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Appendix IV. Gibbs Free Energy and Chemical Potential

# Appendix IV. Gibbs Free Energy and Chemical Potential

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562

Appendix IV

We can represent the total entropy of the universe, Su, as the entropy of

the system under consideration, Ss, plus the entropy of the rest of the

universe, Sr. We can express this in symbols as follows:

S u ẳ Ss ỵ Sr

or

dSu ẳ dSs ỵ dSr

IV:1ị

An increase in Su accompanies all real processes----this is the most succinct

way of stating the second law of thermodynamics. Su is maximum at equilibrium.

The heat absorbed by a system during some process is equal to the heat

given up by the rest of the universe. Let us represent the infinitesimal heat

exchange of the system by dQs. For an isothermal reaction or change, dQs is

simply ÀdQr because the heat must come from the rest of the universe. From

the definition of entropy,1 dS = dQ/T, we can obtain the following relationship:

dSr ẳ

dQr

dQ

dU s ỵ PdV s

ẳ sẳ

T

T

T

IV:2ị

The last step in Equation IV.2 derives from the principle of the conservation

of energy for the case when the only form of work involved is mechanical----a

common assumption in stating the first law of thermodynamics. It is thus

possible to express dQs as the sum of the change in internal energy (dUs) plus

a work term (PdVs). The internal energy (Us) is a function of the state of a

system, i.e., its magnitude depends on the characteristics of the system but is

independent of how the system got to that state. PVs is also a well-defined

variable. However, heat (Qs) is not a function of the state of a system.

As we indicated previously, equilibrium occurs when the entropy of the

universe is maximum. This means that dSu then equals zero. By substituting

Equation IV.2 into the differential form of Equation IV.1, we can express this

equilibrium condition solely in terms of system parameters:





dU s ỵ PdV s

0 ẳ dSs ỵ

T

IV:3ị

or

TdSS ỵ dU s ỵ PdV s ẳ 0

Equation IV.3 suggests that there is some function of the system that has an

extremum at equilibrium. In other words, we might be able to find some

expression determined by the parameters describing the system whose derivative is zero at equilibrium. If so, the abstract statement that the entropy

of the universe is a maximum at equilibrium could then be replaced by a

statement referring only to measurable attributes of the system----easily

measurable ones, we hope.

1. This definition really applies only to reversible reactions, which we can in principle use to

approximate a given change; otherwise, dQ is not uniquely related to dS.

563

IV.B. Gibbs Free Energy

In the 1870s Josiah Willard Gibbs----perhaps the most brilliant thermodynamicist to date----chose a simple set of terms that turned out to have the

very properties for which we are searching. This function is now referred to

as the Gibbs free energy and has the symbol G:

G ¼ U ỵ PV TS

IV:4aị

which, upon differentiating, yields

dG ẳ dU þ PdV þ VdP À TdS À SdT

ðIV:4bÞ

Equation IV.4b indicates that, at constant temperature (dT = 0) and constant pressure (dP = 0), dG is

dG ẳ dU ỵ PdV TdS

at constant T and P

ðIV:5Þ

By comparing Equation IV.5 with the equilibrium condition expressed

by Equation IV.3, we see that dG for a system equals zero at equilibrium at

constant temperature and pressure. Moreover, G depends only on U, P, V, T,

and S of the system. The extremum condition, dG = 0, actually occurs when

G reaches a minimum at equilibrium. This useful attribute of the Gibbs free

energy is strictly valid only when the overall system is at constant temperature and pressure, conditions that closely approximate those encountered

in many biological situations. Thus our criterion for equilibrium shifts from a

maximum of the entropy of the universe to a minimum in the Gibbs free

energy of the system.

IV.B. Gibbs Free Energy

We will now consider how the internal energy, U, changes when material

enters or leaves a system. This will help us derive an expression for the Gibbs

free energy that is quite useful for biological applications.

