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
ẳ
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
ẳ
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
ẳ
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
ẳ
V
dP
ẳ
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
ẳ
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.
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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
adenine/adenosine, 288, 294
adenosine diphosphate (ADP)
see ATP
adenosine triphosphate (ATP),
see ATP
adhesion, 50, 51, 89
adiabatic lapse rate, 426
ADP, see ATP; oxidative
phosphorylation;
photophosphorylation
advection, 446
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
algae, 5, 24; see also Chara;
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
Avogadro’s number, 103, 183, 547
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,
561, 564–569; see also activity,
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
radiationless transitions,
234
resonance transfer, 248–249
vibrational sublevels, 234,
236