Lecture 20 Chemical Potential Reading: Lecture 20, today: Chapter 10, sections A and B Lecture 21, Wednesday: Chapter 10: 10‐17 –end MCB65 3/21/16 1 Pop Question 7 – Boltzmann Distribution Two systems with lowest energy at 0 kBT. Energy levels separated by 3 kBT in system A, and 5 kBT 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 kBT). System A: System B: U = 15kBT p=0.000000291 U = 12kBTUp=0.00000584 = 15kBT p=0.00000030384 U = 9kBT p=0.000117 U = 10kBT p=0.00004509330 U = 6kBTUp=0.0024 = 5kBT p=0.006692438 U = 3kBTUp=0.0473 = 0kBT p=0.993245928 U = 0kBT p=0.9501 Simply apply the Boltzmann distribution: System A: System B: Q = e0+e‐3+e‐6+e‐9+e‐12+e‐15=1.0524 Q = e0+e‐5+e‐10+e‐15=1.0068 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 kBT? Because the contributions to the distribution (and the Q) of higher energy levels becomes MCB65 negligible –i.e. the higher energy levels are unpopulated for all practical purposes. 3/21/16 2 F0F1 ATP synthase performs an unfavorable reaction ~50 kJ/mol stored per ATP ~15,000 kJ/mol per second Credits: John Walker MCB65 http://www.mrc-mbu.cam.ac.uk/research/atp-synthase 3/21/16 3 Today’s goals • Explore the link between concentration and the free energy change G • Chemical potential i Molar free energy G˚i ni T ,P,n ji • Equilibrium constant K: G o RT lnK Q • Reaction quotient Q and mass action ratio Q/K: G RT ln K • Using G and K to look at biological systems: • ATP hydrolysis • Wednesday: acid‐base equilibria, protein folding MCB65 3/21/16 4 Chemical Potential • The chemical potential (i) is defined as the rate of change of the free energy with respect to the number of molecules: G G Units of energy i G(for one molecule of i) Ni Ni (e.g. J) T ,P,N ji • Often in chemical reactions, we use moles (n): G G Units of energy per mole i i (e.g. J mol-1) ni T ,P,n ji • Molar free energy, G°i MCB65 3/21/16 5 Chemical potential at play ‐ diffusion [ ]in = [ ]out spontaneous dNout = -dNin in in out out ΔG <, >, or = 0? equilibrium: ΔG = ? What happens to μin and μout at equilibrium? G dG dN dN 0 in in out out i N i T ,P,N ji in dNin out dNin 0 dG dN (in out )dNin 0 in out MCB65 3/21/16 6 Figure from The Molecules of Life (© Garland Science 2008) Direction of spontaneous change and • System changing towards equilibrium: dG (in out )dNin 0 • For a spontaneous change, dNin and (in‐out) should have opposite signs • If Nin decreases, dNin < 0 • (in‐out) > 0 and in > out • Molecules move spontaneously from regions of high chemical potential to low chemical potential MCB65 3/21/16 7 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: C2 2 1 kBT ln C1 • Where C1 and C2 are the concentrations of molecules MCB65 3/21/16 8 G1 H1 TS1 G2 H2 TS2 G1 G2 A1 A 2 NA1 NA 2 T ,P,NB T ,P,NB H S H S 1 T 1 2 T 2 NA1 NA1 NA 2 NA 2 T ,P,NB T ,P,NB T ,P,NB T ,P,NB MCB65 3/21/16 9 Figure from The Molecules of Life (© Garland Science 2008) For an ideal solution, depends on entropy 2 1 H S H S 2 T 2 1 T 1 NA 2 NA 2 NA1 NA1 T ,P,NB T ,P,NB T ,P,NB T ,P,NB H H ideal solution enthalpy changes are the same 2 1 NA 2 NA1 T ,P,NB T ,P,NB S2 S1 2 1 T T NA 2 NA1 T ,P,NB T ,P,NB MCB65 3/21/16 10 Calculating the entropy: 2 1 • We can use the probabilistic definition of entropy, with three states: N p B (B molecules) 2 M N p A1 (A molecules) 1 M M (N N ) p A1 B (empty gridboxes) 0 M MCB65 3/21/16 11 Figure from The Molecules of Life (© Garland Science 2008) 2 S1 MkB pi ln pi i0 p does not depend on N d/dN = 0 2 A1 A1 S1 MkB p0 ln p0 p1 ln p1 p2 ln p2 NA1 NA1 T ,P,NB M (N A1 N B ) M (N A1 NB ) N A1 NA1 MkB ln ln N A1 M M M M MCB65 3/21/16 12 Applying simple rules for derivatives (chain rule, product rule, etc..) we get: S1 NA1 M (NA1 NB ) 1 T kBTln ln NA1 M M T ,P,NB NB (solvent) >> NA1 (solute) S2 NA 2 M (NA 2 NB ) 2 T kBTln ln NA1 M M T ,P,NB N M N N M N k Tln A 2 ln B ln A1 ln B 2 1 B M M M M N A 2 N A1 C2 kBTln ln kBT ln M M C1 C2 C1 MCB65 3/21/16 13 Molecular diffusion decreases chemical potential C2 2 1 kBT ln C1 • If C2 > C1 then