termodinamica y cinetica de las reacionnes biologicas

8/14/2019 Termodinamica y Cinetica de Las Reacionnes Biologicas

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THERMODYNAMICSNDKINETICS ORTHEBIOLOGICAL CIENCES

GordonG. HammesDepartment of BiochemistryDuke University

GwLEY-eC2rNTERscrENcEA JOHNWILEY& SONS, NC., PUBLICATIONNewYork. chichester weinheim Brisbane singapore Toronto


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The book is printed on acid-free pup"r. @

Copyright @ 2000 by John Wiley & Sons, nc. All rights reserved.

Published simultaneously in Canada.

No part of this publication may be reproduced, stored n a retrieval system or transmitted in any form or

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Copyright Clearance Center, 222 Rosewood Drive, Danvers, MA 01923, (978) 750-8400, ax (978) 750-

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For ordering and customer service. call 1-800-CALL-WILEY.

Library of Congress Catalaging-in-Publication Data:

Hammes. Gordon.. 1934-Thermodynamics and kinetics for the biological sciences/by Gordon G. Hammes.

p. cm.

"Published simultaneously in Canada."Includes bibliographical references and index.ISBN 0-471-31491-l (pbk.: acid-free paper)

1. Physical biochemistry. 2. Thermodynamics. 3. Chemical kinetics.I. Title.

QP517.P49 H35 2000512-dc2l 99-086233

Printed in the United States of America.

1 0 9 8 1 6 5


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I CONTENTS

Preface

1. Heat, Work, and Energy

1.1 Introduction1.2 Temperature1.3 Heat1.4 Work1.5 Definition of Energy1.6 Enthalpy

1.7 Standard States1.8 Calorimetry1.9 ReactionEnthalpies1.10 Temperature Dependence f the Reaction Enthalpy

ReferencesProblems

2. Entropy and Free Energy

2.t Introduction2.2 Statement f the Second Law2.3 Calculation of the Entropy2.4 Third Law of Thermodynamics2.5 Molecular Interpretation of Entropy2.6 Free Energy2.7 ChemicalEquilibria2.8 Pressure and Temperature Dependence of the Free Energy2.9 Phase Changes2.10 Additions to the Free Energy

Problems

3. Applications of Thermodynamics to Biological Systems

3.1 Biochemical Reactions3.2 Metabolic Cycles3.3 Direct Synthesis of ATP3.4 Establishment of Membrane Ion Gradients bv Chemical Reactions

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vt CONTENTS

3.5 Protein Structure3.6 Protein Folding3.7 Nucleic Acid Structures3.8 DNA Meltins3.9 RNAReferencesProblems

Chemical Kinetics

4.1 Introduction

4.2 Reaction Rates4.3 Determination of Rate Laws4.4 Radioactive Decay4.5 Reaction Mechanisms4.6 Temperature Dependence of Rate Constants4.7 Relationship Between Thermodynamics and Kinetics4.8 Reaction Rates Near EquilibriumReferencesProblems

Applications of Kinetics to Biological Systems

5.1 Introduction5.2 Enzyme Catalysis: The Michaelis-Menten Mechanism5.3 u-Chymotrypsin5.4 Protein Tyrosine Phosphatase5.5 Ribozymes5.6 DNA Meltine and RenaturationReferencesProblems

Ligand Binding to Macromolecules

6.1 Introduction6.2 Binding of Small Molecules o Multiple Identical Binding Sites6.3 Macroscopic and Microscopic Equilibrium Constants

6.4 Statistical Effects in Ligand Binding to Macromolecules6.5 Experimental Determination of Ligand Binding Isotherms6.6 Binding of Cro Repressor Protein to DNA6.7 Cooperativity n Ligand Binding6.8 Models for Cooperativity6.9 Kinetic Studies of Cooperative Binding6.10 Allosterism

ReferencesProblems

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Appendixes

1. Standard Free Energies and Enthalpies of Formation at 298 K,l Atmosphere, pHTrand 0.25 M lonic Strength

2. Standard Free Energy and Enthalpy Changes for BiochemicalReactions at 298 K, 1 Atmosphereo pH 7.0, pMg 3.0, and 0.25 MIonic Strength

Structures of the Common Amino Acids at Neutral pH

Useful Constants and Conversion Factors

Index

3.

4.

coNTENTS vii

Ls4

156

157

159

161


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I PREFACE

This book is based on a course hat I have been teaching for the past several years to

first year graduate students n the biological sciences at Duke University. These stu-

dents have not studied physical chemistry as undergraduates and typically have nothad more than a year of calculus. Many faculty believe that an understanding of theprinciples of physical chemistry s important or all students n the biological sciences,and this course s required by the Cell and Molecular Biology Program. The courseconsists of two parts-one devoted o thermodynamics nd kinetics, he other o spec-troscopy. Only the first half of the course s covered in this volume. An introductionto spectroscopy s being planned as a separate olume. One of the reviewers of the pro-posal for this book said hat it was mpossible to teach biology students his material-the reviewer had been trying for many years. On the contrary, I believe the students

that have taken this course have mastered he principles of the subject matter and willfind the knowledge useful in their research.

Thermodynamics and kinetics are introduced with a minimum of mathematics.However, the approach s quantitative and is designed o introduce the student o theimportant concepts hat are necessary o apply the principles of thermodynamics andkinetics to biology. The applications cover a wide range of topics and vary consider-ably in the degree of difficulty. More material is included than is covered n the courseon which the book is based, which will allow the students and instructors to pick andchoose. Some problems are also ncluded, as problem solving is an important part ofunderstanding rinciples.

I am indebted o my colleagues at Duke University for their encouragement and as-sistance. Discussions with them were essential o the production of this book. Specialthanks are due to Professor Jane Richardson and Dr. Michael Word for their assistancewith the color figures. I also want to acknowledge he encouragement nd assistanceof my wife, Judy, during this entire project. would appreciate ny comments or sug-gestions rom the readers of this volume.

Gonoou G. Havlaes

Duke UniversityDurham. North Carolina

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I CHAPTER1

Heat,Work,and Energy

1.1 INTRODUCTION

Thermodynamics s deceptively simple or exceedingly complex, depending on howyou approach t. In this book, we will be concerned with the principles of thermody-namics that are especially useful in thinking about biological phenomena. The empha-sis will be on concepts, with a minimum of mathematics. Perhaps an accuratedescription might be rigor without rigor mortis. This may cause some squirming inthe graves of thermodynamic purists, but the objective is to provide a foundation forresearchers n experimental biology to use thermodynamics. This includes cell biol-ogy, microbiology, molecular biology, and pharmacology, among others. n an idealworld, researchers n these ields would have studied a year of physical chemistry, andthis book would be superfluous. Although most biochemists have his background, tis unusual for other biological sciences o require it. Excellent texts are available thatpresent a more advanced nd complete exposition of thermodynamics cf. Refs. 1 andD.

In point of fact, thermodynamics an provide a useful way of thinking about bio-logical processes and is indispensable when considering molecular and cellularmechanisms. For example, what reactions and cBupled physiological processes repossible? What are he allowed mechanisms nvolved in cell division, in protein syn-

thesis? What are he thermodynamic considerations hat cause proteins, nucleic acids,and membranes to assume heir active structures? t is easy to postulate biologicalmechanisms hat are nconsistent with thermodynamic principles-but just as easy opostulate those that are consistent. Consequently, no active researcher n biologyshould be without arudimentary knowledge of the principles of thermodynamics. Theultimate goal of this exposition s to understand what determines equilibrium in bio-logical systems, and how these equilibrium processes an be coupled ogether o pro-duce living systems, even though we recognize that living organisms are not atequilibrium. Thermodynamics provides

a unifying framework for diverse systems nbiology. Both a qualitative and quantitative understanding re mportant and will bedeveloped.

The beauty of thermodynamics s that a relatively small number of postulates canbe used to develop the entire subject. Perhaps he most important part of this develop-ment is to be very precise with regard to concepts and definitions, without gettingbogged down with mathematics. Thermodynamics s a macroscopic heory, not mo-lecular. As far as thermodynamics s concerned, molecules need not exist. However,we will not be purists in this regard: If molecular descriptions are useful for under-

1


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HEAT, WORK,AND ENERGY

standing or introducing concepts, hey will be used. We wilt not hesitate o give mo-lecular descriptions of thermodynamic results, but we should recognize that these n-terpretations are not inherent in thermodynamics itself. It is important to note,nevertheless, hat large collections of molecules are assumed so that their behavior isgoverned by Boltzmann statistics; hat is, the normal thermal energy distribution is as-sumed. This is almost always the case n practice. Furthermore, thermodynamics isconcerned with time-independent ystems, hat s, systems at equilibrium. Thermody-namics has been extended o nonequilibrium systems, but we will not be concernedwith the formal development of this subject here.

The first step s to define the system. A thermodynamic system s simply that partof the universe n which we are nterested. The only caveat s that the system must belarge relative to molecular dimensions. The system could be a room, it could be abeaker, t could be a cell, etc. An open system canexchange energy and matter acrossits boundaries, or example, a cell or a room with open doors and windows. A closedsystem can exchange energy but not matter, or example, a closed room or box. An iso-lated system can exchange neither energy nor matter, for example, he universe or, ap-proximately, a closed Dewar. We are free to select he system as we choose, but it isvery important that we specify what it is. This will be illustrated as we proceed. Theproperties of a system are any measurable quantities charactenzing he system. Prop-

erties are either extensive, roportional to the mass of the system, or intensive,inde-pendent of the mass. Examples of extensive properties are mass and volume.Examples of intensive properties are temperature, pressure, and color.