The internal energy of a system changes when substances enter or

leave it. For convenience, we will consider a system of fixed volume and

at the same temperature as the surroundings so that there are no heat

exchanges. If dnj moles of species j enter such a system, U increases by

mjdnj, where mj is an intensive variable representing the free energy

contribution to the system per mole of species j entering or leaving.

Work is often expressed as the product of an intensive quantity (such

as mj, P, T, E, and h) times an extensive one (dnj, dV, dS, dQ, and dm,

respectively); that is, the amount of any kind of work depends on both

some thermodynamic parameter characterizing the internal state of the

system and the extent or amount of change for the system. In our current

example, the extensive variable describing the amount of change is dnj,

and mj represents the contribution to the internal energy of the system

per mole of species j. When more than one species crosses the boundary

of our system, which is at constant

volume and the same temperature as

P

the surroundings, the term j mj dnj is added to dU, where dnj is positive

if the species enters the system and negative if it leaves. In the general

564

Appendix IV

case, when we consider all of the ways that the internal energy of a

system can change, we can represent dU as follows:

X

mj dnj

IV:6ị

dU ẳ dQ PdV ỵ

j

We now return to the development of a useful relation for the Gibbs free

energy of a system. When dU as expressed by Equation IV.6 is substituted

into dG as given by Equation IV.5, we obtain

dG ¼ TdS À PdV ỵ

P

ẳ j mj dnj

P

j

mj dnj ỵ PdV TdS

IV:7ị

where dQ has been replaced by TdS. Equation IV.7 indicates that the

particular form chosen for the Gibbs free energy leads to a very simple

expression for dG at constant T and P----namely, dG then depends only on

mj and dnj.

To obtain an expression for G, we must integrate Equation IV.7. To

facilitate the integration we will define a new variable, a, such that dnj is

equal to njda, where nj is the total number of moles of species j present in the

final system; that is, nj is a constant describing the final system. The subsequent integration from a = 0 to a = 1 corresponds to building up the system

by a simultaneous addition of all of the components in the same proportions

that are present in the final system. (The intensive variable mj is also held

constant for this integration pathway, i.e., the chemical potential of species j

does not depend on the size of the system.) Using Equation IV.7 and this

easy integration pathway, we obtain the following expression for the Gibbs

free energy:

R1 P

R

RP

G ¼ dG ¼

j mj dnj ẳ 0

j mj nj da

IV:8ị

R

P

P

1

ẳ j mj nj 0 da ¼ j mj nj

The well-known relation between G and mj’s in Equation IV.8 can also be

obtained by a method that is more elegant mathematically but somewhat

involved.

In Chapter 6 (Section 6.1) we presented without proof an expression for

the Gibbs free energy (Eq. 6.1 is essentially Eq. IV.8) and also noted some of

the properties of G. For instance, at constant temperature and pressure, the

direction for a spontaneous change is toward a lower Gibbs free energy;

minimum G is achieved at equilibrium. Hence, DG is negative for such

spontaneous processes. Spontaneous processes can in principle be harnessed

to do useful work, where the maximum amount of work possible at constant

temperature and pressure is equal to the absolute value of DG (some of the

energy is dissipated by inevitable inefficiencies such as frictional losses, so

ÀDG represents the maximum work possible). To drive a reaction in the

direction opposite to that in which it proceeds spontaneously requires a free

energy input of at least DG (cf. Fig. 2-6).

565

IV.D. Pressure Dependence of mj

IV.C. Chemical Potential

We now

P examine the properties of the intensive variable mj. Equation IV.8

(G ¼ j mj nj ) suggests a very useful way of defining mj. In particular, if we

keep mj and ni constant, we obtain the following expression:

 

 

LG

LG

IV:9ị

mj ẳ

Lnj mi ;ni

Lnj T;P;E;h;ni

where ni and mi refer to all species other than species j. Because mj can

depend on T, P, E (the electrical potential), h (the height in a gravitational

field), and ni, the act of keeping mi constant during partial differentiation is

the same as that of keeping T, P, E, h, and ni constant, as is indicated in

Equation IV.9. Equation IV.9 indicates that the chemical potential of species

j is the partial molal Gibbs free energy of a system with respect to that

species, and that it is obtained when T, P, E, h, and the amount of all other

species are held constant. Thus mj corresponds to the intensive contribution

of species j to the extensive quantity G, the Gibbs free energy of the system.