ln(C2/C1) > 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) MCB65 3/21/16 14 Figure from The Molecules of Life (© Garland Science 2008) Chemical potentials C2 • Switching to molar units: RT ln C1 (multiply Boltzmann constant by Avogadro’s number R = NAkB) • Calculating the chemical potential relative to standard state: o C o C RT ln o RT ln C 1 • ***C/C° (and therefore C/1M) is unitless MCB65 3/21/16 15 What is chemical potential? • Chemical potential is proportional to the logarithm of concentration: C2 • Comparing two solutions: 2 1 kBT ln C1 C C • Comparing to standard state: o RT ln o RT ln C o 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 MCB65 3/21/16 16 What are the concentrations at equilibrium? MCB65 3/21/16 17 Defining a “reaction progress variable”, ξ • Reaction: aA bB cC dD • Change in free energy as the reaction progresses: dG A dnA B dnB C dnC D dnD • 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 MCB65 3/21/16 18 Defining a “reaction progress variable”, ξ • E.g. 2A 1B 1C 2D • 0 < < 1 MCB65 3/21/16 19 Figure from The Molecules of Life (© Garland Science 2008) Defining a “reaction progress variable”, ξ • Reaction: aA bB cC dD • Change in free energy as the reaction progresses: dG A dnA B dnB C dnC D dnD • 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 nA nA (0) a dnA a(d) n n (0) b For a small dn b(d) B B step in the B reaction: nC nC (0) c dnC c(d) nD nD (0) d dnD d(d) MCB65 3/21/16 20 Reaching equilibrium • Substituting into the equation for dG: dni (/)id dG A dnA B dnB C dnC D dnD A a(d) Bb(d) C c(d) D d(d) aA bB cC dD d • At equilibrium, dG = 0 and d can be non‐zero aA bB cC dD 0 aA bB cC dD • Products of chemical potential and stoichiometric MCB65 coefficients are balanced 3/21/16 21 Equilibrium concentrations a c [A] A c o RT ln b d A A 1 B D o [B] B B RT ln 1 [A], etc, refer to the equilibrium concentrations [C] o RT ln C C 1 [A]/1M is dimensionless o [D] D D RT ln a b c d 1 A B C D [A] [B] a o aRT ln b o bRT ln A 1 B 1 [C] [D] c o cRT ln d o dRT ln C 1 D 1 c d o o o o [C] [D] o cC dD aA bB RT ln a b G [A] [B] MCB65 3/21/16 22 Defining the equilibrium constant • We then define the equilibrium constant, K as: o c d G RT lnK [C] [D] eq Keq a b G o [A] [B] RT Keq e Equilibrium constant provides a way to • Keq is measurable determine the concentrations, the extent of reaction, at equilibrium • Keq is unitless • G° is in J/mol and RT is also in J/mol MCB65 3/21/16 23 Extent of reactions at equilibrium • Hydrolysis of ATP: ATP + H2O ADP + Pi + energy • G° = ‐28.7 kJ mol‐1 at pH 7.0 and 10 mM Mg2+ Extent of reaction [ADP][Pi ] K [H2O]/[H2O]° ~ 1 for reaction with [ATP] a smaller G°? G o RT 28 / 2.478 5 K e e 10 ‐2 • [Pi] in cells is maintained at ~10 M, which means at equilibrium [ADP] 107 [ATP] MCB65 3/21/16 24 G –in a situation not at equilibrium dG aA bB cC dD d 0 dG a b c d G d A B C D Figure from The Molecules of Life MCB65 (© Garland Science 2008) 3/21/16 25 Direction of spontaneous change from observed concentrations? MCB65 3/21/16 26 Reaction quotient, Q, describes observed conditions • Combining: o C G aA bB cC dD RT ln o C • We obtain: Go Q Reaction quotient [C]c [D]d G c o d o a o b o RT ln obs obs C D A B [A]a [B]b obs obs G Go RT lnQ Observed, non-equilibrium • Substituting: We get: Q Q G o RT lnK G RT ln 2.3RT log K K MCB65 3/21/16 27 Q/K is the mass action ratio • The ratio Q/K is the mass action ratio 5.8 kJ/mol Q Q G o RT lnK G RT ln 2.3RT log K K • The mass action ratio determines whether a reaction goes forward or backward: Q 1 G < 0, reaction will go forward K Q 1 G > 0, reaction will go backward K MCB65 3/21/16 28 Mass action ratio –Q/K Q Q A B G RT ln 2.3RT log K K mol-1 MCB65 Figure from The Molecules of Life (© Garland Science 2008) 3/21/16 29 G for ATP hydrolysis in cells ATP ADP + Pi + energy K = [ADP][Pi] [ATP] [ADP] • G° = ‐28.7 kJ mol‐1 and at equilibrium: 107 [ATP] • In cells, [ADP]/[ATP] = 10‐3 Q 103 G RT ln ~ RT ln ~ RT ln1010 K 107 ~ 57 kJ mol1 MCB65 3/21/16 30 ATP