1.2 TEMPERATURE

We are now ready to introduce three mportant concepts: emperature, heat, and work.

None of these are unfamiliar, but we must define them carefully so that they can beused as we develop hermodynamics.

Temperature s anobvious concept, as t simply measures ow hot or cold a systemis. We will not belabor its definition and will simply assert hat thermodynamics re-quires a unique temperature scale, namely, the Kelvin temperature scale. The Kelvintemperature scale s related to the more conventional Celsius temperature scale by thedefinition

T*.luin = Zc"lrio,+ 273.16 ( 1 - 1 )

Although the temperature on the Celsius scale s referred to as "degrees Celsius," byconvention degrees are not stated on the Kelvin scale. For example, a temperature of100 degrees Celsius s 373 Kelvin. (Thermodynamics s entirely ogical-some of theconventions used are not.) The definition of thermal equilibriurn s very simple: when

two systems are at the same emperature, hey are at thermal equilibrium.


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1.3 HEAT

1.3 HEAT

Heat flows across he system boundary during a change n the state of the system be-

cause a temperature difference exists between the system and its surroundings. We

know of many examples of heat: Some chemical reactions produce heat, such as hecombustion of gas and coal. Reactions n cells can produce heat. By convention, heatflows from higher temperature o lower temperature. This fixes the sign of the heatchange. t is important to note that this is a convention and s not required by any prin-ciple. For example, f the temperature of the surroundings decreases, eat flows to thesystem, and he sign of the heat change s positive (+). A simple example will illustratethis sign convention as well as the importance of defining the system under consid-

eration.Consider wo beakers of the same size illed with the same amount of water. n one

beaker, A, the emperature s 25oC, and n the otherbeaker, B, the emperature s 75oC.Let us now place the two beakers n thermal contact and allow them to reach thermalequilibrium (50'C). This situation s illustrated n Figure 1- . If the system s definedas A, the temperature f the system ncreases o the heat change s positive. f the sys-tem is defined as B, the temperature of the system decreases o he heat change s nega-tive. If the system s defined as A and B, no heat low occurs across he boundary of

the system, so the heat change s zero This illustrates how important it is to define thesystem before asking questions about what is occurring.The heat change hat occurs s proportional to the temperature difference between

the initial and final states of the system. This can be expressed mathematically as

q = C(Tr- Ti) (r-2)

where q is the heat change, he constant C is the heat capacity,Tris the final tempera-ture, and Z, is the initial temperature. This relationship assumes hat the heat capacityis constant, ndependent of the temperature. n point of fact, the heat capacity oftenchanges as the temperature changes, so that a more precise definition puts this rela-tionship in differential form:

FIGURE 1-1. Illustration of the establishment of thermal equilibrium and importance ofdefining the system carefully. Two identical vessels illed with the same amount of liquid, butat different temperatures, are placed in contact and allowed to reach thermal equilibrium. Adiscussion of this figure is given in the text.


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HEAT,WORK,ANDENERGY

Note that he heat change nd he heat capacity are extensive roperties-the largerthe system he arger he heat capacity and he heat change. emperature, f course,is an ntensive roperty.

1 .4 WORK

The definitionof work s not as simple as hat or heat. Many different orms of workexist, or example, mechanical ork, such as muscle action, and electricalwork, suchas ons crossing harged membranes. e will use arather artificial,but very general,definitionof work that s easilyunderstood. Work s a quantity hat can be ransferred

across he system boundary and can always be converted o lifting and owering aweight n the surroundings. y convention, ork done on a system s positive: hiscoresponds o lowering he weight n the surroundings.

You may recall hat mechanical ork, w, is defined as he product of the orce nthe directionof movement, ,, times he distance moved, , or in differential orm

dw = F*dx (1-4)

(1-s)

Therefore, the work to lower a weight is -mgh,where nz s the mass, g is the gravita-tional constant, and h is the distance he weight is lowered. This formula is generally

useful: For example, mgh s the work required or a person of mass m to walkup a hillof height h.The work required o stretch a muscle could be calculated with Eq. 1-4 fwe knew the force required and the distance he muscle was stretched. Electrical work,for example, s equalto -EIt, where E is the electromotive force, 1is the current, and/ is the time. In living systems, membranes often have potentials (voltages) acrossthem. n this case, he work required or an on to cross he membrane s -zFY,wheree is the valence of the ion, F is the Faraday (96,489 coulombs per mole), and Y is the

potential. A specific example s the cotransport of Na+ and K+, Na+ moving out of thecell and K* moving into the cell. A potential of -70 millivolts is established n theinside so that the electrical work required o move a mole of K* ions to the inside is-(1X96,489X0.07) = -6750 ioules. Y = Youtrid" Yinsid" *70 millivolts.) The nega-tive sign means hat work is done by the system.

Although not very biologically relevant, we will now consider n some detail pres-sure-volume work, ar P-V work. This type of work is conceptually easy o under-stand, and calculations are relatively easy. The principles discussed are generallyapplicable o more complex systems, uch as hose encountered n biology. As a sim-ple exampleof P-V work, consider apiston filled with a gas, as pictured n Figure 1-2.In this case, he force is equal to the external pressure, P"^, times the area, A, of thepiston face, so the infinitesimal work can be written as

dw = -P"*A dx = -P"* dV

If the piston s lowered, work is done on the system and s positive; whereas f the pis-ton is raised work is done bv the svstem and is nesative. Note that the work done on


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1.4 WORK

|ffiIGURE L-2. Schematic epresentation f a piston pushing on the system. "*is the externalpressure, nd Ps5 s the pressure f the system.or by the system by lowering or raising the piston depends on what the external pres-sure is. Therefore, the work can have any value from 0 to -, depending on how theprocess s done. This is a very important point: The work associated with a givenchange n state depends on how the change n state s carried out.

The idea that work depends on how the process s carried out can be illustrated fur-ther by considering the expansion and compression of a gas. The P-V rsotherm or anideal gas s shown in Figure 1-3. An ideal gas s a gas hat obeys he ideal gas aw,PV = nRT (n is the number of moles of gas and R is the gas constant). The behavior of

most gases t moderate pressures s well described y this relationship. Let us considerthe expansion f the gas rom Pr,Vrto P2,V2.lf this expansion s done with the ex-ternal pressure equal to zero, that is, into a vacuum, the work is zero. Clearly this isthe minimum amount of work that can be done for this change n state. Let us nowcarry out the same expansion with the external pressure equal to Pr.In this case, hework is

FIGURE 1-3. A P-V isotherm for an ideal gas. The narrow rectangle with both hatched andopen areas s the work done in going from Pt,Vt to Pt,Vz with an external pressure of P:. Thehatched area s the work done by the system n going from Pr,Vtto Pz,Vz with an externalpressure of Pz. The maximum amount of work done by the system or this change n state s thearea under the curve between PtVt and Pz,Vz.


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HEAT,WORK,ANDENERGY

Fv^w - - J " p " d v - - p 2 ( v 2 - v r )

vI

(1-6)

which is the striped area under the P-V curve. The expansion can be broken intostages; or example, first expand he gas with p"* = p: followed by p"* = pr, asshownin Figure 1-3. The work done by the system s then the sum of the two rectangular ar-eas under the curve. It is clear that as the number of stages s increased, he magnitudeof the work done ncreases. The maximum work that can be attained would set he ex-ternal pressure equal to the pressure of the system minus a small differential pressure,dP,throughout the expansion. This can be expressed s

w^u*=- "avv1

(r-7)

By a similar reasoning process, t can be shown hat for a compression he minimumwork done on the system s

W-in P d V (1-8)

This exercise llustrates wo important points. First, it clearly shows hat the work as-sociated with a change n state depends on how the change n state s carried out. Sec-ond, it demonstrates he concept of a reversible path. When a change n state s carriedout such hat the surroundings and the system are not at equilibrium by an nfinitesimalamount, n this case dP, during the change n state, he process s called reversible. Theconcept of reversibility s only an deal-it cannot be achieved n practice. Obviouslywe cannot really carry out a change n state with only an infinitesimal difference be-tween the pressures of the system and surroundings. We will find this concept veryuseful, nevertheless.

Now let's think about a cycle whereby an expansion s carried out followed by acompression hat returns the system back to its original state. f this is done as a one-stage process n each case, he total work can be written as

wtotul= *"^o* w"o-p (1-e)

wtoiat= P2(V2- Vr) - Pr(Vr - Vr.) (1-10)

=-Ju'v2

wtotat= Pr - P)(Vz- Yl) > 0 ( r - 11 )


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1.5 DEFINITION F ENERGY

In this case, net work has been done on the system. For a reversible process, owever,the work associated with compression and expansion s

wexp P d v(r-r2)

and

--t"v1

tv,w. o * o = - J P d V

v2

( 1 - 1 3 )

so that the total work for the cycle is equal to zero. Indeed, for reversible cycles thenet work is always zero.

To summarize this discussion of the concept of work, the work done on or by thesystem depends on how the change n state of the system occurs. In the real world,changes n state always occur rreversibly, but we will find the concept of a reversiblechange n state o be very useful.

Heat changes also depend on how the process s carried out. Generally a subscript

is appendedto q, for example, ep and qvfor heat changes t constant pressure nd vol-ume, respectively. As a case n point, the heat change at constant pressure s greaterthan that at constant volume if the temperature of a gas s raised. This is because notonly must the temperature e raised, but the gas must also be expanded.

