Lecture 14 Entropy relationship to heat Reading: Lecture 14, today: Chapter 7: 7.20 end Lecture 15, Wednesday: Ref. (2) 2/29/16 1
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 2/29/16 2
Multiplicity (W) The multiplicity of a molecular system is the number of different molecular configurations consistent with the macroscopic parameters that define the system W M! N!( M N)! M is the number of grid boxes N is the number of occupied grid boxes W 81! 6!(81 6)! 6 blue particles, 81 6 = 75 empty squares 2/29/16 3
Multiplicity (W) The multiplicity of a molecular system is the number of different molecular configurations consistent with the macroscopic parameters that define the system W M! N!( M N)! M is the number of grid boxes N is the number of occupied grid boxes A system will spontaneously evolve towards the state of maximal multiplicity Maximum at dw/dn = 0 or dlnw/dn = 0 2/29/16 4
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) 2/29/16 5
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) 2/29/16 6
If we combine systems A and B, what would the combined W be? System A System B System A+B + = 2/29/16 7
W A+B is the product of W A and W B Figure from The Molecules of Life ( Garland Science 2008) 2/29/16 8
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 ) 2/29/16 9
Use of lnw for large systems W rapidly becomes unmanageably large, lnw increases slower with the # of particles 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 2/29/16 10
The hydrophobic effect drives folding The hydrophobic effect is a collective property of solvent and solute molecules Figure from The Molecules of Life ( Garland Science 2008) 2/29/16 11
Statistical definition of entropy Entropy is defined as: S k lnw B Where k B is the Boltzmann constant k B = 1.38 x 10-23 JK -1 = R/N A = 8.314 JK -1 /6.023 x 10-23 S = k B lnw = q/t Entropy is a state function and an extensive property of the system 2/29/16 12
Today s goals Equilibrium state has maximal multiplicity Generalizing multiplicity Statistical and thermodynamic definitions of entropy Reversible processes Relationship between the statistical and thermodynamic definitions of entropy The origin of the Boltzmann constant, k B 2/29/16 13
Generalizing W If there are more than two conditions (i.e. occupied vs. empty grid boxes), then multiplicity generalizes to: M! W! Where M = N i and N i is the number microstates of each condition i i N i W 81! (6!)(8!)(67!) 6 blue particles, 8 red particles, 67 empty squares 2/29/16 14
Thermodynamic definition of entropy Concept of entropy was derived in the 19 th century from the study of heat/steam engines, leading to the definition of the change in entropy: Where q rev is the heat transferred to the system during a reversible transition from one state to another DS has JK -1 units DS q That s why the Boltzmann constant k B is in JK -1 in the statistical definition of entropy rev T 2/29/16 15
Thermodynamic definition of entropy DS q rev T The thermodynamic definition makes entropy measurable Heat is a macroscopic property of a system that can be measured Multiplicity, in contrast, is intuitive for the molecular properties of the system, but not readily measurable 2/29/16 16
Reversible process Reversible process carried out so slowly that the driving force is always nearly in balance with the internal forces Near equilibrium process Small change in the driving force would change the direction of the process Useful concept to calculate the maximum amount of work generated by a process Idealized concept no real process is strictly reversible 2/29/16 17
Ideal gas isothermal expansion In a reversible process, the pressure of the piston (P EXT ) is reduced very gradually Each small step w = P EXT DV For infinitesimally small steps P EXT = P IN dp dw = (P IN dp)dv Figure from The Molecules of Life ( Garland Science 2008) 2/29/16 18
Work corresponds to the area under the curve, or the integration of PdV 2/29/16 19
Entropy change is related to maximum work Let s derive the work done under reversible isothermal expansion: dw Integrating: w w rev rev (P V 2 ( P V V V 2 1 1 IN IN IN dp)dv dp) dv P dv dpdv The dpdv term drops out as you approach reversibility (dp gets infinitesimally small) 2/29/16 20
Entropy change is related to maximum work Let s derive the work done under reversible isothermal expansion: Use the fact that PV = nrt : w rev V 2 V 1 P IN dv V 2 V 1 nrt V dv nrt are constants: w rev nrt V 2 V 1 1 V dv Integral of dv/v = lnv : V 2 w rev nrt ln V 1 2/29/16 21
Recall that in an isothermal expansion, dq = -dw Isothermal expansion example du = dq + dw = 0 because T = constant Therefore, dq = - dw 2/29/16 22
Entropy change is related to maximum work Let s derive the work done under reversible isothermal expansion: Because this is an isothermal process: w q rev rev w q DS rev rev q rev T V 2 nrt ln V Divide by T, and substitute in the thermodynamic definition of entropy: TDS 1 nrln V V 2 Rearranging, the maximum work extracted from a reversible, isothermal process is: 1 2/29/16 23
