Lecture 20. Chemical Potential

Transcription

1 Lecture 20 Chemical Potential Reading: Lecture 20, today: Chapter 10, sections A and B Lecture 21, Wednesday: Chapter 10: end 3/21/16 1

2 Pop Question 7 Boltzmann Distribution Two systems with lowest energy at 0 k B T. Energy levels separated by 3 k B T in system A, and 5 k B T in B. Calculate the partition coefficients for each system, then the probabilities of finding molecules in each energy level (use the energy levels from 0 to 15 k B T). System A: System B: U = 15k B T U = 12k B T p= p= U = 15k B T U = 9k B T p= U = 10k B T U = 6k B T p= U = 5k B T U = 3k B T p= U = 0k B T U = 0k B T p= p= p= p= p= Simply apply the Boltzmann distribution: System A: System B: Q = e 0 +e 3 +e 6 +e 9 +e 12 +e 15 = Q = e 0 +e 5 +e 10 +e 15 = Relationship between Q and the overall distribution of particles? Smaller steps between levels larger Q molecules spread more. Less likely to find molecules at any given level Why limit to <15 k B T? Because the contributions to the distribution (and the Q) of higher energy levels becomes negligible i.e. the higher energy levels are unpopulated for all practical purposes. 3/21/16 2

3 F0F1 ATP synthase performs an unfavorable reaction ~50 kj/mol stored per ATP ~15,000 kj/mol per second Credits: John Walker 3/21/16 3

4 Today s goals Explore the link between concentration and the free energy change Chemical potential i G Molar free energy n G i i T,P,n j i Equilibrium constant K: G o RT lnk Reaction quotient Q and mass action ratio Q/K: G RT ln Q K Using G and K to look at biological systems: ATP hydrolysis Wednesday: acid base equilibria, protein folding 3/21/16 4

5 Chemical Potential The chemical potential ( i ) is defined as the rate of change of the free energy with respect to the number of molecules: i G N i T,P,N j i G N i G (for one molecule of i) Units of energy (e.g. J) Often in chemical reactions, we use moles (n): i G n i T,P,n j i G i Units of energy per mole (e.g. J mol -1 ) Molar free energy, G i 3/21/16 5

6 Chemical potential at play diffusion in out spontaneous ΔG <, >, or = 0? equilibrium: ΔG =? What happens to μ in and μ out at equilibrium? [ ] in = [ ] out dn out = -dn in in out i G N i dg dn T,P,N j i dg in dn in out dn out 0 in dn in out dn in 0 ( in out )dn in 0 Figure from The Molecules of Life ( Garland Science 2008) in out 3/21/16 6

7 Direction of spontaneous change and System changing towards equilibrium: dg ( in out )dn in 0 For a spontaneous change, dn in and ( in out ) should have opposite signs If N in decreases, dn in < 0 ( in out ) > 0 and in > out Molecules move spontaneously from regions of high chemical potential to low chemical potential 3/21/16 7

8 Chemical potential and concentration In an ideal dilute solution, molecules do not influence each other and the enthalpy is independent of concentration. Assumption commonly used in biochemistry For an ideal dilute solution, we ll show that the difference in chemical potential is related to the ratio of concentrations: 2 1 k B T ln C 2 C 1 Where C 1 and C 2 are the concentrations of molecules 3/21/16 8

9 G 1 H 1 TS 1 G 2 H 2 TS 2 A1 G 1 N A1 H 1 N A1 T,P,N B T S 1 T,P,N B N A1 T,P,N B A 2 G 2 N A 2 H 2 N A 2 T,P,N B T S 2 T,P,N B N A 2 T,P,N B Figure from The Molecules of Life ( Garland Science 2008) 3/21/16 9

10 For an ideal solution, depends on entropy 2 1 H 2 T S 2 N A 2 T,P,N B N A 2 T,P,N B H 1 N A1 T S 1 T,P,N B N A1 T,P,N B ideal solution enthalpy changes are the same H 2 H 1 N A 2 N T,P,N B A1 T,P,N B 2 1 T S 2 N A 2 T S 1 T,P,N B N A1 T,P,N B 3/21/16 10