Although this discussion of gases seems ar removed rom biology, the conceptsand conclusions eached are quite general and can be applied to biological systems.The only difference is that exact calculations are usually more difficult. It is useful toconsider why this s true. n the case of ideal gases, simple equation of state s known,PV = nRT, that s obeyed quite well by real gases under normal conditions. This equa-tion is valid because as molecules, on average, re quite fn apartand heir energeticinteractions can be neglected. Collisions between gas molecules can be approximatedas billiard balls colliding. This situation obviously does not prevail in liquids and sol-ids where molecules are close together and the energetics of their interactions can-not be neglected. Consequently, imple equations of state do not exist for liquidsand solids.

1 .5 DEFINITIONOF ENERGY

The first law of thermodynamics s basically a definition of the energy change asso-ciated with a change n state. t is based on the experimental observation hat heat andwork can be interconverted. Probably the most elegant demonstration of this is the ex-perimental work of James Prescott Joule n the late 1800s. He carried out experimentsin which he measured he work necessary o turn a paddle wheel in water and the con-comitant rise in temperature of the water. With this rather primitive experiment, hewas able to calculate the conversion factor between work and heat with amazing ac-


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HEAT,WORK,ANDENERGY

curacy, namely, o within 0.27o. The first law states hat the energy change, AE, asso-ciated with a change n state s

Furthermore, the energy change s the same regardless of how the change n state scarried out. In this regard, energy clearly has quite different properties than heat andwork. This is true for both reversible and rreversible processes. ecause of this prop-erty, the energy (usually designated he internal energy in physical chemistry text-books) is called a state function. State functions are extremely important inthermodynamics, both conceptually and practically.

Obviously we cannot prove the first law, as t is a basic postulate of thermodynam-ics. However, we can show that without this law events could occur that are contraryto our experience. Assume, for example, that the energy change n going from state 1to state 2 is greater than the negative of that for going from state 2 to 1 because hechanges n state are carried out differently. We could then cycle between these twostates and produce energy as each cycle is completed, essentially making a perpetualmotion machine. We know that such machines do not exist. consistent with the firstlaw. Another way of looking at this law is as a statement of the conservation of energy.

It is important that thermodynamic variables are not just hypothetical-we mustbe able to relate them to laboratory experience, hat is, to measure hem. Thermody-namics is developed here for practical usage. Therefore, we must be able to relate theconcepts o what can be done n the laboratory. How can we measure energy changes?If we only consider P-V work, the first law can be written as

A4*YnA ^ E = q + w

"v^L E = q _ ) ' p " * d V

V,

If the change n state s measured at constant volume, then

(1 -14 )

(1- l s )

( 1 - 1 6 ),E = q,

At first glance, it may seem paradoxical that a state function, the energy change, s

equal o a quantity whose magnitude depends on how the change n state s carried out,namely, he heat change. However, n this instance we have specified how the changein state s to occur, namely, at constant volume. Therefore, if we measure he heatchange at constant volume associated with a change n state, we have also measuredthe energy change.

Temperature s an especially important variable in biological systems. f the tem-perature s constant during a change n state, he process s isothermal. On the otherhand, f the system s insulated so that no heat escapes or enters he system during the

change n state q - O), he process s adiabatic.


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1.6 ENTHALPY

1.6 ENTHALPY

Most experiments in the laboratory and in biological systems are done atpressure, ather than at constant volume. At constant pressure,

constant

(1-17),E= ep- P(Vz- V,l

or

E r- E r = e p - P ( V z - Vr )

The heat change t constant ressure an be written as

ep= (Ez+PV) - (Et+ PV) (1-19)

This relationship an be simplified by defining a new state unction, he enthalpy, H:

H = E + P V (r-20)

The enthalpy is obviously a state function since E, P, and V are state unctions. Theheat change at constant pressure s then equal to the enthalpy change:

q r= LH = Hz- Ht

( 1 - 1 8 )

(r-2r)For biological reactions and processes, we will usually be interested n the enthalpychange ather than the energy change. t can be measured experimentally by determin-ing the heat change at constant pressure.

As a simple example of how energyand enthalpy can be calculated, et's considerthe conversion of liquid water to steam at 100"C and 1 atmosphere pressure, hat is,

boiling water:

Hr.o(/,l atm, 00'c)+ Hro(g,1 atm, 00"c) e_zz)The heat equired or this process, AF1 = e is 9.71 kilocalories/mole. What is AE forthis process? This can be calculaterl as ollows:

A , E = L H - A ( P W- ^ H - PAV

LV= vr- vr - 2L.4liters/mole 18.0x 10-3 iters/mol - pvn= RT

A,E= N{ - RT _g7lo -2(373) = g970 calories/mole

Note that the Kelvin temperature must be used n thermodynamic calculations and thatAFl is significantly greater han AE.


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1O HEAT,WoRK,ANDENERGY

Let's do a similar calculation or the melting of ice nto liquid water

HrO(s, 273 K, 1 atm) + HrO( Zlg K, I atm) (r-23)

In this case he measured eat change, LH (= ei is 1.44 kilocalories/mole. The cal-culation of AE parallels the previous calculation.

L E = L H - P LV

LV = Vr

Vr= 18.0 milliliters/mole - 19.6 milliliters/mole = -1.6milliliters/mole

P LV = -1.6 ml.atm = -0.04 calorie

LE - 1440 + 0.04 = 1440 calones/mole

In this case M and Al1are essentially the same. n general, hey do not differ greatlyin condensed media, but the differences can be substantial n the gas phase.

The two most common units for energy are the calorie and the joule. (One calorieequals 4.184 oules.) The official MKS unit is the oule, but many research ublica-tions use the calorie. We will use both in this text. in order to familiarize the studentwith both units.

1.7 STANDARD TATES

Only changes n energy states can be measured. Therefore, it is arbitrary what we setas the zerofor the energy scale. As a matter of convenience, a common zero has beenset or both the energy and enthalpy. Elements n their stablest oms at25"C (298 K)and 1 atmosphere are assigned an enthalpy of zero. This is called a standard state andis usually written as Hlnr. The superscript means 1 atmosphere, and the subscript sthe temperature n Kelvin.

As an example of how this concept s used, consider the formation of carbon tetra-chloride from its elements:

C (graphitel + 2 Cl2G) + CCl4 4

Nr = Hinr1..ro1- |* 2Hinr1r,t

(r-24)

LIr - Hinr1""ro)


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1.8 CALORIMETRY 1 1

The quantity Hlnrrr.., is called the heat of formation of carbon tetrachloride. Tablesof heats of formation"are available for hundreds of compounds and are useful in cal-culating the enthalpy changes associated with chemical eactions cf. 3,4).

In the case of substances f biological interest n solutions, he definitions of stand-ard states and heats of formation are a bit more complex. tn addition to pressure andtemperature, other factors must be considered such as pH, salt concentration, metal onconcentration, etc. A universal definition has not been established. n practice, t isbest to use heats of formation under a defined set of conditions, and ikewise to definethe standard state as these conditions. Tables of heats of formation for some com-pounds of biological interest are given in Appendix 1 (3). A prime is often added tothe symbol for these heats of formation (FIr") to indicate the unusual nature of the

standard state. We will not make that distinction here, but it is essential hat a consis-tent standard state s used when making thermodynamic calculations for biologicalsystems.

A useful way of looking at chemical eactions s as algebraic equations. A charac-teristic enthalpy can be assigned o each product and reactant. Consider the "reaction"

a A + b B + c C + d D

For this reaction, MI = Hproducts Hreactants, f

(1-2s)

A,H= dHo+ cHc- aHo- bH,

where the H, are molar enthalpies. At298 K and 1 atmosphere, he molar enthalpiesof the elements are zero, whereas or compounds, he molar enthalpies are equal to the

heats of formation, which are tabulated. Before we apply these considerations o bio-logical reactions, a brief digression will be made o discuss how heats of reactions aredetermined experimentally.

1.8 CALORIMETRY

The area of science concerned with the measurement of heat changes associated withchemical reactions is designated as calorimetry. Only a brief introduction is givenhere, but it is important to relate the theoretical concepts o laboratory experiments.To begin this discussion we will return to our earlier discussion of heat changes andthe heat capacity, Eq. 1-3. Since he heat change depends on how the change n stateis carried out, we must be more precise n defining the heat capacity. The two mostcommon conditions are constant volume and constant pressure. The heat changes nthese cases an be written as

dqv= dE = CvdT (r-26)


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12 HEAr'oRK'ND NERGY*ii., tz #r,a rl .1r e ( |

-4

dqp= dH = CP dT

A moreexact mathematical treatment of these definitions would make use of partial

derivatives, but we will avoid this complexity by using subscripts o indicate what is

held constant. These equations can be integrated to give

L=dH(r-27)

cv dr

ATP + H2O =: ADP + Pt

.T^LE=J

Tl

J^^H lT1

(1-28)

(r-2e)

(1-30)

cP dr

Thus, heat changes can readily be measured f the heat capacity is known' The heat

capacity of a substance can be determined by adding a known amount of heat to the

substance and determining the resulting increase n temperature. The known amount

of heat s usually added electrically since his permits very precise measurement'Re-

call that the electrical heat is lR, where 1is the current and R is the resistance of the

heating element.) If heat s added epeatedly n small increments over alarge tempera-ture range, he temperature dependence f the heat capacity can be determined' Tabu-

lations of heat capacities are available and are usually presented with the temperature

dependence escribed as a power series:

C p = a + b T + r T 2 + ' ' '

where a, b, c,. . . are constants determined by experiment'

For biological systems, wo typesof calorimetry are commonly done-batch cal-

orimetry and scanning calorimetry. In batch calorimetry, the reactants aremixed to-

gether and the ensuing temperature rise (or decrease) is measured'A simple

experimental setup s depicted in Figu." 1-4, where the calorimeter is aDewar flask

and the temperature ncrease s measured by a thermocouple or thermometer'

For example, f we wished o measure he heat change or the hydrolysisof adeno-

sine S'-triphosphate ATP)'

a solution of known ATP concentration would be put in the Dewarat a defined pH'

metal ion concentration, buffer, etc. The reaction would be initiated byadding a small

amount of adenosine riphosphatase ATPase), an enzyme that efficientlycatalyzes

the hydrolysis, and the subsequent emperature ise measured' Theenthalpy of reac-

tion can be calculated from the relationship

(1-31)

L H = C p LT (r-32)


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1.8 CALORIMETRY 1 3

FIGURE L-4. Schematic representation of a simple batch calorimeter. The insulated vessel sfilled with a solution of ATP in a buffer containing salt and Mg'*. The hydrolysis of ATP is

initiated by the addition of the ATPase enzyme, and the subsequent rise in temperature ismeasured.

