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:

dStotal  dSsys  dSsurr  0 More generally:

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  TS MCB65 3/11/16 15 Introducing Gibbs free energy (G)

Returning to the first law of thermodynamics: dU  dq  dw If the process is carried out reversibly, then TdS is the 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

dU PdV Substituting dS   sys  sys into dS  dS  0 surr T T sys surr 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

ATP  H2O  ADP  Pi

1 G  G(ADP  Pi )  G(ATP  H 2O)  28 kJ mol

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 be derived from a series of intermediate reactions, or thermodynamic cycle:

 G (Z )4 kJ mol-1-1 A  B  C  D ffG(Z )4 kJ mol Z

 1  1  f G (X )  3 kJ mol G (Z)  5 kJ mol

 1 Textbook 9.8-9.12 –  GG ((YY))2 kJ2 kJ mol mol1 X C  Dff  X Y example with glucose

 1 1 1 1  f G (Z)  3 kJ mol  2 kJ mol  5 kJ mol  4 kJ mol MCB65 3/11/16 27 Free energy and work

The “free” in free energy signifies that it corresponds to the energy that is “free” to do work

The work can be expansion work or non-expansion work

So far we have only considered expansion work, let’s consider non-expansion work

MCB65 Figure from The Molecules of Life (© Garland Science 2008) 3/11/16 28 Free energy can be extracted as work

We will show that the Gibbs free energy change corresponds to the maximum possible non-expansion work that can be extracted from a system for a spontaneous process: w  G The work done by the system has a negative value, and G ≤ 0 for a  w  G spontaneous process Therefore the magnitude of work is less than or equal to the magnitude of the free

energy change MCB65 3/11/16 29 Deriving non-expansion work at constant pressure

The system can do some non-expansion work, dwnon-exp

dU sys  dwnonexp  dwsys  (dU surr  dwsurr )

The expansion work cancels out on both sides

dU surr  dU sys  dwnonexp

Substituting dUsurr into our equation from slide 17, we get: dU dw PdV dS   sys  nonexp  sys surr T T T

dHsys MCB65 3/11/16 30 Deriving non-expansion work at constant pressure From the definition of enthalpy, the 1st and 3rd terms become dH dw dS   sys  nonexp surr T T

We use this to substitute dSsurr: dSsys  dSsurr  0 dH dw dS  sys  nonexp  0 sys T T Multiplying both sides by T and rearranging (dropping subscripts):

TdS  dH  dwnonexp  0

dwnonexp  dH TdS

MCB65 dwnonexp  dG 3/11/16 31 Free energy is available to do work The “free” in free energy signifies that it corresponds to the energy that is “free” to do work

w  G

wnonexp  H TS

TS is the amount of energy “lost” to entropy

MCB65 3/11/16 32 Some concepts to remember The Gibbs free energy, G = H – TS, is extensive and additive and a state function

The change in Gibbs free energy is the non-expansion work that can be extracted from a spontaneous process at constant pressure

G = H - TS ≤ 0 for a spontaneous process at constant pressure

TS is the amount of energy “lost” to entropy

The standard state has concentrations at 1 M, pressure at 1 atm, and in biochemistry, [water] = 55 M and [protons] 10-7 M. The molar free energy change is then labeled G°

The standard free energy of formation, G°, of the most stable form of elemental compounds is 0 MCB65 3/11/16 33 Visualizing, Analyzing and Presenting Macromolecular Structures with PyMOL Free 2-Day nanocourse taught by Rachelle

March 15-16, 2016 Time: 10am-noon and 2-4pm each day Location: Pierce Hall 301 (29 Oxford St)

Course website: https://canvas.harvard.edu/courses/11408

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MCB65 3/11/16 34