Lecture 13. Multiplicity and statistical definition of entropy

Transcription

1 Lecture 13 Multiplicity and statistical definition of entropy Readings: Lecture 13, today: Chapter 7: Lecture 14, Monday: Chapter 7: end 2/26/16 1

2 Today s Goals Concept of entropy from statistical thermodynamics Bulk properties of large collections of events/molecules Multiplicity Macro and microstates Statistical definition of the second law of thermodynamics For a spontaneous process, the multiplicity of the system increases Statistical treatment of multiplicity explains macroscopic properties back to the ideal gas expansion 2/26/16 2

3 Hemoglobin and probability Oxygen binding molecule. Its quaternary structure is a tetramer composed of two α- and β- subunits. Assume that for each heme: p(bound)=1/2 p(unbound)=1/2 2/26/16 3

4 Independent events ½ ½ ½ ½ The probability of two independent events is the product of their individual probabilities. ½ ½ ½ ½ p = ½ x ½ x ½ x ½ = 1/16 ½ ½ ½ ½ Therefore, for each hemoglobin, the probability of obtaining any given outcome is 1/16 2/26/16 4

5 Heme position Multiplicity What if all you care about is the number of heme groups that either have or do not have oxygen bound? Aggregate the relevant possible outcomes together # of desired outcomes p total# of possible outcomes Number of desired outcomes is the multiplicity, W 2/26/16 5

6 Multiplicity of a molecular system The multiplicity of a molecular system is the number of different molecular configurations consistent with the macroscopic parameters (e.g. temperature, volume, concentration) that define the system A microstate is one particular configuration of molecules that is consistent with the global macroscopic parameters that define a particular state of the system The multiplicity is the number of microstates 2/26/16 6

7 Heme position Multiplicity 4 bound & 0 unbound W = 1 p = 1/16 = bound & 1 unbound W = 4 p = 4/16 = bound & 2 unbound W = 6 p = 6/16 = bound & 3 unbound W = 4 p = 4/16 = bound & 4 unbound W = 1 p = 1/16 = /26/16 7

8 How to compute multiplicity When we get to a large number of outcomes / microstates, it gets tedious to classify and count all of these For the binomial case (i.e. two possible outcomes, such as bound/unbound), we can generalize this as: W M! N!( M N)! Where N is the number of bound hemes obtained in a total of M hemes 2/26/16 8

9 Calculating multiplicity using factorials Calculate W for 2 bound & 2 unbound in a hemoglobin: Calculate W for 4 bound& 0 unbound in a hemoglobin: Note: 0! = 1 (the product of no numbers at all is 1) 2/26/16 9

10 Ways of binding oxygen Four hemes, labeled 1 through 4 (numbers represented by different colors) In how many possible ways can the hemes be chosen? 4 choices x 3 choices x 2 choices x 1 choice 4!=24 2/26/16 10

11 Multiplicity of 2 bound & 2 unbound 2 crimson bound sites 2 blue unbound sites 2/26/16 11

12 Multiplicity of 2 bound & 2 unbound 2 crimson bound sites 2 blue unbound sites In the 24 outcomes, there are several instances of different outcomes with the same bound/unbound pattern We re over-counting these instances By how much? There are 2! ways of rearranging the bound sites and 2! ways of rearranging the unbound sites W = 4!/(2! x 2!) = 6 2/26/16 12

13 Hemoglobin multiplicity Outcome Example W M! N!( M N)! 4 bound & no unbound 3 bound & 1 unbound 2 bound & 2 unbound W W W 4! 4!0! 4! 31!! 4! 2!2! bound & 3 unbound W 4! 1!3! 4 0 bound & 4 unbound W 4! 0!4! 1 2/26/16 13

14 Over-counting in a bigger dataset 2 hemoglobins, 8 hemes, 5 bound and 3 unbound Number of ways to pick the 8 sites in random order: 8! Two microstates of the same outcome Ways to arrange 5 bound states: 5 x 4 x 3 x 2 x 1 = 120 = 5! Same for the 3 unbound states: 3! Over-counting for this red-blue pattern: 5! x 3! Multiplicity = 8!/5!3! = M!/(N!(M-N)!) Where M = # hemes, N = # bound states and (M-N) = # unbound states 2/26/16 14

15 Multiplicity (W) Multiplicity (W) Multiplicity (W) Multiplicity (W) 4 Hemes (1 Hemoglobin) As M increases 10 Hemes Bound O 2 molecules 200 Hemes Bound O 2 molecules 1000 Hemes Bound O 2 molecules Bound O 2 molecules Modified from The Molecules of Life ( Garland Science 2008) 2/26/16 15

16 250 hemoglobins-1000 hemes M = 1000, N = 500 oxygen-bound hemes The multiplicity of this outcome is: W 1000! 500!500! Stirling s approximation is a useful mathematical tool to calculate values for large factorials ln n! nln n n Where n is a large number 2/26/16 16

17 Using Stirling s approximation M = 1000, N = 500 oxygen-bound hemes Applying a natural log to both sides: Then: lnw lnw ln W ln(1000!) ln(500!) ln(500!) ln(1000!) 2ln(500!) 1000! 500!500! 1000! 500!500! Applying Stirling s approximation: ln n! nln n n lnw 1000ln (500ln ) W = e 694 ~ /26/16 17

18 Probability of N=500? W 500 ~ Number of possible outcomes: ~ p 500 = multiplicity / # of possible outcomes ~ / ~ 1 Essentially all sequences of 1000 will contain an equal number of bound and unbound hemoglobin proteins 2/26/16 18