The heat capacity of the system s calculated by putting a known amount of heat intothe system hrough an electrical heater and measuring he temperature ise of the sys-tem. The enthalpy change calculated s for the number of moles of ATP in the system.

Usually the experimental result is reponed as a molar enthalpy, that is, the enthalpychange or a mole of ATP being hydrolyzed. This result can be obtained by dividingthe observed enthalpy change by the moles of ATP hydrolyzed. Actual calorimetersare much more sophisticated han this primitive experimental setup. The calorimeteris well insulated, mixing is done very carefully, and very precise temperature meas-urements are made with a thermocouple. The enthalpy changes or many biologicalreactions have been measured, but unfortunately this information is not convenientlytabulated in a single source. However, many enthalpies of reaction can be derived

from the heats of formation in the table in Appendix 1.Scanning calorimetry is a quite different experiment and measures he heat capac-

ity as a function of temperature. In these experiments, a known amount of heat isadded to the system through electrical heating and the resulting temperature rise ismeasured. Very small amounts of heat are used so the temperature changes are typi-cally very small. This process s repeated automatically so that the temperature of thesystem slowly rises. The heat capacity of the system s calculated or each heat ncre-ment as qrlLT, and the data are presented as a plot of C, versus Z.

This method has been used, or example, to study protein unfolding and denatura-tion. Proteins unfold as he temperature s raised, and denaturation usually occurs overa very ntrrow temperature ange. This is illustrated schematically n Figure 1-5, wherethe fraction of denatured protein, fo, is plotted versus he temperature along with theconesponding plot of heat capacity, Co, versus temperature.

As shown Figure l-5, the plot of heat capacity versus emperature s a smooth,slowly rising curve for the solvent. With the protein present, a peak n the curve occursas the protein is denatured. The enthalpy change associated with denaturation s thearea under the peak (striped area = lCp dT).In some cases, he protein denaturation

Thermometer


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1 4 HEAT,WORK,ANDENERGY

FIGURE 1'5. Schematic representation of the denaturation of a protein and the resultingchange n heat capacity, Cp. In (a) the fraction of denatured protein, Tb, is shown as a functionof temperature, 7. In (b) the heat capacity, as measured by scanning calorimetry, is shown as afunction of temperature. The lower curve is the heat capacity of the solvent. The hatched areais the excess heat capacity change due to the protein denaturing and is equal to AIl for theunfolding.

may occur in multiple stages, in which case more than one peak can be seen in the heat

capacity plot. This is shown schematically in Figure 1-6 for a two-stage unfoldingprocess.

The enthalpies associated with protein unfolding are often interpreted n molecularterms such as hydrogen bonds, electrostatic interactions, and hydrophobic interac-tions. It should be borne in mind that these nterpretations are not inherent in thermo-dynamic quantities, which do not explicitly give information at the molecular level.Consequently, such nterpretations should be scrutinized very critically.

FIGURE L-6. Schematic epresentation of a calorimeter scan n which the denaturation occursin two steps. The hatched area permits the sum of the enthalpy changes o be determined, andthe individual enthalpies of the unfolding reactions can be determined by a detailed analysis.As in Figure 1-5, Cp is the measured heat capacity, and Z is the temperature.

T(b)

T(a)

\


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1.9 REACTION NTHALPIES 1 5

1.9 REACTION ENTHALPIES

We now return to a consideration ofreaction enthalpies. Because he enthalpy s a state

function, it can be added and subtracted or a sequence f reactions-it does not matterhow the reaction occurs or in what order. In this regard, chemical reactions can be con-sidered as algebraic equations. For example, consider he reaction cycle below:

LHtA ----------+B

I A

t lLH) | | LH" | |- ' o

V I

c ---------+DLH3

If these eactions are written sequentially, t can readily be seen how the enthalpies arerelated.

A - + C L H ,

C - + D L H . '

D - + B L H o

A -+ B LH, - A^Hr+A,H,+ A,Ho

This ability to relate enthalpies of reaction n reaction cycles n an additive fashion soften called Hess's Law, although it really is derived from thermodynamic principlesas discussed. We will find that this "law" is extremely useful, as t allows determina-tion of the enthalpy of reaction without studying a reaction directly if a sequence ofreactions s known that can be added o give the desired reaction.

As an illustration, we will calculate the enthalpy of reaction for the transfer of aphosphoryl group from ATP to glucose, a very important physiological reaction cata-lyzedby the enzyme hexokinase.

Glucose + ATP - ADP + Glucose-6-phosphate (1-33)

The standard enthalpy changes or the hydrolysis of these our compounds are givenin Table 1-1. These data are or very specific conditions: T =298 K, P = I atm, pH =7.0, pMg = 3, and an ionic strength of 0.25 M. The ionic strength s a measure of thesalt concentration hat takes nto account he presence of both monovalent and divalent


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1 6 HEAT.WORK.ANDENERGY

TABLE 1.1

Reaction MIinrGJ/mol)

ATP + HzO(l)


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1.10 TEMPERATURE EPENDENCE F THE REACTION NTHALPY 17

It is important, therefore, hat the equation for the reaction under consideration be ex-

plicitly stated. The myokinase reaction could be written as

ADP+|a r r+ jnue

In this case, he reaction enthalpy per mole would be one-half of that reported or Eq. 1-34.

1.10 TEMPERATURE EPENDENCE FTHEREACTION NTHALPY

In principle, the enthalpy changes as he pressure and temperature change. We will notworry about the dependence f the enthalpy on pressure, as t is usually very small forreactions n condensed phases. The temperature dependence of the enthalpy is givenby Eq. I-27. This can be used directly to determine he temperature dependence of re-action enthalpies. f we assume he standard state enthalpy s known for each reactant,then the temperature dependence of the enthalpy for each reactant, , is

(1-35)

(1-36)

(r-37)

Hr,i= Hlnr,, f,nrc P,ir

If we apply this relationship to the reaction enthalpy for the generalized eaction of Eq.l-25, we obtain the following:

A,Hr= cHr,c+ dHr,- aHr^- bHr,u

JLIIr= Minr+ ) LCp dT

298

with

and

More generally,

LHlrr= cHlss,c+ H;s8,D oHirr,o- bHln*u

LCp cCp,c+ dCr,r- aCp,t- bCp,u

LHr-Mro*fr,or, o,

Equation 1-37 is known as Kirchhoff's Law. It can also be stated n differential form:


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1 8 HEAT,WORK,ANDENERGY

d LHIdT = LCr (1-38)

It is important to remember that this discussionof the temperature dependence of thereaction enthalpy assumes hat the pressure s constant.

The conclusion of these considerations of reaction enthalpies s that available tabu-lations are often sufficient to calculate the reaction enthalpy of many biological reac-tions. Moreover, if this is done at a standard emperature, he reaction enthalpy at othertemperatures can be calculated f appropriate nformation about the heat capacities sknown or estimated. For most chemical reactions of biological interest, the tempera-ture dependence of the reaction enthalpy is small. On the other hand, for processessuch as protein folding and unfolding, the temperature dependence s often significantand must be taken into account n data analysis and thermodynamic calculations. Thiswill be discussed urther in Chapter 3.

The first law of thermodynamics, namely, the definition of energy and its conser-vation, is obviously of great mportance n understanding he nature of chemical reac-tions. As we shall see, however, the first law is not sufficient to understand whatdetermines chemical equilibria.

REFERENCES

I. Tinoco, Jr., K. Sauer, andJ. C. Wang,Physical Chemistry: Principles andApplicationsto the Biological Sciences, 3rd edition, Prentice Hall, Englewood Cliffs, NJ, 1995.

D. Eisenberg and D. Crothers,Physical ChemistrywithApplications to the Life Sciences,Benjamin/Cummings, Menlo Park, CA, 1979.

3. The NBSTables of Thermodynamic Properties, D. D. Wagman et al., eds.,"/. Phys.Chem. Ref. Data, 11, Suppl. 2, 1982.

4. D. R. Stull, E. F. Wesffum, Jr., and G. C. Sinke, The Chemical Thermodynamics ofOrganic Compounds, Wiley, New York,1969.

5. R. A. Alberty,Arch. Biochem. Biophys.353, 116 (1998).

PROBLEMS

1-1. When a gas expands apidly through a valve, you often feel the valve get colder.

