before the quiz… - umd department of physics - umd...
TRANSCRIPT
Before the Quiz…
Make sure you have SIX pennies
If you have more than 6, please share with your neighbors
There are some additional pennies in the baskets up front – please be sure to return them after class!!!
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Physics 132- Fundamentals of Physics for Biologists II
Statistical Physics and Thermodynamics
Questions from Thermodynamics and Statistical Physics
How do we describe isolated, closed, open systems?
How can systems with thermal energy have zero momentum?
If the laws of thermodynamics are LAWS (i.e. fundamental principles--not a model), then why will it be expressed in a consistent form in biology, chemistry, and physics?
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Questions from The 1st Law of Thermodynamics
I'm still confused on the usage of signs. How do we determine if a certain energy is positive or negative?
I get confused as to what is considered work done by a system and work done on the system, is there some examples we could go over that explains the difference between the two?
The reading mentions that the total energy can be expressed as E=KE+PE+U(internal). Why is potential energy included twice in the form of PE and U(internal)?
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Questions from Enthalpy
Why do we care more about the change in enthalpy than the enthalpy itself?
Does enthalpy not exist when volume is held constant?
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A simple 6 atom system
http://besocratic.colorado.edu/CLUE-Chemistry/CLUE-STUDENT/chapter-1/London%20Disperson%20Force/1.2-interactions-2.html
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Where is energy in this system?
Kinetic Energy KEPotential energy PE: Internal energy of a System Related to interactions (forces) within the System Can turn into KE (or other energy) when the
objects in the system move Stored in INTERACTION (line between objects)7
Kinetic Energy
Potential Energy
KE
PE
First Law of Thermodynamics
Total amount of energy in a system is unchanged, unless energy is put into the system or taken out of the system across the boundaries.
The first law allows us to tackle two questions:How does energy move within the system?How can the energy of the system change?If the first law states that energy is conserved, how do we apply this law to an open system when energy is being exchanged? **
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Suppose an isolated box of volume 2V is divided into two equal compartments. An ideal gas occupies half of the container and the other half is empty. The partition is quickly removed without friction. How does the energy of the gas at the end compare to the energy of the gas at the start?A. The internal energy increasesB. The internal energy decreasesC. The internal energy stays the
sameD. There is not enough
information to determine the answer
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Classifying Energy
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Macroscopic kinetic energy requires all molecules to move in the same direction, i.e., exhibit coherent motion. Finally the objects in our system also have chemical energy associated with the internal kinetic and potential energies of the electrons in atoms and the binding of atoms into molecules.† We will write the total result of both of these internal energies as Uinternal. With this, the total energy of our system can be written
E = KE + PE + Uinternal†
Moving object with many parts
vaveKT=(1/2) m vT
2vT=vave+vrand
KT= (1/2) m vave2 + (1/2) m vrand
2
KT = Kave + Krand
macroscopic internal
ZOOM out – the six atoms interact with their surrounding
The whole system can move and have kinetic energy
The whole system can have potential energy
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EARTH
AIR molecules
Total energy of a macroscopic object
Exchanges of energy between the object and the rest of the universe
Chemistry: Chemical processes usually do not involve motion of the whole object. Thus the macroscopic potential and kinetic energy usually do not change:
12
E KE PE U
E QWArbitrary sign choice:
Work done by the object on outside
Internal energy
Physics 131
U Q W
Macroscopic ObjectKinetic and Potential Energy
Suppose an isolated box of volume 2V is divided into two equal compartments. An ideal gas occupies half of the container and the other half is empty. The partition separating the two halves has some friction F when it slides and is gradually pushed away by the gas. How does the energy of the gas at the end compare to the energy of the gas at the start?
A. The internal energy increasesB. The internal energy decreasesC. The internal energy stays the
sameD. There is not enough
information to determine the answer
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W is work which you learned about in the first semester
Q is heat – Internal Energy that flows from one system to another when the systems are free to exchange energy– To understand how internal energy flows
from one system to another system, we should first understand how internal energy moves within the system
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How does internal energy move within the system?
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http://besocratic.colorado.edu/CLUE-Chemistry/CLUE-STUDENT/chapter-1/London%20Disperson%20Force/1.2-interactions-2.html
How does energy move within the system?
physical description – forces and motion of colliding atoms/molecules – The loss of energy of one of the colliding
atoms/molecules equals the gain of the other colliding atom/molecule
Statistical description – what happens on average in many collisions or other interactions
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Statistical Description:Thermal Equilibrium
Internal energy resides in “bins” , KE or PE associated with degrees of freedom.
Thermodynamic equilibrium is dynamic –Energy moves from bin to bin, changes keep happening in each bin, but total energy remains unchanged.
Key Assumption - All sharing arrangements among bins are equally likely to occur.
Let’s build a simple model of sharing energy
Total amount of energy is conserved, Energy is divided into small chunks, shared among bins.