Expansion work is all entropy w rev q rev TDS For an ideal gas in isothermal conditions, the only work it can perform is expansion work Since T > 0 when expansion work is done, w rev < 0 DS > 0, therefore entropy increases Furthermore, the maximum work i.e. under reversible process conditions only depends on the temperature and entropy, not on the energy of the system This will lead to the idea of free energy. 2/29/16 24
Statistical vs. thermodynamics definitions of entropy The number of grid boxes increases from M to M+D We can calculate the corresponding change in W using: lnw ln M! N!( M number of grid boxes = av 1 N)! N ln M N number of grid boxes = av 2 for a large number of (second version of Stirling s approximation when M >> N) Figure from The Molecules of Life ( Garland Science 2008) atoms, N 2/29/16 25
For a system: Difference in multiplicity lnw M are units of volume (i.e. M=aV) Difference after expansion: ln M1 av 1 N ln N ln N N W 1 ln lnw N ln M N av 2 av 2 lnw1 N ln N ln N N W 1 2 lnw1 N ln av 2 N N ln N av V V 2 1 M 2 av2 lnw2 N ln N ln N N 1 2/29/16 26
Definitions of entropy are equivalent Let s derive the resulting change in entropy, DS: From the last slide: DS S 2 S1 kb lnw2 kb ln W DS kb(lnw 2 lnw1 ) k B N ln V V 2 1 1 N is the number of atoms = nn A : DS nn A k B ln V V 2 1 k B *N A = R, the gas constant: DS nrln V V 2 1 The two definitions are equivalent! 2/29/16 27
Direction of spontaneous change Increase in entropy indicates the direction of spontaneous change An isolated system will not spontaneously go from a state with high entropy to lower entropy Only if the system is coupled to an external agent (i.e. surroundings) with at least a corresponding increase in entropy can it go spontaneously to a state with lower system entropy The combined entropy of the system and the surroundings must always increase This is a statement of the second law of thermodynamics 2/29/16 28
Conditions for equilibrium Another way of stating the second law of thermodynamics is: S sys + S surr = maximal (at equilibrium) or ds sys + ds surr = 0 (at equilibrium) 2/29/16 29
Semi-permeable membranes Initial state: Semi-permeable membranes allow some molecules to pass through and not others Membrane permeable to blue particles, not red particles We ll assume the volume is divided into 1000 elements Biological membranes Dialysis in the lab (e.g. Anfinsen experiment) Dialysis in cases of kidney disease How will the blue particles partition? Figure from The Molecules of Life ( Garland Science 2008) 2/29/16 30
Entropy of the initial state Entropy on the left: S k L B lnw L ln M! N!( M N!) 1000! 500!500! 693.7 Entropy on the right: S k R B 693.7 Entropy is additive, so the total entropy is: S k T B S k L B S k R B 1387.4 2/29/16 31
If blue particles cross the membrane Let s say 10 blue particles cross the membrane Entropy on the left: S k L B lnw L ln N red! N blue M!!( M N T!) 1000! (500!)(10!)(490!) 740.1 Entropy on the right: S k R B 1000! 490!510! 692.9 Before Entropy S is additive, so the R total entropy 693.7 k is: B S k T B S k L B S k R B 1433.0 N red = 500 N blue = 490 N blue = 10 2/29/16 32
Mixing increases entropy So, with the particles completely segregated: S total = 1387.4k B With 10 blue particles crossing the barrier: S total = 1433.0k B The segregated system was not at equilibrium once the semi-permeable membrane was introduced. 2/29/16 33
Entropy of mixing in semi-permeable membrane system Max entropy on the right Max entropy on the left Curve looks shallow because this is lnw It s the combined entropy that is important in driving spontaneous change Figure from The Molecules of Life ( Garland Science 2008) 2/29/16 34
Increasing the size of the system Max entropy when 330,000 blue particles have crossed to the left Figure from The Molecules of Life ( Garland Science 2008) 2/29/16 35
Increasing the size of the system With 10 6 particles of each kind, just a 1% difference from maximum: W ( N W ( N blue, left blue, left 330000) e 340000) will never realistically be observed in such a system 100 2.5x10 43 Figure from The Molecules of Life ( Garland Science 2008) 2/29/16 36
Some concepts to remember Entropy is a function of multiplicity Additive property Extensive property State function Statistical definition of entropy: S = k B lnw k B makes the statistical definition of entropy consistent with the historical thermodynamic definition Thermodynamic definition DS = q rev / T Where q rev is the heat transferred to the system during a reversible process Second Law of Thermodynamics: ds sys + ds surr > 0 for a spontaneous process ds sys + ds surr = 0 at equilibrium 2/29/16 37
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