In Chapters 2 and 3 we argued that mj depends on T, aj (aj = gjcj; Eq. 2.5),

P, E, and h in a solution, and that the partial pressure of species j, Pj, is also

involved for the chemical potential in a vapor phase. We can summarize the

two relations as follows (see Eqs. 2.4 and 2.21):

¼ mj ỵ RT ln aj ỵV j P ỵ zj FE ỵ mj gh

mliquid

j

IV:10aị

Pj

ỵ mj gh

Pj

IV:10bị

mvapor

ẳ mj ỵ R T ln

j

The forms for the gravitational contribution (mjgh) and the electrical one

(zjFE) can be easily understood. We showed in Chapter 3 (Section 3.2A) that

for

RT ln aj is the correct form for the concentration term in mj. The reasons





the forms of the pressure terms in a liquid (V j P) and in a gas [RT ln Pj =PÃj ]

are not so obvious. Therefore, we will examine the pressure dependence of

the chemical potential of species j in some detail.

IV.D. Pressure Dependence of mj

To derive the pressure terms in the chemical potentials of solvents, solutes,

and gases, we must rely on certain properties of partial derivatives as well as

on commonly observed effects of pressure. To begin with, we will differentiate the chemical potential in Equation IV.9 with respect to P:

 

Lmj

LP T;E;h;ni ;nj

"

#

 

L LG

¼

LP Lnj T;P;E;h;ni

T;E;h;ni ;nj

"

#

 

L LG

Lnj LP T;E;h;ni ;nj

T;P;E;h;ni

IV:11ị

566

Appendix IV

where we have reversed the order for partial differentiation with respect

to P and nj (this is permissible for functions such as G, which have welldefined and continuous first-order partial derivatives). The differential

form of Equation IV.4 (dG = dU + PdV + VdP À TdS À SdT) gives us a

suitable form for dG.

P If we substitute dU given by Equation IV.6

(dU ¼ TdS PdV ỵ j mj dnj , where dQ is replaced by TdS) into this

expression for the derivative of the Gibbs free energy, we can express

dG in the following useful form:

X

mj dnj

IV:12ị

dG ẳ VdP SdT ỵ

j

Using Equation IV.2 we can readily determine the pressure dependence

of the Gibbs free energy as needed in the last bracket of Equation

IV.11----namely, ðLG=LPÞT;E;h;ni ;nj is equal to V by Equation IV.12. Next, we

have to consider the partial derivative of this V withÀrespectÁto nj (see the last

equality of Eq. IV.11). Equation 2.6 indicates that LV=Lnj T;P;E;h;ni is V j, the

partial molal volume of species j. Substituting these partial derivatives into

Equation IV.11 leads to the following useful expression:

 

Lmj

ẳ Vj

IV:13ị

LP T;E;h;ni ;nj

Equation IV.13 is of pivotal importance in deriving the form of the pressure

term in the chemical potentials of both liquid and vapor phases.

Let us first consider an integration of Equation IV.13 appropriate for a

liquid. We will make use of the observation that the partial molal volume of a

species in a solution does not depend on the pressure to any significant

extent. For a solvent this means that the liquid generally is essentially incompressible. If we integrate Equation IV.13 with respect to P at constant T,

E, h, ni, and nj with V j independent of P, we obtain the following relations:

Z

Lmj

dP ẳ

LP

Z

mj

Z

mliquid

j

dmj ẳ mliquid

mj

j

Z

IV:14ị

V j dP ẳ V j dP ẳ V j P ỵ constant

where the definite integral in the top line is taken from the chemical potential of species j in a standard state as the lower Rlimit up to the general mj in a

liquid as the upper limit. The integration of V j dP leads to our pressure

term V j P plus a constant. Because the integration was performed while

holding T, E, h, ni, and nj fixed, the “constant” can depend on all of these

variables but not on P.