Each bin can have an arbitrary number of chunks(but the total number of chunks for all bins is fixed).
We are going to count, in how many ways this slicing of energy into chunks can be done.
Each way of slicing is assumed to be equally likely.
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This simple physical model of sharing of energy will give you a new way to think
about the following concepts that are used in chemistry and biology
Entropy – what is it, and why does it always seem to increase?
Temperature - Why do two bodies in thermal equilibrium have the same temperature?
Boltzmann distribution - why does the probability of having a certain energy decrease exponentially?
All in a few lectures…2/7/2014 Physics 132
( ) B
Ek TP E e
Lets test this model with a simple experiment: Let’s represent energy chunks with pennies
1. Everyone starts with 6 pennies! (i.e. 6 chunks of energy)
2. Turn to a random group of neighbors and “interact” (random exchange of chunks of energy)
1. Interaction means to pool pennies of all people2. One person takes a random amount, a second person
takes a random amount from the rest (or nothing if nothing is left).
20
Before we start: How many pennies do you expect to end up with?
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Rank Responses1 62 53 44 05 76 Other
(Raise your hand if more than 9)
6 5 4 0 7
42%
25%
8%8%
17%
How many pennies do you have? (raise your hand if you have more than 9)
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Rank Responses1 32 43456 Other
3 4
50%50%
Now lets calculate! Two energy “bins” divide up 19
chunks of energy randomly
After a “collision”, the energy is RANDOMLY divided between Bin 1 and Bin 2
Q: How many ways can this be done?
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Tota
l ene
rgy
BIN
1B
IN 2
19 chunks
Giving leftover energy to Bin 2Bin 2 has the following energy
How many ways can this happen
0 11 12 13 14 1…19 1
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TA & LA
TOTAL number of “Scenarios”: 20
Students fill in!
Students fill in!
Giving Leftovers to Bin 3Bin 3 has the following energy (1&2 have the rest)
How many ways can this happen
0 201 182 173 164 15
19 1
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Whiteboard, TA & LA
TOTAL number of “Scenarios”: 210
Students fill in!
Students fill in!
Students fill in!
Three bins – m chunks of energy
Bin 3 has the following energy
How many ways can this happen
0 m+11 m2 m-13 m-24 m-3
m 1
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TA & LA
TOTAL number of “Scenarios”:
Students fill in!
Students fill in!
212
m Students fill in!
Now let’s add Bin 4
Bin 4 has the following energy
How many ways can this happen
01234
m
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TA & LA
TOTAL number of “Scenarios”:
21 2m Students fill in!
2 2m 21 2m
22 2m
23 2m
1
31 3 2m Students fill in!
Students fill in!
Now let’s consider n Bins
2/7/2014 28Physics 132Whiteboard,
TA & LA
n bins
2 bins:
3 bins:
4 bins:
212
m
( 1)m
313 2
m
111 !
nmn
Students fill in!
How many different ways can m chunks of energy be shared by n=10 bins
N (m) ; mn1
(n 1)!
0
2 108
4 108
6 108
8 108
1 109
0 5 10 15 20 25 30 35 40
N(m
)
m
There are many ways!
Increases fast with m.
Maybe we should plot it differently?
Suggestions?
For n=10 bins
Plot the log of N(m)S(m) ln N(m)
0
5
10
15
20
25
0 5 10 15 20 25 30 35 40
S(m
)
m
Remember:
m represents total internal energy
For n=10 bins
S(m) is EntropyS(m) ln N(m)
0
5
10
15
20
25
0 5 10 15 20 25 30 35 40
S(m
)
m
Remember:
m represents total internal energy
Entropy increases with internal energy
Do you remember an equation that contains both entropy and energy? Write on whiteboard!
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Add 11th Bin and Assume 40 shared packets
How many ways can the 11th bin have m packets?
N (40 m) ; (40 m)9
(9)!
0
2 108
4 108
6 108
8 108
1 109
0 5 10 15 20
N(4
0-m
)
m
model
Boltzmann
N(40 m) ; N(40)exp(
mT
)
can show
where
1T
dS(m)dm m40
U TS
This is the definition of temperature !Whiteboard, TA &
LA
Quiz 1 Info
Average 6.3 St. Dev 2.9
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0
5
10
15
20
25
30
35
0 1 2 3 4 5 6 7 8 9 10
Frequency (020x)
Frequency (020x)
Correct: 23 D BE
0
10
20
30
40
50
0 1 2 3 4 5 6 7 8 9 10
Fequency (Both)
Fequency (Both)
Questions from Thermodynamic Equilibrium and Equipartition
Why does thermodynamic equilibrium equate to death in living organisms? How does 1/2kT describe energy for all degrees of freedom? What does it mean to say " in biological systems it's the fact that energy always TENDS towards equilibrium“?
