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!!! 2/7/2014 Physics 132 1
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? 2/7/2014 Physics 132 3
Questions from The 1 st 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)? 2/7/2014 Physics 132 4
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? 2/7/2014 Physics 132 5
A simple 6 atom system http://besocratic.colorado.edu/clue-chemistry/clue- STUDENT/chapter-1/London%20Disperson%20Force/1.2- interactions-2.html 1/23/13 Physics 132 6
Where is energy in this system? Kinetic Energy KE Potential energy PE: Internal energy of a System Kinetic Energy KE Potential Energy PE 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
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? ** 2/7/2014 Physics 132 8
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 increases B. The internal energy decreases C. The internal energy stays the same D. There is not enough information to determine the answer Follow - on How quick is quickly? The internal energy incre... 10% The internal energy decr... 34% The internal energy stays... 56% There is not enough inf... 2/7/2014 Physics 131 9 0%
Classifying Energy 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 U internal. With this, the total energy of our system can be written E = KE + PE + U internal v T =v ave +v rand K v T =(1/2) m v 2 T ave K T = (1/2) m v ave2 + (1/2) m v rand 2 Moving object with many parts K T = K ave macroscopic + K rand internal 2/7/2014 Physics 132 10
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 AIR molecules EARTH 2/7/2014 Physics 132 11
Total energy of a macroscopic object Macroscopic Object Kinetic and Potential Energy E KE PE U Exchanges of energy between the object and the rest of the universe E Q W Internal energy Arbitrary sign choice: Work done by the object on outside Chemistry: Chemical processes usually do not involve motion of the whole object. Thus the macroscopic potential and kinetic energy usually do not change: Physics 131 U Q W 12
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 increases B. The internal energy decreases C. The internal energy stays the same 55% D. There is not enough 26% information to determine the answer 20% 0% The internal energy incre... The internal energy decr... The internal energy stays... There is not enough inf... 2/7/2014 Physics 131 13
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 2/7/2014 Physics 132 14
How does internal energy move within the system? http://besocratic.colorado.edu/clue-chemistry/clue-student/chapter- 1/London%20Disperson%20Force/1.2-interactions-2.html 2/7/2014 Physics 132 15
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 2/7/2014 Physics 132 16
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. Physics 132 17
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. Physics 132 18
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 PE decrease exponentially? All in a few lectures 2/7/2014 Physics 132 E kt e ( ) B
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 people 2. 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? Rank Responses 1 6 2 5 3 4 4 0 5 7 6 Other (Raise your hand if more than 9) 42% 25% 17% 8% 8% 2/7/2014 Physics 131 21 6 5 4 0 7
How many pennies do you have? (raise your hand if you have more than 9) Rank Responses 1 3 2 4 3 4 5 6 Other 50% 50% 2/7/2014 Physics 131 22 3 4
Now lets calculate! Two energy bins divide up 19 chunks of energy randomly 19 chunks Total energy BIN 1 BIN 2 After a collision, the energy is RANDOMLY divided between Bin 1 and Bin 2 Q: How many ways can this be done? 2/7/2014 Physics 132 23
Giving leftover energy to Bin 2 Bin 2 has the following energy 0 1 1 1 2 1 3 1 4 1 19 1 TOTAL number of Scenarios : 20 How many ways can this happen Students fill in! Students fill in! Whiteboard, TA & LA 2/7/2014 Physics 132 24
Giving Leftovers to Bin 3 Bin 3 has the following energy (1&2 have the rest) 0 20 1 18 2 17 3 16 4 15 How many ways can this happen Students fill in! Students fill in! 19 1 TOTAL number of Scenarios : 210 Students fill in! 2/7/2014 Whiteboard, 25 Physics 132 TA & LA
Three bins m chunks of energy Bin 3 has the following energy 0 m+1 1 m 2 m-1 3 m-2 4 m-3 How many ways can this happen Students fill in! Students fill in! m 1 TOTAL number of Scenarios : m 1 2 Students fill in! Whiteboard, TA & LA 2/7/2014 Physics 132 26 2
Now let s add Bin 4 Bin 4 has the following energy 0 1 2 3 4 How many ways can this happen Students m 1 2 fill 2 in! Students m 2 2fill in! m 1 2 2 m 2 2 2 m 3 2 2 m TOTAL number of Scenarios : Whiteboard, TA & LA 2/7/2014 Physics 132 27 1 3 m 1 3 2 Students fill in!