This is an adiabatic expansion(q =

0). Calculate the decreasen temperature of

1.0 mole of ideal gas as t is expanded rom 0.20 to 1.00 iter under he condi-tions given below. Assume a constant volume molar heat capacity, Cr, of ln.Note that the energy, E, of an ideal gas depends only on the temperature: t isindependent of the volume of the system.A. The expansion is irreversible with an external pressure of 1 atmosphere

and an initial temperature of 300 K.B. The expansion s reversible with an initial temperature of 300 K.C. Calculate A^E or the changes n state described n parts A and B.

l .

2.


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PROBLEMS 1 9

D. Assume he expansion s carried outisothermally at 300 K, rather han adi-abatically. Calculate the work done if the expansion is carried out irre-versibly with an external pressure of 1.0 atmosphere.

E. Calculate the work done if the isothermal expansion s carried out revers-ibly.

F. Calculate q andAE for the changes n state described n parts D and E.

l-2. A. Calculate he enthalpy change or the conversion of glucose CuHrrOu(s)land oxygen [Oz(g)] to CO2(ag) and HzO(/) under standard conditions.The standard enthalpies of formation of glucose(s), CO2(ag), and HzO(l)are 304.3, -98.7, and -68.3 kcaUmol, espectively.

B. When organisms metabolize glucose, approximately 50Vo of the energyavailable s utilized for chemical and mechanical work. Assume 25Voof thetotal energy from eating one mole of glucose can be utilized to climb amountain. How high a mountain can a 70 kg person climb?

1-3. Calculate the enthalpy change for the oxidation of pyruvic acid to acetic acidunder standard onditions.

2 CH3COCOOH(I) +O2(B)

+2 CHTCOOH(I) +2CO2G)

The heats of combustion of pyruvic acid and acetic acid under standard condi-tions are -227 kcaUmol and -207 kcaVmol, respectively. Heats of combustionare determined by reacting pyruvic or acetic acid with Or(g) to give HzO(l)and CO2(g). Hint: First write balanced chemical equations or the combustionprocesses.

l-4. Calculate the amount of water (in liters) that would have to be vaporized at40'C (approximately body temperature) o expend the2.5 x 106calories of heatgenerated by a person n one day (commonly called sweating). The heat of va-porization of water at this temperature s 574 caUg.We normally do not sweatthat much. What's wrong with this calculation? f l%oof the energy producedas heat could be utilized as mechanical work, how large a weight could be lifted1 meter?

1-5. A. One hundred milliliters of 0.200 M ATP is mixed with an ATPase n a De-war

at 298 K, 1 atm, pH 7.0, pMg 3.0, and0.25 M ionic strength. The tem-perature of the solution ncreases 1.48 K. What is Afl' for the hydrolysisof ATP to adenosine 5'-diphosphate ADP) and phosphate? Assume theheat capacity of the system s 418 J/K.

B. The hydrolysis reaction can be written as

AT P + H 2 O = A D P + P,


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20 HEAT,WORK, NDENERGY

Under the same conditions, the hydrolysis of ADP,

A D P + H r O A M P + P t

has a heat of reactiono LHo, of -28.9 kJ/mol. Under the same conditions,

calculate AI1" for the adenylate kinase reaction:

2 ADP + AMP + ATP

1-6. The alcohol dehydrogenase eaction,

NAD + Ethanol + NADH + AcetaldehYde

removes ethanol from the blood. Use the enthalpies of formation in Appendix

1 to calcul ateL,H" for this reaction. If 10.0 g of ethanol (a generous martini)is

completely converted to acetaldehyde by this reaction, how much heat ispro-

duced or consumed?


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I CHAPTER2

Entropyand Free Energy

2.1 INTRODUCTION

At the outset, we indicated the primary objective of our discussion of thermodynamicsis to understand chemical equilibrium in thermodynamic terms. Based on our discus-sion thus far, one possible conclusion is that chemical equilibria are governed by en-ergy considerations and that the system will always proceed o the lowest energy state.This idea can be discarded quite quickly, as we know some spontaneous eactions pro-duce heat and some require heat. For example, the hydrolysis of ATP releases heat,M]*= -30.9 kJ/mol, whereas ATP and AMP are formed when ADP is mixed withmyokinase, yet LHige= +2.0 kJ/mol under dentical conditions. The conversion of liq-uids to gases equires heat, hat is, A11is positive, even at temperatures bove he boiling

point. Clearly the lowest energy state s not necessarily he most stable state.What factor is missing? (At this point, traditional treatments of thermodynamics

launch nto a discussion f heat engines, a topic we will avoid.) The missing ngredientis consideration of the probability of a given state. As a very simple illustration, con-sider three balls of equal size that are numbered 1, 2, 3. These balls can be arrangedsequentially n six different ways:

t23 132 2r3 231 3r2 32r

The energy state of all of these urangements s the same, yet it is obvious that the prob-ability of the balls being in sequence 123) s 1/6, whereas he probability of the ballsbeing out of sequence s 5/6. In other words, the probability of a disordered state smuch greater than the probability of an ordered state because a larger number of ar-rangements of the balls exists n the disordered state.

Molecular examples of this phenomenon can readily be found. A gas expandsspontaneously nto a vacuum even though the energy state of the gas does not change.This occurs because he larger volume has more positions available for molecules, soa greater number of arrangements, or more technically microstates, of molecules arepossible. Clearly, probability considerations re not sufficient by themselves. f thiswere the case, he stable state of matter would always be a gas. We know solids andliquids are stable under appropriate conditions because hey are energetically favored;that is, interactions between atoms and molecules result in a lower energy state. Thereal situation must involve a balance between energy and probability. This is a quali-tative statement of what determines he equilibrium state of a system, but we will beable to be much more quantitative than this.

21


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22 ENTROPY ND FREE ENERGY

The second aw states hat disordered states are more probable than ordered states.This is done by defining a new state unction, entropy, which is a measure of the dis-order (or probability) of a state. Thermodynamics does not require this interpretationof the entropy, which is quasi-molecular. However, this is a much more intuitive wayof understanding entropy than utilizing the traditional concept of heat engines. Themore disordered a state, or the larger the number of available microstates, he higherthe entropy. We already can see a glimmer of how the equilibrium state might be de-termined. At constant entropy, the energy should be minimized whereas at constantenergy, the entropy should be maximized. We will return to this topic a little later.First, we will define the entropy quantitatively.

2.2 STATEMENT FTHESECOND AW

A more formal statement of the second aw is to define a new state unction, the en-tropy, S, by the equation

(2-r)

(2-2)

ds=+

^s=J+The temperature scale n this definition is Kelvin. This definition is not as straightfor-

ward as that for the energy. Note that this definition requires a reversible heat change,

er",et d4r"u,yet entropy is a stateunction. At first glance, his seems quite paradoxi-

.at. th" meaning of this is that the entropy change must be calculated by finding a re-

versible path. However, all reversible paths give the same entropy change, and the

calculated entropy change s correct even f the actual change n state s carried out ir-

reversibly. Although this appears o be somewhat confusing, consideration of some

examples will help in understanding his concept.The second aw also ncludes mportant considerations about entropy: For a revers-

ible change n state, he entropy of the universe s constant, whereas or an irreversible

change n state, he entropy of the universe ncreases.

Again, the second aw cannot be proved, but we can demonstrate hat without this1aw, events could ffanspire that are contrary to our everyday experience. Two exam-

ples are given below.Without the second aw, a gas could spontaneously ompress Let's illustrate his

by considering the isothermal expansion of an ideal gas, V, to Vrwith constant Z. The

entropy change s

As=J @q,,/D Q/n J 0n,",= ,"/T(2-3)


31/180


32/180

24 ENTROPYANDREE NERGY

dS"+ dSn= dq (llTn- 1lT")


33/180

2.3 CALCULATIONFTHEENTROPY 25

siderations, ut we will consider one example o illustrate the process. Let us deter-mine the entropy change or the following process:

HrO(/,298 K, 1 arm) -) HrO(g 298 K, 1 atm)

This change s not a reversible change n state, as we know that the normal boilingpoint of water at 1 atmosphere s 373 K. A possible eversible cycle that would gofrom the initial state o the final state at constant pressure s

(2-1 )

(2-12)

(2-13)

The entropy change for the bottom process, which is reversible, is simply NIIT =97101373 = 26 callmol.K). The entropy change or the left-hand side of the square s(Eq.2-8)

AS = Cr ln(TrlT1) = 18 1n(373/298) 4 cal/(mol'K)

and for the right-hand side of the square s

AS = Coln(TrlTt) = 8.0ln(2981373) -1.8 cal/(mol.K)

The entropy changes or these hree reversible processes an be added o give the en-tropy change or the change n state given in Eq. 2-ll:28 caV(mol'K).

An alternative reversible path that can be constructed owers the pressure o theequilibrium vaporpressure of water at298 K. The corresponding constanttemperature

cvcle is

HrO(/,298 K, I atm) + HrO(g 298 K, 1 atm)

JTHzO(1,373 , 1 atm) -)HrO(g 373 K, I atm)

HrO((,298 K, 1 atm) + Hro(g, 298 K, I atm)

JT\O(/,298K,0.0313 atm) + HrO(g,298 ,0.0313 tm)

In this case, we would have to calculate the change n entropy as the pressure s low-ered and raised. This can easily be done but is beyond the scope of this presentationof thermodynamics. The point of this exercise s to illustrate how entropy changes canbe calculated for irreversible as well as reversible processes and multiple reversibleprocesses an be found.