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Questions from The 2nd Law of Thermodynamics: A Probablistic Law
I don't completely understand the general connection between probability and entropyI'm still a little confused about the difference between a macrostateand a microstate. They both pertain to the same situation but how are they different?I don't understand the statement "it is entirely possible that one part of the universe will exhibit an entropy decrease during a spontaneous process while the rest of the universe exhibits a larger increase in entropy, such that the overall entropy in the universe has increased." Could you provide some sort of analogy/visual to clarify this?
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Why are two systems that can share energy at the same
temperature?
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Two systems each with 10 bins and a total of m chunks of energy
A Bexchange of chunks
mAmB
Conservation of energy mA mB m
Number of states with a given mA, mB N(mA ) N(mB )ST ln(N(mA ) N(mB )) S(mA ) S(mB )
Entropy is additive & most likely division has highest entropy
Likelyhood of mA
0
1 1012
2 1012
3 1012
4 1012
5 1012
0 5 10 15 20 25 30 35 40
N(m
A)*
N(m
B)
mA
Equal sharing is most likely
Range of values shrinks as number of bins goes up.
most likely
range
Condition for Maximum EntropyA Bexchange
of chunksmA mB
SA (mA ) SB (mB )
no longer identical systemsmA mB m
ST SA (mA ) SB (mB m mA )
Total entropy maximum when dST
dmA
0
TA TBTemperatures equal
Result
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What is entropy? S = ln (N) Why does entropy increase ? Max S most likely What is temperature? 1/T = dS/dU Why do two bodies in thermal equilibrium have the
same temperature? Condition for maximum total S Where does the Boltzmann distribution come from?
1 Bin sharing with many Bins
Ta Da!
Enthalpy
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Enthalpy
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Enthalpy
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Entropy – an extensive measure of how well energy is spread in a system.
Entropy measures– The number of microstates
in a given macrostate– The amount that the energy of a system is
spread among the various degrees of freedom
Change in entropy upon heat flow
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Foothold ideas:Entropy
S kB ln(W )
S QT
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Foothold ideas:The Second Law of Thermodynamics
Systems spontaneously move toward the thermodynamic (macro)state that correspond to the largest possible number of particle arrangements (microstates).– The 2nd law is probabilistic. Systems show fluctuations –
violations that get proportionately smaller as N gets large. Systems that are not in thermodynamic equilibrium will
spontaneously transform so as to increase the entropy.– The entropy of any particular system can decrease as
long as the entropy of the rest of the universe increases more.
The universe tends towards states of increasing chaos and uniformity. (Is this contradictory?)
A small amount of heat Q flows out of a hot system A (350K) into a cold system B (250K). Which of the following correctly describes the entropy changes that result? (The systems are thermally isolated from the rest of the universe.)
A. |∆SA | > |∆SB|B. |∆SB | > |∆SA|C. |∆SA| = |∆SB|D. It cannot be
determined from the information given
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Suppose a small amount of heat Q flows from a system A at low temperature (250K) to a system B at high temperature (350K). Which of the following must be true regarding the entropy of the rest of the universe during this process?A. It increases by an amount
greater than (|∆SA| - |∆SB|)B. It increases by an amount
less than (|∆SA| - |∆SB|)C. It decreasesD. It stays the sameE. It cannot be determined
from the information given
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Suppose an isolated box of volume 2V is divided into two equal compartments. An ideal gas occupies half of the container and the other half is empty. When the partition separating the two halves of the box is removed and the system reaches equilibrium again, how does the new internal energy of the gas compare to the internal energy of the original system?
A. The internal energy increasesB. The internal energy decreasesC. The internal energy stays the
sameD. There is not enough
information to determine the answer
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Suppose an isolated box of volume 2V is divided into two equal compartments. An ideal gas occupies half of the container and the other half is empty. When the partition separating the two halves of the box is removed and the system reaches equilibrium again, how does the new pressure of the gas compare to the pressure of the original system?
A. The pressure increasesB. The pressure decreasesC. The pressure stays the
sameD. There is not enough
information to determine the answer
A. The entropy increasesB. The entropy decreasesC. The entropy stays the sameD. There is not enough
information to determine the answer
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Suppose an isolated box of volume 2V is divided into two equal compartments. An ideal gas occupies half of the container and the other half is empty. When the partition separating the two halves of the box is removed and the system reaches equilibrium again, how does the new entropy of the gas compare to the entropy of the original system?
Suppose an isolated box of volume 2V is divided into two equal compartments. An ideal gas occupies half of the container and the other half is empty. When the partition separating the two halves of the box is removed and the system reaches equilibrium again, how does the new entropy of the gas compare to the entropy of the original system?1. The entropy increases2. The entropy decreases3. The entropy stays the
same4. There is not enough
information to determine the answer
Does the volume affect entropy?
1. In an ideal gas, atoms take up no volume so atoms have infinitely many microstates they may choose
2. Each atom has a finite volume, so the number of microstates increases with the available volume
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