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Lecture 19 Defining Free Energy

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Lecture 19 Defining Free Energy Lecture 19 Defining free energy Reading: Lecture 19, today: Chapter 9 Lecture 20, Monday 3/25: Chapter 10, section A and B MCB65 3/11/16 1 Today’s goals We now have in hand most of the essential thermodynamic concepts But we have to refer to the surroundings and the total energy and entropy values It would be much better if we could concern ourselves only with the system under study itself We’ll develop the concept of free energy as a new state function that provides the direction of spontaneous change We’ll see how the free energy describes the maximal amount of non-expansion work that can be extracted from a spontaneous process We’ll start with a brief review of temperature MCB65 3/11/16 2 Definition of temperature 1 U S Slope of the tangents: T S V ,N U V ,N Systems at lower temperature will undergo a larger increase in entropy for the same energy input T2 Therefore, when in thermal contact, system at higher temperature will transfer heat to system at lower temperature to reach thermal T < T equilibrium T1 1 2 MCB65 Figure from The Molecules of Life (© Garland Science 2008) 3/11/16 3 Systems in thermal contact U A U B T S A SB VA ,N A VB ,NB At equilibrium, the temperatures will be the same same slope of S vs. U But the energies UA and UB are not necessarily equal Neither are the entropies SA and SB necessarily the same MCB65 3/11/16 4 Temperature is always ≥ 0 For a systems with unlimited energy levels, increasing the energy of the system will also lead to an increase in the entropy in the system More energy levels are populated U ve T 0 S ve Absolute 0 Kelvin would be reached if all molecular motions could be frozen, and energy and entropy were both 0 MCB65 Figure from The Molecules of Life (© Garland Science 2008) 3/11/16 5 Some concepts to remember The Boltzmann distribution: u j e kBT Larger partition function Q (smaller p energy separation) spreads the j Q molecules across more energy levels The temperature is defined as the slope of the energy vs. entropy: U T S V ,N Absolute 0 Kelvin is defined as the temperature that would be reached if all molecular motions could be frozen, and energy and entropy were both 0 Systems at lower T undergo a larger increase in entropy for the same energy input, therefore a system at higher T will transfer heat to system at lower temperature to reach thermal equilibrium MCB65 3/11/16 6 System at equilibrium Test tube = system Water bath = surroundings At a microscopic level, movements of molecules is unpredictable and chaotic Global properties of the system at equilibrium are stable with time MCB65 Figure from The Molecules of Life (© Garland Science 2008) 3/11/16 7 Perturbing the system Adding more molecules and heating the system will move it away from equilibrium, and the system will reach a new equilibrium over time. How do we determine what the new equilibrium will be? MCB65 Figure from The Molecules of Life (© Garland Science 2008) 3/11/16 8 System at equilibrium What we’ve seen so far is that a system is at equilibrium when the entropy of both the system and the surroundings is maximal At equilibrium: More generally: dStotal dSsys dSsurr 0 dSsys dSsurr 0 MCB65 Figure from The Molecules of Life (© Garland Science 2008) 3/11/16 9 Perturbing the system Can we determine what the new equilibrium will be just from the properties of the system itself? We’ll define the free energy of the system Always decreases in a spontaneous process and is at a minimum at equilibrium Will enable us to predict the direction of spontaneous change from the properties of the system MCB65 Figure from The Molecules of Life (© Garland Science 2008) 3/11/16 10 System at constant P and T Systems at constant volume and temperature are simplest – covered in the book We’ll go straight to biologically relevant systems at constant pressure and temperature Surroundings are so large that the temperature of the system can be maintained constant – isothermal process Frictionless piston is introduced to let us calculate the expansion work done by the system (by measuring the change in volume) MCB65 Figure from The Molecules of Life (© Garland Science 2008) 3/11/16 11 d d What we’ve seen so far is that a system is at equilibrium when the entropy of both the system and the surroundings is maximal At equilibrium: More generally: dStotal dSsys dSsurr 0 dSsys dSsurr 0 MCB65 Figure from The Molecules of Life (© Garland Science 2008) 3/11/16 12 Introducing Gibbs free energy (G) In our system, there is heat exchange and: dqsys dqsurr From the first law of thermodynamics: dU dq dw dU sys dwsys dU surr dwsurr The expansion work by the system and surroundings are equal in magnitude but opposite in