In principle, the entropy can be calculated from statistical considerations.Boltzmann derived a relationship between he entropy and the number of microstates,N:

S = f t s l n N (2-r4)


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where ku is Boltzmannos onstant, 1.38 x 10-23 /K. It is rarely possible o determinethe number of microstates although the number of microstates could be calculated

from Eq.2-I4 if the entropy is known. For a simple case, such as the three numberedballs with which we started our discussion f the second aw, this calculation can ead-ily be done. The disordered system has 3 microstates and an entropy of 1.51 x 10-23J/K. Any ordered sequence-for example 1,2,3, -has only 1 microstate, so N = 1,and S = 0. Since the disordered state has a higher entropy, this predicts that the ballswill spontaneously disorder, and that an ordered state s extremely unlikely.

2.4 THIRD AWOF THERMODYNAMICS

We will not dwell on the third law as the details are of little consequence n biology.The important fact for us is that the third law establishes azero for the entropy scale.Unlike the energy, entropy has an absolute scale. The third law can be stated as fol-lows: The entropy of perfect crystals of all pure elements and compounds s zero atabsolute zero. The tricky points of this law are the meanings of "perfect" and "pure,"

but we will not discuss his in detail. It is worth noting that a perfect crystal has onemicrostate, ffid therefore an entropy of zero according to Eq. 2-I4.

The absolute standard entropy can bedetermined from measurements of the tem-

perature dependence of the heat capacity using the relationship

c PdT/T(2-1s)

Here the entropy at absolute zero has been assumed o be zero in accord with the third

law. A plot of CplT versus 7 gives a curve such as hat in Figure 2-l.T\e area under

the curve is the absolute standard entropy. Tables of ^St'nt re readily available (cf.

100 200 300T (K)

FIGURE 2-1. Aplot of the constant-pressure eat capacity divided by the temperature CplT,

versus the temperature, I, for graphite. The absolute entropy of graphite at 300 K is the area

under the curve up to the dashed ine (Eq. 2-15).If phase changes occur as the temperature s

changed, the enffopy changes associated with the phase changes must be added to the area

under the curve of the CrlT versus T plot.

298sin, J

0

=c{v6 oE

r g 4. r.F- a ro -

ct

o


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2.5 MOLECULAR NTERPRETATION F ENTROPY 27

Refs. 3 and 4, Chapter 1). The entropy at temperatures other than 298 K can be calcu-lated from the relationship

si =si'st cP dr/T (2-r6)

For a chemical reaction,

(2-r7)

J+J

298

.TNr"* J

298asz = LCPdT/T

These relationships are analogous o those used for the enthalpy. Indeed, entropies ofreactions can be calculated n a similar fashion to enthalpy changes.

2.5 MOLECULARNTERPRETATIONF ENTROPY

We will now consider a few examples of absolute entropies and entropy changes orchemical reactions, and how they might be interpreted n molecular terms. The abso-lute entropies or water as a solid,liquid, and gas at273 K and 1 atmosphere, re 41.0,63.2, and 188.3 J/(mol'K), respectively. The molecular nterpretation of these num-bers is straightforward, namely, that solid is more ordered than liquid, which is moreordered than gas; therefore, the solid has the lowest entropy and the gas the highest.

Standard entropy changes or some chemical reactions are given in Table 2-1. Asexpected, he entropy change s negative for the first two reactions n the gas phase asthe number of moles of reactants s greater han the number of moles of products. Ofcourse, hese eactions could have been written in the opposite direction. The entropy

change would then have the opposite sign. At first glance, the result for the third re-action in the table is surprising, as he number of moles of reactants s greater han thenumber of moles of products. This results because he solvent must be ncluded n themolecular interpretation of the observed entropy change. Ions in water interactstrongly with water so that highly ordered water molecules exist around the ions.

TABLE 2.1

Reaction ASfi, [J/(mol.K)]

H ( g ) + H ( g ) = H z ( g )2Hz@) + Oz(g) = 2 H2O(g)H*(aq) + OH-(aq) = H2O(l)Cytidine 2'-monophosphate + Ribonuclease A


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28 ENTROPYAND FREE ENERGY

When the neutral species s formed, these highly ordered water molecules become essordered. Thus, the entropy change or water is very positive, much more positive thanthe expected entropy decrease or H+ and OH- when they form water. This simple ex-ample indicates that considerable care must be exercised in interpreting entropychanges or chemical reactions n condensed media. The entropy of the entire systemunder consideration must be taken nto account. The final entry is for the binding of aligand to an enzyme.In this case, no reasonable nterpretation of the entropy changeis possible as hree factors come into play: the loss n entropy as wo reactants becomea single entity, the changes n the structure of water, and structural changes n the pro-tein. The fact that the enffopy change s comparable o the value expected or the com-bination of two molecules to produce one molecule is simply fortuitous. Extreme

caution should be exercised n making molecular interpretations of thermodynamicchanges n complex systems.

We have established all of the thermodynamic principles necessary o discusschemical equilibrium. We will now apply these principles to develop a general rame-work for dealing with chemical reactions.

2.6 FREE ENERGY

In a sense we have reached our goal. We have developed hermodynamic criteria forthe occurrence of reversible and rreversible (spontaneous) processes, amely, for theuniverse, AS = 0 for reversible processes nd must be greater han zero for irreversible

)t

---4 processes. Unfortunately, this is not terribly useful, as we are nterested n what is hap-

"ipening in the system and require criteria that are easily applicable to chemical reac-

i tions. This can be achieved by defining a new thermodynamic state function, the

i Gibbs free energy:

i o-H-rs , , t , lAY (' '-rE;# 'J : Lls'u tki (2-1i (J. Willard Gibbs developed the science of therniodynamics virtually single handed

i around the turn of the century at Yale University. To this day, his collected works are

i pnzedpossessions n the libraries of people seriously interested n thermodynamics.), At constant emPerature,

rAG= AH- f AS

= ep- ZAS = ep- er", (2-19)

- - - - - i ' - , 1 , 5 .. l ? f l , r. A | \ =O f o " - t t ' 1 t V ' . 0 ( a ( t i { 's t n ( ( A - \ 0

t or a reverstDle rocess, p= er",so AG = 0. For rreversible rocesses, he situationis a bit more complex as he sign of the heat change must be considered. or an endo-thermic rocess, pl er"uso G < 0. For an exothermic rocess, he absolute alueofqris greater han he absolute alue of er",so again, AG < 0. This s the free energy

"ituttg"for the system nddoes ot nvolve he surroundings. e now have developed


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2.6 FREE ENERGY 29

criteria that tell us if a process occurs spontaneously. f LG < 0, the change n stateoccurs spontaneously, whereas f AG > 0, the reverse change n state occurs sponta-neously. f AG = 0, the system s at equilibrium (at constant pressure nd emperature).

As with the energy and enthalpy, only differences n free energy can be measured.Consequently, he zero of the free energy scale s arbitrary. As for the enthalpy, thezero of the scale s taken as the elements n their stable state at I atmosphere and 298K. Again, analogous o the enthalpy, tables of the free energies of formation of com-pounds are available so that free energy changes or chemical reactions can be calcu-lated (cf. Refs. 3-5 in Chapter 1). Referring back to the reaction n Eq. I-25,

LGln,= rGlss.c+ G]rr'- oGlg,o- bGlnr,u (2-20)

where the free energies on the righrhand side of the equation are free energies permole. We will discuss he temperature ependence f the free energy a bit later, but itis useful to remember that at constant emperature and pressure,

L G = N I - T L S (2-21)

Standard state free energy changes are available for biochemical reactions although

comprehensive abulations do not exist, and the definition of "standard state" s morecomplex than in the gas phase, as discussed reviously. Standard state ree energiesof formation for some substances f biochemical nterest are given in Appendix 1.

As an illustration of the concept of free energy, consider he conversion of liquidwater to steam:

HzO(/) -r HrO(g)

At the boiling point, NI =97I0 caVmol and AS =26 eu. Therefore,

(2-22)

AG= 9710-267 (2-23)

At equilibrium, AG = 0 and Eq.2-23 gives T = 373 K, the normal boiling point ofwater. If T > 373 K, LG < 0, and the change n state s spontaneous, hereas f T 0, and the reverse process, ondensation, ccurs spontaneously.

Calculation of the free energy for a change n state s not always straightforward.

Because he entropy is part of the definition of the free energy, a reversible processmust always be found to calculate the free energy change. The free energy change, ofcourse, s the same egardless f how the change n state s accomplished ince he freeenergy s a state unction. As a simple example, again consider he change n state nEq.2-11. This is not a reversible process since he two states are not in equilibrium.This also s not a spontaneous hange n state so that AG > 0. In order o calculate hevalue of AG, we must think of a reversible path for carrying out the change n state.The two cycles n Eqs. 2-12 and2-13 againcan be used. n both cases, he bottom re-action s an equilibrium process, o AG = 0. To calculate AG for the top reaction, we


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only need to add up the AG values for the vertical processes. These can be calculatedfrom the temperature upper cycle) and pressure lower cycle) dependence f G. Suchfunctional dependencies will be considered shortly. It will be a useful exercise or

thereader o carry out the complete calculations.