Now let s consider n Bins 2 bins: 3 bins: 4 bins: ( m 1) m 1 2 2 m 1 32 3 n bins n Students fill 1 in! m 1 n 1! Whiteboard, TA & LA 2/7/2014 Physics 132 28
How many different ways can m chunks of energy be shared by n=10 bins 1 10 9 8 10 8 For n=10 bins N(m) ; m n 1 (n 1)! N(m) 6 10 8 4 10 8 There are many ways! Increases fast with m. 2 10 8 Maybe we should plot it differently? 0 0 5 10 15 20 25 30 35 40 m Suggestions?
Plot the log of N(m) S(m) ln N(m) S(m) 25 20 15 10 For n=10 bins Remember: m represents total internal energy 5 0 0 5 10 15 20 25 30 35 40 m
S(m) 25 20 15 10 5 S(m) is Entropy S(m) ln N(m) Remember: m represents total internal energy Entropy increases with internal energy 0 0 5 10 15 20 25 30 35 40 m
Do you remember an equation that contains both entropy and energy? Write on whiteboard! 2/7/2014 Physics 132 32
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)! 1 10 9 8 10 8 model can show N(40 m) ; N(40)exp( m T ) where N(40-m) 6 10 8 4 10 8 2 10 8 Boltzmann 1 T ds(m) dm m 40 U T S This is the definition of temperature! 0 0 5 10 15 20 Whiteboard, m TA & LA
Quiz 1 Info Average 6.3 St. Dev 2.9 35 30 25 20 15 Frequency (020x) Frequency (020x) Correct: 23 D BE 10 5 0 0 1 2 3 4 5 6 7 8 9 10 Fequency (Both) 50 40 30 20 Fequency (Both) 10 0 0 1 2 3 4 5 6 7 8 9 10 2/7/2014 Physics 132 34
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? 2/7/2014 Physics 132 35
Questions from The 2 nd Law of Thermodynamics: A Probablistic Law I don't completely understand the general connection between probability and entropy I'm still a little confused about the difference between a macrostate and 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? 2/7/2014 Physics 132 36
Why are two systems that can share energy at the same temperature? 2/7/2014 Physics 132 37
Two systems each with 10 bins and a total of A m A m chunks of energy exchange of chunks B m B Conservation of energy m A m B m Number of states with a given m A, m B N(m A ) N(m B ) S T ln(n(m A ) N(m B )) S(m A ) S(m B ) Entropy is additive & most likely division has highest entropy
Likelyhood of m A 5 10 12 most likely Equal sharing is most likely Range of values shrinks as number of bins goes up. N(m A )*N(m B ) 4 10 12 3 10 12 2 10 12 1 10 12 range 0 0 5 10 15 20 25 30 35 40 m A
Condition for Maximum Entropy A m A exchange of chunks B m B S A (m A ) S B (m B ) no longer identical systems m A m B m S T S A (m A ) S B (m B m m A ) Total entropy maximum when Temperatures equal T A T B ds T dm A 0
Result 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! 2/7/2014 Physics 132 41
Enthalpy 2/7/2014 Physics 132 42
Enthalpy 2/7/2014 Physics 132 43
Enthalpy 2/7/2014 Physics 132 44
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 Foothold ideas: Entropy S k B ln(w ) S Q T 2/6/13 Physics 132 45
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 2 nd 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?) 2/6/13 Physics 132 46
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. S A > S B B. S B > S A C. S A = S B D. It cannot be determined from the information given 95% 1% 4% 0% It cannot be determine... 2/7/2014 Physics 131 47
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 ( S A - S B ) B. It increases by an amount less than ( S A - S B ) C. It decreases D. It stays the same E. It cannot be determined from the information given It increases by an amoun... 25% It increases by an amount... 32% It decreases 4% 34% 4% It stays the same It cannot be determined... 2/7/2014 Physics 131 48
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 increases B. The internal energy decreases C. The internal energy stays the same D. There is not enough information to determine the answer 25% 7% 68% 0% The internal energy incre... The internal energy decr... The internal energy stays... There is not enough inf... 2/7/2014 Physics 131 49
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 increases B. The pressure decreases C. The pressure stays the same D. There is not enough information to determine the answer 9% 83% 8% 0% 2/7/2014 Physics 131 50 The pressure increases The pressure decreases The pressure stays the same There is not enough informati..
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? A. The entropy increases B. The entropy decreases C. The entropy stays the same D. There is not enough 70% information to determine the answer 17% 14% 0% The entropy increases The entropy decreases The entropy stays the same There is not enough inf... 2/7/2014 Physics 131 51
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 increases 2. The entropy decreases 3. The entropy stays the same 4. 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 1/23/13 Physics 132 53
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