Equation IV.14 indicates that the chemical potential of a liquid contains

a pressure term of the form V j P. The other terms (mÃj , RT ln aj, zjFE, and

mjgh; see Eq. IV.10a) do not depend on pressure, a condition used throughout this text. The experimental observation that gives us this useful form for

mj is thatV j is generally not influenced very much by pressure; for example, a

liquid is often essentially incompressible. If this should prove invalid under

567

IV.D. Pressure Dependence of mj

certain situations, V j P would then not be a suitable term in the chemical

potential of species j for expressing the pressure dependence in a solution.

Next we discuss the form of the pressure term in the chemical

potential of a gas, where the assumption of incompressibility that we

used for a liquid is not valid. Our point of departure is the perfect or

ideal gas law:

P j V ¼ nj R T

ðIV:15Þ

where Pj is the partial pressure of species j, and nj is the number of moles

of species j in volume V. Thus we will assume that real gases behave like

ideal gases, which is generally justified

applications. Based

À for biological

on Equations IV.15 and 2.6 ẵV j ẳ LV=Lnj T;P;E;h;ni , the partial molal

volume V j for gaseous

Pspecies j is RT/Pj. We also note that the total

pressure P is equal to j Pj, where the summation is over all gases present

(Dalton’s law of partial pressures); hence, dP equals dPj when ni, T, and V

are constant. When we integrate Equation IV.13, we thus find that the

chemical potential of gaseous species j depends on the logarithm of its

partial pressure:

Z mvapor

Z

Z

Z

j

Lmj

RT

Ã

dP ¼

dmj ¼ mvapor

m

V

dP

dPj

j

j

j

LP

Pj

mj

IV:16ị

ẳ R T ln Pj ỵ constant

where the constant can depend on T, E, h, and ni but not on nj (or Pj). In

particular, the “constant” equals ÀR T ln Pj ỵ mj g h, where P*j is the saturation partial pressure for species j at atmospheric pressure and some particular temperature. Hence, the chemical potential for species j in the vapor

) is

phase (mvapor

j

mvapor

ẳ mj ỵ R T ln Pj R T ln Pj ỵ mj g h ẳ mj ỵ R T ln

j

Pj

ỵ mj g h

Pj

IV:17ị

which is essentially the same as Equations 2.21 and IV.10b.

We have defined the standard state for gaseous species j, mÃj , as the

chemical potential when the gas phase has a partial pressure for species j

(Pj) equal to the saturation partial pressure (PÃj ), when we are at atmospheric

pressure (P = 0) and the zero level for the gravitational term (h = 0), and for

some specified temperature. Many physical chemistry texts ignore the gravitational term (we calculated that it has only a small effect for water vapor;

see Chapter 2, Section 2.4C) and define the standard state for the condition

when Pj equals 1 atm and species j is the only species present (P = Pj). The

chemical potential of such a standard state equals mÃj À R T ln PÃj in our

symbols.

The partial pressure of some species in a vapor phase in equilibrium with

a liquid depends slightly on the total pressure in the system----loosely speaking, when the pressure on the liquid is increased, more molecules are

squeezed out of it into the vapor phase. The exact relationship between

568

Appendix IV

the pressures involved, which is known as the Gibbs equation, is as follows

for water:

