors
2 nearest neighbors
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?
Hint, how many particles are in a mole?
Ebond = n-n/2 (tot # of particles) (-ε)
A.
B.
C.
D.
E.
-1.204x1024ε
-2.408x1024ε
1.204x1024ε
2.408x1024ε
-4.816x1024ε
Ok ready to start another model?
KE + PE = Etot = Ebond + Ethermal
We’ve talked about everything except Eth at the
microscopic level… so guess what we’re going to
cover next?
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?
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…
Hmmm, no spring.
Atom
MONOTOMIC gases
Hmmm, no spring.
Atom
What MUST be equal
to zero?
A.PE
B.KE
C.Ebond
D.Ethermal
E. PE and Ebond
MONOTOMIC gases
Hmmm, no spring.
Atom
What MUST be equal
to zero?
D. PE and Ebond
KE + PE = Etot = Ebond + Ethermal
For gases:
KE = Ethermal = Etot
Gases have different ‘ways’ to have
energy than liquids and gases!
This brings us to MODES
Mode: A ‘way’ for a particle to store energy.
Each mode contains (½ kb T) of energy
where kb is Boltzmann’s constant: kb = 1.38x10-23 J/K,
and T = Temperature in Kelvin
But more on this value later……
Modes of an atom in monoatomic gas
Every atom can move in three directions
Gas
No bonds, i.e. no springs
3 KEtranslational modes
0 PE modes
Modes of an atom in solid/liquid
Every atom can move in three directions
3 KEtranslational modes
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, Ethermal 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 :
Ethermal per mode = (1/2) kBT
Liquids and Solids
Gas
Let’s calculate the Thermal Energy of a
mole of monatomic gas at 300K….
• Ethermal per mode = (1/2) kBT
• = ½ (1.38x10-23J/K)(300K) (# of modes per particle)
• = ½ (1.38x10-23J/K)(300K)(3)(6.02x1023)
• = 3.74 kJ
What about the Total KE for a monatomic gas?
(# of particles)
How does the KE compare to the
Ethermal of a monatomic gas?
A.
B.
C.
D.
E.
KE>Ethermal
KE<Ethermal
KE=Ethermal
Depends on the Substance
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
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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
Potential Energy between two atoms
PE
Repulsive:
Atoms push apart as they
get too close
Flattening:
atoms have negligible forces
at large separation.
separation
r
Distance between the atoms