19 Multiplicity (W) Other outcomes still have (vanishingly small) probability Important to keep in mind that the probabilities are still a distribution 1000 Hemes Bound O 2 molecules For 1000 hemes (250 hemoglobin molecules), p(n=0) ~ 1/ Figure from The Molecules of Life ( Garland Science 2008) 2/26/16 19

20 Probability and multiplicity The relative probability of two outcomes is the ratio of their respective probabilities Example, 1000 hemes, N=500 vs. N=600 p p W W It s also the ratio of their multiplicity! In our example: ln (W 500 /W 600 ) = = W W W 500 /W 600 = e 20 = ~5x10 8 Again, an equal number of bound and unbound is by far the most likely outcome 2/26/16 20

21 Maximal multiplicity when the derivative is 0 How can we find the maximal multiplicity? When the slope or derivative of the function is 0 and (N 0) or (N M) That is: W = f(n) and dw/dn = 0 is a condition for maximal multiplicity 2/26/16 21

22 lnw W W and lnw have the same maximum Bound O 2 molecules So is dlnw/dn = 0 Stirling s approximation will be extremely helpful Bound O 2 molecules Modified from The Molecules of Life ( Garland Science 2008) 2/26/16 22

23 Maximal multiplicity How can we find the maximal multiplicity? For the binomial case, it can be derived, using Stirling s approximation, that: d lnw dn ln N ln( M N) At a maximum: d lnw/dn = 0, such that lnn = ln(m-n) N = M-N 2N = M M = N/2 2/26/16 23

24 Maximal multiplicity leads to maximal entropy What we ve learned from hemoglobin: The observed outcome of an experiment involving a large number of trials is the one that corresponds to maximum multiplicity This is broadly applicable beyond hemoglobin Hopefully, it s starting to sound like the 2 nd law of thermodynamics! Much of thermodynamics is about finding the conditions where multiplicity is maximal 2/26/16 24

25 Back to molecules: starting with an ideal gas Isothermal expansion example from Lecture 9 Here du = dq + dw = 0 because T = constant Therefore, dq = - dw Even though du = 0, there is a change in the system We stated that entropy drives the change Figure from The Molecules of Life ( Garland Science 2008) 2/26/16 25

26 Back to molecules: starting with an ideal gas Let s now consider that this compressed gas is in fact made of particles, gas molecules More space is available to these molecules when the gas expands What happens to the multiplicity of the system? Figure from The Molecules of Life ( Garland Science 2008) 2/26/16 26

27 Multiplicity in a molecular system The multiplicity of a molecular system is defined as the number of different configurations or conformations of the component particles that are equivalent Note: in reality, a configuration includes the description of kinetic energy of each molecule. Here, we ll focus only on the position component Ideal gas: particles have positions in the system volume Grid boxes of arbitrary size, small enough to uniquely define the position of the atoms Figure from The Molecules of Life ( Garland Science 2008) 2/26/16 27

28 Calculating multiplicity There are 49! ways to put 49 particles in 49 boxes Figure from The Molecules of Life ( Garland Science 2008) 2/26/16 28

29 6 particles in a box Similar to the case of bound and unbound hemes: M = number of boxes N = number of red particles (M N) = number of empty boxes ( blue particles ) W M! N!( M N)! 49! 6!(49 6)! Figure from The Molecules of Life ( Garland Science 2008) 2/26/16 29

30 Multiplicity increases as volume increases Double the volume of the box and the multiplicity is now: 98! W 6!(98 6)! Figure from The Molecules of Life ( Garland Science 2008) lnw W ln 98! ln 6! ln 92! 98ln ln 720 (92ln 92 92) /26/16 30

31 Atoms are likely to spread The two configurations shown below are equally likely in a larger space But from our calculations, there are ~1.4x10 7 configurations equivalent to the one on the left, and ~10 9 configurations to the one on the right We re therefore ~100 times more likely to see the atoms on both sides of the box Figure from The Molecules of Life ( Garland Science 2008) 2/26/16 31

32 Equilibrium state of a system A microstate is a particular configuration of the atoms or molecules in a system Instantaneous snapshot A state is a global description of the system Uses bulk properties like temperature, pressure, number of molecules The multiplicity of a state is the number of corresponding microstates The higher the multiplicity, the higher the likelihood of observing that state The system will spontaneously evolve towards the state with highest multiplicity 2/26/16 32

33 Maximal multiplicity defines equilibrium state Sudden increase in volume of the system Over time, the atoms spread to increase multiplicity, until they are evenly distributed, which has maximal multiplicity The multiplicity of the system drives the expansion Figure from The Molecules of Life ( Garland Science 2008) 2/26/16 33

34 If we combine systems A and B, what would the combined W be? 2/26/16 34

35 W A+B is the product of W A and W B Figure from The Molecules of Life ( Garland Science 2008) 2/26/16 35

36 lnw is an additive and extensive property Additive System A System B W A W B W=W A W B ln W = ln W A + ln W B Extensive System 1 System 2 W=W 1 2x W=(W 1 ) 2 lnw=2ln(w 1 ) 2/26/16 36

37 Use of lnw for large systems W rapidly becomes unmanageably large, lnw increases slower with the # of particles Notice lnw is an extensive and additive property of the system Combining systems A and B to yield (A+B) W A+B = W A x W B lnw A+B = lnw A + lnw B lnw is a state function, it depends only on the parameters of the present state of the system 2/26/16 37

38 Statistical definition of entropy Entropy is defined as: S k W B ln Where k B is the Boltzmann constant k B = 1.38 x JK -1 = R/N A = JK -1 /6.023 x Entropy is a state function of the system 2/26/16 38

39 Some concepts to remember Multiplicity (W) is the number of microstates for a given conformation of the system A system will spontaneously evolve towards the state of maximal multiplicity Entropy is a function of multiplicity Additive property Extensive property State function 2/26/16 39