Lln Pwv ị V w

RT

LP

IV:18ị

or

LPwv V w

¼

LP

V wv

where the second equality follows from the derivative of a logarithm

[L ln u/Lx = (1/u)(Lu/Lx)] and the ideal or perfect gas law [PwvV = nwvRT

(Eq. IV.15), so LV=Lnwv ¼ V wv ¼ RT=Pwv ]. Because V w is much less than is

V wv , the effect is quite small (e.g., at 20 C and atmospheric pressure,

V w ¼ 1:8 Â 10À5 m3 molÀ1 and V wv ¼ 2:4 Â 10À2 m3 molÀ1 ). From the first

equality in Equation IV.18, we see that V w dP equals RTd ln Pwv. Hence,

if the chemical potential of the liquid phase (mw) increases by V w dP as an

infinitesimal pressure is applied, then an equal increase, RTdln Pwv,

occurs in mwv (see Eq. IV.10b for a definition of mwv), and hence we will

still be in equilibrium (mw = mwv). This relation can be integrated to give

RT ln Pwv ẳ V w P ỵ constant, where the constant is RT ln P0wv and P0wv is the

partial pressure of water vapor at standard atmospheric pressure; hence,

RT ln Pwv =P0wv is equal to V w P, a relation used in Chapter 2 (see Section

2.4C). We note that effects of external pressure on Pwv can be of the

same order of magnitude as deviations from the ideal gas law for water

vapor, both of which are usually neglected in biological applications.

IV.E. Concentration Dependence of mj

We will complete our discussion of chemical potential by using Equation

IV.17 to obtain the logarithmic term in concentration that is found for mj in a

liquid phase. First, it should be pointed out that Equation IV.17 has no

concentration term per se for the chemical potential of species j in a gas

phase. However, the partial pressure of a species in a gas phase is really

analogous to the concentration of a species in a liquid; e.g., PjV = njRT for

gaseous species j (Eq. IV.15), and concentration means number/volume and

equals nj/V, which equals Pj/RT.

Raoult’s law states that at equilibrium the partial pressure of a particular

gas above its volatile liquid is proportional to the mole fraction of that

solvent in the liquid phase. A similar relation more appropriate for solutes

is Henry’s law, which states that Pj in the vapor phase is proportional to the

Nj of that solute in the liquid phase. Although the proportionality coefficients in the two relations are different, they both indicate that Pvapor

j

depends linearly on N solution

. For dilute solutions the concentration of species

j

j, cj, is proportional to its mole fraction, Nj (this is true for both solute and

changes from one equilibrium condition to ansolvent). Thus when Pvapor

j

because mliquid

is equal to mvapor

at

other, we expect a similar change in csolution

j

j

j

equilibrium.

In particular, Equation IV.17 indicates that mvapor

depends on





j

R T ln Pj =PÃj , and hence the chemical potential of a solvent or solute

IV.E. Concentration Dependence of mj

569

should contain a term of the form RT ln cj, as in fact it does (see Eqs. 2.4 and

IV.10). As we discussed in Chapter 2 (Section 2.2B), we should be concerned

about the concentration that is thermodynamically active, aj (aj = g jcj; Eq.

2.5), so the actual term in the chemical potential for a solute or solvent is

RT ln aj, not RT ln cj. In Chapter 3 (Section 3.2A), instead of the present

argument based on Raoult’s and Henry’s laws, we used a comparison with

Fick’s first law to justify the RT ln aj term. Moreover, the Boyle–Van’t Hoff

relation, which was derived assuming the RT ln aj term, has been amply

demonstrated experimentally. Consequently, the RT ln aj term in the chemical potential for a solute or solvent can be justified or derived in a number

of different ways, all of which depend on agreement with experimental

observations.

Index

Page numbers in boldface refer to figures or structural formulas (entry also usually mentioned on that text page).