11 Calculating the entropy: 2 1 We can use the probabilistic definition of entropy, with three states: p 2 N B M (B molecules) p 1 N A1 M p 0 M (N A1 N B ) M (A molecules) (empty gridboxes) Figure from The Molecules of Life ( Garland Science 2008) 3/21/16 11

12 S 1 N A1 S 1 Mk B Mk B T,P,N B 2 i 0 p i ln p i p 2 does not depend on N A1 d/dn A1 = 0 p N 0 ln p 0 p 1 ln p 1 p 2 ln p 2 A1 Mk B M (N A1 N B ) N A1 M ln M (N A1 N B ) N A1 M M ln N A1 M 3/21/16 12

13 Applying simple rules for derivatives (chain rule, product rule, etc..) we get: 1 T S 1 N A1 2 T S 2 N A1 T,P,N B k B T ln N A1 ln M (N A1 N B ) M M N B (solvent) >> N A1 (solute) k B T ln N A 2 ln M (N A 2 N B ) M M T,P,N B 2 1 k B T ln N A 2 ln M N B M M ln N A1 ln M N B M M k B T ln N A 2 M ln N A1 k M B T ln C 2 C 1 C 2 C 1 3/21/16 13

14 Molecular diffusion decreases chemical potential 2 1 k B T ln C 2 C 1 If C 2 > C 1 then ln(c 2 /C 1 ) > 0 and is positive The A molecules in Region 2 have a higher chemical potential Makes sense molecules will move spontaneously from Region 2 (high concentration) to Region 1 (low concentration) Figure from The Molecules of Life ( Garland Science 2008) 3/21/16 14

15 Chemical potentials Switching to molar units: RT ln C 2 C 1 (multiply Boltzmann constant by Avogadro s number R = N A k B ) Calculating the chemical potential relative to standard state: o RT ln C C o o RT ln C 1 ***C/C (and therefore C/1M) is unitless 3/21/16 15

16 What is chemical potential? Chemical potential is proportional to the logarithm of concentration: Comparing two solutions: 2 1 k B T ln C 2 C 1 Comparing to standard state: o RT ln C C o o RT ln C 1 In mechanics, the direction of spontaneous change is always towards a reduction in potential energy Similarly, in thermodynamics, the direction of spontaneous change is always towards a reduction in Gibbs free energy The partial molar Gibbs free energy (G i) of a type of molecule (i) is its chemical potential ( i ) 3/21/16 16

17 What are the concentrations at equilibrium? 3/21/16 17

18 Reaction: Defining a reaction progress variable, ξ aa bb cc dd Change in free energy as the reaction progresses: dg A dn A B dn B C dn C D dn D These terms are NOT independent. To account for their coupling, we define the reaction progress variable (ξ or xi ) which is a measure of how far the reaction has progressed 3/21/16 18

19 E.g. Defining a reaction progress variable, ξ 2A 1B 1C 2D 0 < < 1 Figure from The Molecules of Life ( Garland Science 2008) 3/21/16 19

20 Reaction: Defining a reaction progress variable, ξ aa bb cc dd Change in free energy as the reaction progresses: dg A dn A B dn B C dn C D dn D These terms are NOT independent. To account for their coupling, we define the reaction progress variable (ξ) which is a measure of how far the reaction has progressed n A n A (0) a n B n B (0) b n C n C (0) c n D n D (0) d For a small step in the reaction: dn A a(d ) dn B b(d ) dn C c(d ) dn D d(d ) 3/21/16 20

21 Reaching equilibrium Substituting into the equation for dg: dn i ( / )id dg A dn A B dn B C dn C D dn D A a(d ) B b(d ) C c(d ) D d(d ) a A b B c C d D d At equilibrium, dg = 0 and d can be non zero a A b B c C d D 0 a A b B c C d D Products of chemical potential and stoichiometric coefficients are balanced 3/21/16 21

22 Equilibrium concentrations A o A RT ln [A] 1 B o B RT ln [B] 1 C o C RT ln [C] 1 D o D RT ln [D] 1 a A b B [A], etc, refer to the equilibrium concentrations [A]/1M is dimensionless a A b B c C d D c c d D a o A art ln [A] b o 1 B brt ln [B] 1 c o C crt ln [C] d o 1 D drt ln [D] 1 c o C d o D a o A b o B RT ln [C]c [D] d [A] a [B] b G o 3/21/16 22