sign, therefore cancel out: dU sys dU surr MCB65 3/11/16 13 Reminder… Ideal gas isothermal expansion In a reversible process, the pressure of the piston (PEXT) is reduced very gradually Each small step dw = – PEXTdV MCB65 Figure from The Molecules of Life (© Garland Science 2008) 3/11/16 14 Reminder… Entropy change is related to maximum work Let’s derive the work done under reversible isothermal expansion: Because this is an isothermal process: V 2 qrev wrev nRT ln V 1 q V S rev nRln 2 T V1 wrev qrev TS MCB65 3/11/16 15 Introducing Gibbs free energy (G) Returning to the first law of thermodynamics: If the process is carried out reversibly, thendU TdS dqis the dw heat transferred and –PdV is the work done: dU TdS PdV The equation is a restatement of the first law of thermodynamics that incorporates entropy dU PdV Rearranging to isolate dS: dS T T MCB65 3/11/16 16 Eliminating variables of the surrounds From: dSsys dSsurr 0 We want to re-work this definition of the 2nd law to get an expression only dependent on the properties of the system dU PdV From the last slide: dS surr surr surr T T We know that dVsys dVsurr therefore: dU PdV dS surr sys WRITE ON BOARD surr T T Also, dU sys dU surr and now, substituting: The change in entropy of the dU sys PdVsys dS surrounds is now expressed surr MCB65 T T using variables of the 3/11/16system 17 Eliminating variables of the surrounds dS dS 0 dUsys PdVsurr Substituting dS sys sys into surr T T dU PdV dS sys sys 0 sys T T Rearranging: TdSsys dU sys PdVsys 0 Recall that we defined enthalpy as H sys U sys PVsys And the change in enthalpy at constant pressure: dH sys dU sys PdVsys Which leads to: TdSsys dH sys 0 or dH sys TdSsys 0 MCB65 3/11/16 18 Defining Gibbs free energy as a new state function We now have an expression containing only system variables: dH sys TdSsys 0 Let’s define the Gibbs free energy, a new state function of the system: G H TS At constant temperature, a small change in G (dG): dG dH TdS SdT dH TdS 0 Therefore, we have a new condition for a spontaneous process at constant pressure that dG 0, and dG 0 at equilibrium MCB65 3/11/16 19 “G” is at a minimum at equilibrium Changes in the system parameters that result in a reduction in the free energy, G, will occur spontaneously MCB65 Figure from The Molecules of Life (© Garland Science 2008) 3/11/16 20 “G” is only stable at equilibrium A system will tend to remain stable at a minimum value of free energy, but will tend to move away from a maximum value MCB65 3/11/16 21 Gibbs free energy is useful in biology The Gibbs free energy is extremely useful in biological processes, which typically occur at constant pressure, to determine the direction of spontaneous change. For a reaction: ATP H2O ADP Pi The free energy change will be: G G(products) G(reactants) G G(ADP Pi ) G(ATP H2O) Do you expect free energy to be an extensive or an intensive property? MCB65 3/11/16 22 Free energy is an extensive property Free energy is an extensive property of the system To simplify things, we’ll use the molar free energy as the standard free energy, G° G is a function of many state variables, such as temperature, pressure, etc Let’s standardize before we start stating some values MCB65 Figure from The Molecules of Life (© Garland Science 2008) 3/11/16 23 Standard state in biochemistry We’ll define the standard state, which for biochemical reactions, corresponds to: 1 M concentration 1 atm pressure Standard free energy change = G° (Typically expressed at 298 K) A couple of exceptions in biochemistry: Water: standard state is pure water; i.e., [H2O] = 55 M Protons: standard state is pH 7; i.e., [H+] = 10-7 M *** This is different from other branches of chemistry Sometimes denoted as G°’ for the standard biochemistry state MCB65 3/11/16 24 G° for ATP hydrolysis 1 G G(ADP Pi ) G(ATP H 2O) 28 kJ mol ATP H2O ADP Pi Standard molar free energy change G° at 1 M, 1 atm and 298 K G° is a hypothetical concept, i.e. G° = -28 kJ mol-1 for ATP hydrolysis corresponds to complete hydrolysis of 1 mol of ATP That is, this reaction will not spontaneously proceed to completion Will reach equilibrium before completion because of the concentration dependence of G MCB65 3/11/16 25 Standard free energy of formation fG° G f G (product) f G (reactant) all products all reactants fG°, the standard free energy of formation, is arbitrarily set to 0 for elemental molecules in their most stable form This is OK because we are usually only interested in changes MCB65 Figure from The Molecules of Life (© Garland Science 2008) 3/11/16 26 Free energy is a state function G is a state function, therefore independent of path and additive Example: Z is made from the following elemental molecules: A B C D f G(Z ) Z This may not be a measurable reaction in practice but can
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