A

Ames/A, 377, 394–396, 397, 403,

419, 420

abscisic acid (ABA), 373, 427,

429

abscission, leaf, 456, 478, 485

absolute humidity, 386

absolute zero, 18, 326

absorbance, 216

absorptance, 325, 329

leaf, 328, 332, 352

absorption band, 214, 218, 266

absorption coefficient, 214–216

foliar, 454–456

molar, 216

pigments, 216, 221, 234, 239,

243, 244, 266, 267

absorption spectrum, 192, 209, 213

carotenoids, 240

Chl a, 234

phycobilins, 243

phytochrome, 221

absorptivity, 325

accessory pigments, 238, 246,

248–249, 296

acclimation, photosynthesis, 426,

431

action spectrum, 209, 214, 219

O2 evolution, 256–257, 258

phytochrome, seed

germination, 222–223

activation energy, 135–136

active transport, 129, 130–131,

138, 141, 144

carriers, 149–150, 152

energy required, 142–143,

293

membrane resistance, 131

Michaelis–Menten

formalism, 149–151

Na–K pump, 143, 149

Nitella, 140–142

phloem, 483

proton, 127, 131, 148,

371–372

activity, thermodynamic, 61, 63,

106–107

activity coefficient, xv, 61, 85, 115

ions, 106–108

water, 67, 70

activity (concentration),

chemical potential, xv, 60–61,

108, 569

see ATP

see ATP

phosphorylation;