23 Defining the equilibrium constant We then define the equilibrium constant, K as: K eq [C]c [D] d [A] a [B] b G o RT lnk eq K eq e G o RT K eq is measurable Equilibrium constant provides a way to determine the concentrations, the extent of reaction, at equilibrium K eq is unitless G is in J/mol and RT is also in J/mol 3/21/16 23

24 Extent of reactions at equilibrium Hydrolysis of ATP: ATP + H 2 O ADP + P i + energy G = 28.7 kj mol 1 at ph 7.0 and 10 mm Mg 2+ Extent of reaction for reaction with a smaller G? K [ADP][P i] [ATP] [H 2 O]/[H 2 O] ~ 1 K e G o RT e 28 / [P i ] in cells is maintained at ~10 2 M, which means at equilibrium [ADP] [ATP] 107 3/21/16 24

25 G in a situation not at equilibrium dg a A b B c C d D d 0 dg d a A b B c C d D G Figure from The Molecules of Life ( Garland Science 2008) 3/21/16 25

26 Direction of spontaneous change from observed concentrations? 3/21/16 26

27 Reaction quotient, Q, describes observed conditions Combining: G a A b B c C d D We obtain: G c o C d o D a o o A b B G o o RT ln C C o RT ln [C] c obs [A] a obs Q d [D] obs b [B] obs Reaction quotient G G o RT lnq Observed, non-equilibrium Substituting: We get: G o RT lnk G RT ln Q K 2.3RT log Q K 3/21/16 27

28 Q/K is the mass action ratio The ratio Q/K is the mass action ratio 5.8 kj/mol G o RT lnk G RT ln Q K 2.3RT log Q K The mass action ratio determines whether a reaction goes forward or backward: Q K 1 Q K 1 G < 0, reaction will go forward G > 0, reaction will go backward 3/21/16 28

29 Mass action ratio Q/K A B G RT ln Q K 2.3RT log Q K mol -1 Figure from The Molecules of Life ( Garland Science 2008) 3/21/16 29

30 G for ATP hydrolysis in cells ATP K = [ADP][P i ] [ATP] ADP + P i + energy G = 28.7 kj mol 1 and at equilibrium: [ADP] [ATP] 107 In cells, [ADP]/[ATP] = 10 3 G RT ln Q K ~ 57 kj mol 1 ~ RT ln ~ RT ln /21/16 30

31 ATP synthesis is not spontaneous ATP ADP + P i + energy K = [ADP][P i ] [ATP] G = 28.7 kj mol 1 and at equilibrium: [ADP] [ATP] 107 How to drive ATP synthesis? From the chemical potential concept Increasing the concentration of ADP and P i could lead to ATP synthesis Impractical for the cell 3/21/16 31

32 An example of non expansion work Using a gradient of molecules across a membrane to synthesize ATP Unfavorable reaction: (ADP+P i ) F (ATP) F* Favorable reaction: B(high) B(low) (ADP+P i ) F + B (ATP) F* B Figure from The Molecules of Life ( Garland Science 2008) 3/21/16 32

33 F0F1 ATP synthase uses proton chemical potential to synthesize ATP Credits: John Walker 3/21/16 33

34 Oxidative phosphorylation in mitochondria Figures from The Molecules of Life ( Garland Science 2008) 3/21/16 34

35 U = q+ w, and q ~ 0 (under idealized conditions) U = w F 0 F 1 ATP synthase couples ATP synthesis to a transmembrane proton gradient (ADP+P i ) F + H + (ATP) F* H + U is positive because ATP is produced K = [(ATP) F* H + ] [(ADP P i ) F][H + ] w is positive, corresponding to work done on the system by the surroundings Chemical work as a consequence of transferring B molecules from high to low concentration, decreasing the free energy of B, and storing this energy into synthesized ATP Figure from The Molecules of Life ( Garland Science 2008) 3/21/16 35

36 Some concepts to remember The chemical potential,, describes the rate of change of the free energy with respect to concentrations The equilibrium constant, K eq, provides a link between free energy and the concentrations at equilibrium The mass action ratio, Q/K, is related to the reaction free energy change, G, determining the driving force for the reaction 3/21/16 36