2 .7 CHEMICALEQUILIBRIA

Although we now have developed criteria for deciding whether or not a process willoccur spontaneously, hey are not sufficient for consideration of chemical reactions.We know that chemical reactions are generally not "a11 r nothing" processes; nstead,

an equilibrium state s reached where both reactants and products are present. We willnow derive a quantitative relationship between he free energy change and the concen-trations of reactants and products. We will do this in detail for the simple case of idealgases, nd by analogy or reactions n liquids.

The starting point for the derivation is the definition of free energy and ts total de-rivative:

G = H - Z , S = E + P V- T S

dG = dE + P dV + V dP - T dS - S dT

Since dE = dq + dw - T dS - P dVfor a reversible process,

(2-24)

(2-2s)

d G _ V d P _ S d T (2-26)

(Although this relationship was derived for a reversible process, t is also valid for an

irreversible process.) ,et us now consider a chemical reaction of ideal gases at con-stant temperature. For one mole of each gas component,

d G = V d P - RT d P l P (2-27)

We will refer all of our calculations o a pressure of 1 atmosphere or each component.In thermodynamic erms, we have selected 1 atmosphere s our standard state.If Eq.2-27 is now integrated from Po = 1 atmosphere o P,

dG = RT dPlP

G = G" + RT ln(P/Ps) = Go + RZ ln P (2-28)

We will now return to our prototype reaction, Eq. l-25, and calculate the free en-ergy change. The partial pressures of the reactants are given in parentheses:


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2.7 CHEMICAL QUILIBRIA 31

aA(P ^) + bB(PB) == cC(P") + dD(Po) (2-29)

AG = cG"+ dGo- aGn - bGu- cG[+ dG;-"G;-

bG]+ cRTln P.+ dRT n Po

- aRT lnPo - bRT lnP"

(2-30)

The AG is the free energy for the reaction in Eq. 2-29 when the system s not at equi-librium. At equilibrium, at constant temperature and pressure, AG = 0, and Eq. 2-30becomes

LG" = (2-3r)

Here the subscript e has been used to designate equilibrium and K is the equilibriumconstant.

We now have a quantitative relationship between the partial pressures of the reac-tants and the standard ree energy change, AGo. The standard ree energy change s aconstant at a given temperature and pressure but will vary as he temperature and pres-sure change. f AG" < 0, then K > 1, whereas f AG'> 0, K < l. A common mistakeis to confuse the free energy change with the standard ree energy change. The free

energy change s always equal to zero at equilibrium and can be calculated from Eq.2-30 when not at equilibrium. The standard ree energy change s a constant repre-senting he hypothetical reaction with all of the reactants and products at a pressure of1 atmosphere. t is equal to zero only if the equilibrium constant ortuitously is 1.

Biological reactions do not occur in the gas phase. What about free energy n solu-tions? Conceptually there s no difference. The molar free energy at constant empera-ture and pressure can be written as

G = G " + RT l n ( c l c s ) (2-32)where c is the concentration and co s the standard state concentration. A more correcttreatment would define the molar free energy as

G = G " + RT l n a (2-33)

where a is the thermodynamic activity and is dimensionless. However, the thermody-namic activity can be written as a product of an activity coefficient and the concentra-

AG=AG"+-rtffi)

- orr"(W-) =- RrtnK\P^PL


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32 ENTRoPYANDREE NERGY

tion. The activity coefficient can be included in the standard ree energy, Go, whichgives rise to Eq.2-32. We need not worry about his as ong as he solution conditionsare clearly defined with respect o salt concentration, pH, etc. The reason t is not ofgreat concern is that all of the aforementioned complications can be included in thestandard ree energy change since, n practice, the standard ree energy s determinedby measuring the equilibrium constant under defined conditions.

Finally, we should note that the free energy per mole at constant emperature andpressure s called the chemical potential, p, in more sophisticated reatments of ther-modynamics, but there is no need to introduce this terminology here.

If we take the standard state as 1 mole/ liter, then the results parallel to Eqs. 2-28,2-30, and2-31 arc

G = G " + R Z l n c (2-34)

(2-3s)

(2-36)

Equations 2-34to2-36 summarizethethermodynamic elationships necessary o dis-

cuss chemical equilibria.Note that, strictly speaking, he equilibrium constant s dimensionless as all of the

concenffations are ratios, the actual concentration divided by the standard state con-

centration. However, practically speaking, t is preferable to report equilibrium con-

stants with the dimensions implied by the ratio of concentrations n F;q. 2-36. The

equilibrium constant s determined experimentally by measuring concentrations, and

attributing dimensions to this constant assures hat the correct ratio of concentrations

is considered and that the standard state s precisely defined.

Consideration of the free energy also allows us to assess ow the energy and en-tropy are balanced o achieve he final equilibrium state. Since AG = NI - ZAS at con-

stani Zand P, it can be seen hat a change n state s spontaneous f the enthalpy change

is very negative and/or the entropy change is very positive. Even if the enthalpy

change s unfavorable (positive), the change n state will be spontaneous f the T AS

term is very positive. Similarly, even if the entropy change s unfavorable (Z A,S ery

negative), he change n state will be spontaneous f the enthalpy change s sufficiently

negative. Thus, the final equilibrium achieved s abalance between he enthalpy ( Ift

and the entropy (f A$.

AG:AG.+t{rffi)

AG'=Rr"[,ft)-],rn^u


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2 . 8 PRESSUREANDTEMPERATUREDEPENDENCEOFTHEFREEENERGY 3

2.8 PRESSURE NDTEMPERATURE EPENDENCE FTHEFREEENERGY

We will now return to the pressure and temperature dependence f the free energy. At

constant emperature, he pressure dependence f the free energy ollows directly from

W.2-26, namely,

d G = V d P (2-37)

This equation can be integrated if the pressure dependence of the volume is known,

as for an ideal gas. For a chemical reaction, Eq.2-37 can be rewritten as

d L G = LV d P (2-38)

where AVis the difference n volume between he products and reactants. The pressure

dependence of the equilibrium constant at constant emperature ollows directly,

d L G = - RT d I n K = LV d P (2-3e)

dln K LV= - -dP RT

(2-40)

For most chemical reactions, he pressure dependence of the equilibrium constant squite small so that it is not often considered n biological systems.

Equilibrium constants, however, frequently vary significantly with temperature. Atconstant pressure, Eq.2-26 gives

d G = - S d T (2-4r)

d L G = - L S d T (2-42)

Returning to the basic definition of free energy at constant emperature and pressure,

LG = LH - ZAS = NI + T(d LG/dT)

This equation can be divided by T'and rearranged s ollows:

-LG /72 + @ LG / dT) T = -LH /72


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34 ENTROPY NDFREE ENERGY

d(LG/T) Nr= - _dT T2

(2-43)

Equation 2-43 is an mportant thermodynamic relationship describing the remperaruredependence f the free energy at constant pressure and s called the Gibbs-Helmholtzequation. The temperature dependence of the equilibrium constant ollows directly:

d(LG" T) ^ d ln Kdr

= -rrdT

AII"

rf LH" is independent of temperature, Eq. 2-44 caneasily be integrated:

AH"dT----------RT"

dln K- =dT

r"d t n K = l

. 7 ,

(2-44)

(Nf /R)(72- Tr) (2-4s)T,T,

When carrying out calculations, the difference between reciprocal temperaturesshould never be used directly as t introduces a large error. Instead he rearrangementin Eq. 2-45 should be used n which the difference between two temperatures occurs.With this equation and a knowledge of LHo, the equilibrium constant can be calcu-lated at any temperature f it is known at one temperature.

What about the assumption that the standard enthalpy change s independent oftemperature? his assumption s reasonable or many biological reactions f the tem-perature range s not too large. In some cases, he temperature dependence annot beneglected and must be included explicitly in canying out the integration of F;q. 2-44.The temperature dependence f the reaction enthalpy depends on the difference n heatcapacities between he products and reactants s given by Eq. 1-38.

Examples of the temperature dependence of equilibrium constants are displayed n

Figure 2-2. As predicted by Eq. 2-45, a plot of ln K versus IT is a straight ine with aslope of -NI"/R. The data presented are or the binding of DNA to DNA binding pro-teins (i.e., Zn fingers). The dissociation onstants or binding are quite similar for bothproteins at22"C,1.08 x 10-e M (WTl protein), and 3.58 x 10-e M (EGRI protein).However, as ndicated by the data n the figure, the standard enthalpy changes are op-posite n sign, +6.6 kcaVmol and -6.9kcaVmo1, espectively. Consequently, he stand-ard entropy changes are also quite different, 63.3 eu and 15.3 eu, respectively. Thesedata ndicate that there are significant differences n the binding processes, espite hesimilar equilibrium constants.

,"8)=+Ii-t)=


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2.9 PHASE CHANGES 35

3.3 3.4 3.5rooor (r1)

FIGURE 2-2. Temperature dependence of the equilibrium binding constant, K for the bindingof DNA to binding proteins ("zinc fingers"), WT1 (o) and EGRI (o), to DNA. Adapted withpermission rom T. Hamilton, F. Borel, and P. J. Romaniuk, Biochemistry 37,2051 (1998).Copyright O 1998 American Chemical Society.

2.9 PHASE CHANGES

The criterion that LG = 0 at equilibrium at constant emperature and pressure s quitegeneral. For chemical reactions, his means hat the free energy of the products s equalto the free energy of the reactants. For phase changes of pure substances, his meansthat at equilibrium the free energies of the phases are equal. If we assume wo phases,A and B, Eq. 2-26 gives

dGo- V^dP - S^dT - dGt= VBdP - SBdT

Rearrangement ivesii , irt: ,i . .= .,,r i 1i

-i-

dP _ASBA (2-46)dT AVuo

where ASuo SB So and AVuo = Vn- yA. (All of these quantities are assumed o beper mole for simplicity.) This equation s often referred to as the Clapeyron equation.Note that the entropy change can be written as

AS"e = A,HstlT (2-47)

Equation 2-46 gives the slope of phase diagrams, plots of P versus T that are usefulsummaries of the phase behavior of a pure substance. As an example, the phase dia-gram of water s given in Figure 2-3.