photophosphorylation

aerenchyma, 20

Agave, 356, 410, 421–422, 424

air, 550

composition, 387

density, 53, 549

thermal conductivity,

546–547

viscosity, 549

air boundary layer, see boundary

layer

air gap, root-soil, 490–492

air packets, see eddy; eddy

diffusion coefficient

albedo, 325

alcohols, reflection coefficients,

162, 169

Chlorella; Nitella

excitation transfer, 249

pigments, 233, 240, 241,

245, 258

altitude, see elevation

amino acids, 22, 478

anion, mobility, 117–119

anode, 60

antennae, pigments, 245, 258,

259, 272

antiporter, 145, 301, 309

aphid stylet, 478

apoplast, 10, 32, 83, 470, 476

aquaporin, 145, 146, 375

Arrhenius equation, 136

Arrhenius plot, 136–137

atmosphere, transmittance, 325

atmospheric CO2, 188, 189, 230,

405, 432, 499

atmospheric pressure/diffusion

coefficients, 20

atomic orbitals/theory, 196

ATP, 19, 23, 46, 58, 271, 289

active transport, 143, 293

bonds, 288, 292

energy currency, 183, 278,

286, 292–293

formation reaction, 230, 276,

287–293, 298

proton ratio/involvement

chloroplasts, 300–301

mitochondria, 309–310

synthase (ATPase), 148,

288, 302, 308

turnover, 278

ATP synthase (ATPase), see ATP

B

bacteria, 25, 233, 241, 251, 259,

293, 301, 307

bacteriochlorophyll, 233, 251, 259

bandwidth, absorption band, 237

basidiomycetes, transpiration,

391

Beer’s law, 176, 215–217, 453

bicarbonate, 398–399

biochemical reaction, 149, 228,

230, 255, 290

biodiversity, 497, 499

biosphere, energy flow, 177, 278,

310, 313, 315

blackbody, 185, 190, 191, 311,

326–327, 329

blue light, 181, 182–183, 185, 200,

216, 352, 372

bleaching (absorption band), 260

bluff body, 335

boundary layers, 338–339

heat flux density, 341

Boltzmann energy distribution/

factor, 100, 132–133, 134,

184, 185, 235

vibrational sublevels, 209, 211

bond energy, chemical, 183, 231

Bouguer–Lambert–Beer law, 216

boundary layer, 26, 318, 333

air, 336–339, 377

conductance/resistance/flux

density, 364, 368–370,

383, 388

cylinder, 338–339

571

572

flat plate, 336–337, 343

leaf, 318, 336–338, 373

sphere, 338–339

Bowen ratio, 448

Boyle–Van’t Hoff relation, 44,

74–76

chloroplasts, 76–78, 167–168

irreversible

thermodynamics, 167–168

Brownian movement, 12, 185

buoyancy, 334, 344, 426, 444

bundle sheath cells, 408, 409, 477

C

C3, 303

enzymes, 406, 409

photosynthesis, 404, 406, 407,

408, 409, 410, 413

PPF, 405, 419

WUE, 425, 429–432, 446

C4, 303

anatomy, 408, 477

enzymes, 409

photosynthesis, 404, 408–409,

410, 413

PPF, 419

WUE, 425, 429–432, 446

C13/C12, 410

cactus, 342, 353–354, 355, 495

calcium, 33, 127, 148, 289, 373

callose, 477

calomel electrode, 286

Calvin (Calvin–Benson) cycle,

407, 409

CAM, 73, 409, 410, 421–422, 423,

424–425, 432

cambium, vascular, 8, 9

candela (candle), 185, 186, 312,

553

canopy, 442, 451

capacitance, 104

membrane, electrical,

104–106, 130

water storage, 438, 492–495

leaf, 390, 493, 496

tree trunk, 492–493

capacitor, 104–105

capillary/capillary rise, 50, 51–53,

461

contact angle, 50, 51–52

height, 44, 52–53

xylem, 53–54

carbohydrate, 230–231, 313

carbon dioxide, 12, 20

atmospheric level, 188, 189,

230, 405, 432, 499

cellular conductance, 419

compensation point, 412–414

concentration

Index

above canopy, 447

leaf, 413, 416

plant community,

452, 456–459

units, 391, 405

conductance/resistance,

392–393, 394, 399–403,

404, 416, 418–419

diffusion coefficient, 20, 393,

397, 545

elevated, 432, 497–499

fixation, see photosynthesis

flux density

above canopy, 443,

446–447

leaf, 364, 416–418

plant community,

451, 456–459

Michaelis–Menten,

404, 408

partition coefficient,

398–399

permeability coefficient, 397,

401, 500

photosynthesis, 230–231,

253

processing time,

254–255

solubility, 398–399

units, 391, 405

carbonic anhydrase, 398, 408

␣-carotene, 240

␤-carotene, 240, 241, 249

absorption spectrum, 240

carotenes, 240

carotenoids, 238–242, 245

absorption bands, 240

photochemistry/

photoprotection,

241–242

carrier, 24, 144–145, 149–150,

151

Casparian strip, 9, 10, 470

cathode, 60

cation, mobility, 117–119

cavitation, 54, 473, 489

cell sap, osmotic pressure, 68–69,

77, 80, 333

cellular conductance, 419

cellulose, 3, 33

microfibrils, 32, 33, 37, 39

Young’s modulus, 39, 40

cell growth, 44, 93–95

cell wall, 3, 4, 10, 31–32

composition, 33–34

diffusion across, 34–35,

399–400

Donnan potential, 127–129

elasticity, 2, 39–40, 80–81

hydrostatic pressure in, 32,

88–90

interstices, 4, 32, 34

water relations, 53,

70, 88–90, 385, 387,

474–475

microfibrils, 32, 33, 37, 39

middle lamella, 32, 35, 470

permeability, 34–36, 399–400

pits, 35

plastic extension, 40, 94

Poiseuille flow, 475–476

pressure, 88–91, 127

primary, 32–33, 35, 470

resistance, CO2, 393, 399–400

secondary, 32–33, 35, 37, 470

stress-strain relations, 37–39

water, 34, 53, 70, 89–91, 127

water potential, 78, 88–91

yield threshold, 94

Young’s modulus, 39, 40

Celsius, 18

central vacuole, 4, 72–74, 81

CFo/CF1, 299, 302, 303

channel, membrane, 145–148,

299, 308

potassium, 147–148, 371–372

Chara, 5, 38, 39

growth, 93–94

membranes, 110–111, 162

charge number, 103

chelate, 289

chemical energy/electrical

energy, 283–285

chemical potential, 44, 56–66,

102–103, 113, 115–116, 279,

thermodynamic;

concentration; electrical

term; gravitational term;

pressure;

standard

state

protons, 297–301, 307–309

water, 70

water vapor, 84–87

chemical reaction/conventions,

280–281, 290

chemiosmotic hypothesis

chloroplasts, 299–301

mitochondria, 307–309

chilling-sensitive plants, 136–137

Chl a, 232, 233

absorption spectrum,

233–234, 237

excitation transfers, 246

fluorescence, 234–236, 251

photosystems, 245, 258–259

234

resonance transfer, 248–249

vibrational sublevels, 234,

236

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Appendix IV. Gibbs Free Energy and Chemical Potential

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