The lines in Figure 2-3 indicate when two phases are in equilibrium and coexist.Only one phase exists n the open areas, and when the two lines meet, three phases arein equilibrium. This is called he triple point and can only occur atapressure of 0.006


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FIGURE 2-3. Schematic representation of the phase diagram of water with pressure, P, andtemperature, Z, as variables. The phase diagram is not to scale and is incomplete, as severaldifferent phases of solid water are known. If volume is included as a variable, athree-dimensional phase diagram can be constructed.

atmosphere and 0.01"C. The slopes of the lines are given by Eq. 2-46. The number of

phases that can coexist is governed by the phase rule,

f = c - p + 2 (2-48)

where/is the number of degrees of freedom, c is the number of components, and p is

the number of phases. This rule can be derived from the basic laws of thermodynam-ics. For a pufe substance, he number of components s 1, so,f - 3 - p.In the open

spaces, ne phase exists, so that the number of degrees f freedom s 2, which means

that both Z and P can be varied. Two phases coexist on the lines, which gives/= l.Because only one degree of freedom exists, at a given value of P, Z is fixed and vice

versa. At the triple point, threephases

coexist,and there are no degrees of freedom,

which means hat both P andT are ixed.The study of phase diagrams s an important aspect of thermodynamics, and some

interesting applications exist in biology. For example, membranes contain phos-

pholipids. If phospholipids are mixed with water, they spontaneously orm bilayer

vesicles with the head groups pointing outward toward the solvent so that the interior

of the bilayer is the hydrocarbon chains of the phospholipids. A schematic repre-

sentation of the bilayer is given in Figure 2-4. The hydrocarbon chain can be either

disordered, at high temperatures, or ordered, at low temperatures, as indicated sche-

matically in the figure. The transformation from one form to the other behaves as aphase ransition so that phase diagrams can be constructed for phospholipids. More-

over, the phase diagrams depend not only on the temperature but on the phospholipid

composition of the bilayer. This phase ransition has biological implications in that the

fluidity of the hydrocarbon portion of the membrane strongly affects the transport and

mechanical properties of the membranes. Phase diagrams or phospholipids and phos-

pholipid-water mixtures can be constructed by a variety of methods, ncluding scan-

ning calorimetry and nuclear magnetic resonance.

IIIII

0.01"c


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PROBLEMS 37

N:I

I, ffi[[[-ffi[[x(b)\ /\ ' - ------ '(a)

FIGURB 2-4. Schematic representation of the vesicles of phospholipids that are formed whenphospholipids are suspended n water (a). Here the small circles represent he head groups andthe wiggly lines the two hydrocarbon chains. A phase change can occur in the side chains froma disordered to an ordered state as sketched n (b).

2.10 ADDITIONS TO THE FREE ENERGY

As a final consideration n developing the concept of free energy, we will return to ouroriginal development of equilibrium, namely,Eq.2-33. In arriving at this expressionfor the molar free energy at constant temperature and pressure, he assumption wasmade that only P-V work occurs. This is not true in general and for many biological

systems n particular. If we went all the way back to the beginning of our development ofthe free energy changes associated with chemical equilibria (Eq. 2-26) and derived the mo-lar free energy at constant emperafure and pressure, we would find that

G = G " + RT l n c + l u , r r u " (2-4e)where w-u* is the maximum (reversible) non P-V work. For example, or an ion ofcharge zin a potential field Y, w-u* is zFY, where F is the Faraday. We have usedthis relationship in Chapter I to calculate the work of moving an ion across a mem-

brane with an mposed voltage.In terms of transport of an on across a membrane, hisassumes he ion is being transported from the inside to the outside with an externalmembrane potential of Y.) This extended hemical potential is useful for discussingion transport across membranes, s we shall see ater. t could also be used, or exam-ple, for considering molecules n a gravitational or centrifugal field. (See Ref. 1 inChapter 1 for a detailed discussion of this concept.)

We are now ready to consider some applications of thermodynamics o biologicalsystems n more detail.

PROBLEMS

2-1. One mole of an ideal gas nitially at 300 K is expanded rom 0.2 to 1.0 liter.Calculate AS for this change n state f it is carried out under the following con-ditions. Assume a constant volume heat capacity or the gas, Cr, of ]R.

A. The expansion s reversible and sothermal.


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38 ENTROPYND REE NERGY

B. The expansion s irreversible and sothermal.C. The expansion s reversible and adiabqtic (q = 0).D. The expansion is irreversible with an external pressure of I atmosphere

and adiabatic.

2-2. The alcohol dehydrogenase eaction, which removes ethanol from your blood,proceeds according to the following reaction:

NAD+ + Ethanol + NADH + Acetaldehyde

Under standard conditions (298 K, 1 atm, pH 7.0, pMg 3, and an ionic strengthof 0.25 M), the standard enthalpies and free energies of formation of the reac-tants are as ollows:

tI" (kJ/mol) G" (kJ/mol)

NAD*NADHEthanolAcetaldehyde

-10 .3-4t.4

-290.8-2r3 .6

r059.1rr20.r

63.024.1

A. Calculate LG", LHo, and A,So or the alcohol dehydrogenase eaction understandard onditions.

B. Under standard conditions, what is the equilibrium constant for this reac-tion? Will the equilibrium constant ncrease or decrease s he temperatureis increased?

2-3. The equilibrium constant under standard onditions (1 atm, 298K, pH 7.0) forthe reaction catalyzed by fumarase,

Fumarate + HrO r--malate

is 4.00. At 310 K, the equilibrium constant s 8.00.

A. What is the standard ree energy change, LG", for this reaction?B. What is the standard enthalpy change, LHo, or this reaction? Assume he

standard enthalpy change s independent of temperature.C. What is the standard entropy change, ASo, at 298K for this reaction?D. If the concentration of both reactants s equal to 0.01 M, what is the free

energy change at 298 K? As the reaction proceeds to equilibrium, willmore fumarate or l-malate form?

E. What is the free energy change for this reaction when equilibrium isreached?

2-4. The following reaction is catalyzedby the enzyme creatine kinase:


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A.

PROBLEMS 39

Creatine + ATP S Creatine phosphate + ADP

Under standard conditions (1 atm, 298 K, pH 7.0, pMg 3.0 and an ionicstrength of 0.25 M) with the concentrations of all reactants equal to 10 mM,the free energy change, AG, for this reaction is 13.3 kJ/mol. what is thestandard ree energy change, AGo, for this reaction?What is the equilibrium constant or this reaction?The standard enthalpies of formation for the reactants are as follows:

B.C.

CreatineCreatine hosphate

ATPADP

-540 kJ/mol- 1510kJlmol-2982

kJ/mol-2000 kJ/mol

What is the standard enthalpy change, LII', for this reaction?D. What is the standard entropy change, A^So,or this reaction?

2-5. A. It has been proposed that the reason ce skating works so well is that thepressure rom the blades of the skates melts the ice. Consider this proposalfrom the viewpoint of phase equilibria. The phase change n question is

HrO(s) = H2O(/)

Assume that LH for this process s independent of temperature and pres-sure- nd is equal to 80 calJg.The change n volume, Lv, is about -9.1 x10-s Llg. The pressure exerted s the force per unit area. For a 180 poundperson and an area for the skate blades of about 6 square nches, the pres-sure s 30lb/sq. in. or about 2 atmospheres. With this information, calculatethe decrease n the melting temperature of ice caused by the skate blades.(Note that 1 cal = 0.04129 L.atm.) Is this a good explanation for why iceskating works?

B. A more efficient way of melting ice is to add an inert compound such asurea. We will avoid salt o save our cars.) The extent o which the freezingpoint is lowered can be calculated by noting that the molar free energy ofwater must be the same n the solid ice and the urea solution. The molarfree energy of water in the urea solution can be approximated asGriquia RT ln X*u,.n where X*ut".

s the mole fraction of water in the solu-tion. The molar free energy of the solid can be written as Gi,,o. Derive anexpression or the change n the melting temperature of ice by equating thefree energies n the two phases, differentiating the resulting equation withrespect o temperature, ntegrating from a mole fraction of 1 (pure solvent)to the mole fraction of the solution, and noting that ln X*ute.= ln(1 _ 4r"u)= -Xur"u. (This relationship is the series expansion of the logarithm forsmall values of 4rru. Since the concentration of water is about 55 M, thisis a good approximation.) With the relationship derived, estimate the de-


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40 ENTROPYND REE NERGY

crease n the melting temperature of ice for an 1 M urea solution. The heat

of fusion of water s 1440 callmol.

What is the maximum amount of work that can be obtained from hydrolyzingI mole of ATP to ADP and phosphate at 298 K? Assume that the concentra-

tions of all reactants re 0.01 M and AGo s -32.5 kJ/mol. If the conversion of

free energy to mechanical work is 1007o efficient, how many moles of ATP

would have o be hydrolyzedto lift a 100 kg weight I meter high?

2-6.


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I CHAPTER3

Applications f Thermodynam

termodinamica y cinetica de las reacionnes biologicas

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