J. Phys. Chem. 1995, 99, 11028-11034
11028
Standard Transformed Formation Properties of Carbon Dioxide in Aqueous Solutions at Specified pH Robert A. Alberty Department of Chemistry, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139 Received: February 20, 1995@
When a biochemical reaction is studied at a specified pH, the transformed Gibbs energy G' provides the criterion of equilibrium, and the apparent equilibrium constant F yields the standard transformed reaction Gibbs energy ArG'". The Legendre transformation is required to make pH a natural variable. The concentrations of reactants in the expression for the apparent equilibrium constant are sums of concentrations of species, and the reactants each have a standard transformed Gibbs energy of formation A&'" and a standard transformed enthalpy of formation AfH" at the specified T, P, pH, and ionic strength. These formation properties of a reactant can be calculated if the standard Gibbs energies and standard enthalpies of formation of the species involved are known. Here the standard formation properties of the species C02, H2CO3, HCO3-, and C032- are calculated using data from the NBS tables and ArG" and A T Pfor the reaction H2O C02 = H2C03. These values are then used to calculate ArG'"(TotCO2) and AfH'O(TotC02), where TotCO2 represents the sum of the four species in aqueous solution at a specified pH. The same values for the standard transformed properties of TotCOz can be calculated from the NBS tables alone, but the advantage of using information on the hydration reaction in addition is that the equilibrium concentrations of the species C02 and H2CO3 can be calculated and are of interest kinetically. This analysis leads to values for a pH-dependent Henry's law constant. The application of these ideas to the formate dehydrogenase reaction is discussed in detail.
+
Introduction Apparent equilibrium constants K' of biochemical reactions involving carbon dioxide are usually expressed in terms of the total concentration [TotCOa] of the species C02, H2C03, HC03-, and C032- in solution at the specified T, P,pH, and I (ionic strength), but they can also be expressed in terms of the partial pressure of C02(g) at equilibrium. For example, the biochemical equation for the formate dehydrogenase reaction (EC 1.2.1.2)' at specified pH is H 2 0 4- Formate
+ NAD,,
= TotCO,
+ NADred
(1)
where NAD,, represents oxidized nicotinamide adenine dinucleotide and NADred represents the reduced form; these are each single species in the neutral pH range. Formate represents the sum of the acid and anion forms. The expression for the apparent equilibrium constant for reaction 1 is
F=
[TotCOJ [NAD,] [Formate][NAD,,]
Alternatively, the biochemical equation can be written without the H20 on the left side by using Formate -I- NAD,, = c02(g)
+ NADred
(3)
discussed later. Apparent equilibrium constants are functions of the pH, and they may also be functions of the free concentrations of metal ions, which are bound by the reactants. Biochemical equations like eqs 1 and 3 are written in terms of sums of species, and they do not balance hydrogen and metal ions since the pH and free concentration of a metal ion are specified, although they do balance other elements2 When the pH is taken to be an independent variable, the Gibbs energy G does not provide the criterion of equilibrium, and it is necessary to define a transformed Gibbs energy G' with a Legendre transform involving the specified chemical potential of the hydrogen i ~ n . ~When - ~ the pH is specified, the various species of a reactant become pseudoisomers with the same transformed chemical potential.6 When carbon dioxide is a reactant, the sum of species, represented by TotC02, has a standard transformed Gibbs energy of formation AG'" and a standard transformed enthalpy of formation A$€", which can be calculated at the desired T, P , pH, and I (ionic strength) from the standard thermodynamic properties of the four aqueous species involved.
Definition of a Transformed Gibbs Energy G The fundamental equation for the Gibbs energy at specified T and P for an aqueous solution containing the various species of C02 in water and hydrogen ion is given by
with apparent equilibrium constant (4)
where the standard state concentration and pressure are represented by co = 1 M and PO = 1 bar. These apparent equilibrium constants are related by a pH-dependent Henry's law constant @Abstractpublished in Advance ACS Abstracts, June 15, 1995.
where pi is the chemical potential of species i and ni is its amount. The thermodynamic properties are for a specified ionic strength. A biochemical reaction is studied as a specified pH, and so it is necessary to make a Legendre transform' to define a transformed Gibbs energy that has pH as a natural variable.
0022-3654/95/2099-11028$09.00/0 0 1995 American Chemical Society
Standard Transformed Formation Properties of C02
J. Phys. Chem., Vol. 99,No. 27, 1995 11029
The objective in making the Legendre transform in the present case is to make C02, H2C03, HC03-, and co32-pseudoisomers. Pseudoisomers have the characteristic that at specified values of the natural variables of G' the distribution of pseudoisomers is independent of the concentrations of other reactants. However, C02, H2C03, HC03-, and C032- cannot be made to be pseudoisomers unless p(H2O) is brought into the Legendre transform because C02 differs from the other three species in that it lacks H20. The transformed Gibbs energy G' of the system is therefore defined by G' = G - n'(water)p(H,O)
- n'(H)p(H+)
(6)
where n'(water) is given by (7) and n'(H) is the total amount of hydrogen in the system,
Chi -
NH,O(
i)p(H20) - NH(i)p(H+)]ni= >:ni
(9) where the transformed chemical potential of species i is defined by
The transformed chemical potentials of the four species involving C02 are given by P'(C02) = P(C02)
+ P(H2O) + 2P(H+>
$(H2C03)
= p(H2C03) - 2p(Hf>
P'(HCO,-)
= P(HCO,-)
- P(H+)
(1 1)
p(H2C03)= p'(TotC0,)
+ 2p(Hf)
+ 2p(Hf)
(16)
p(CO?-) = p'(TotC0,) (18) When these four equations are substituted into eq 5 , the
+ (19)
where n'(TotC0,) = n(C02)
+ n(H2C03)+ n(HC03-) + ~ ( c o , ~ - ) (20)
n'(water) = n(H20) - n(C02)
HI = n ( ~ ++) ~(Hco,-)
(21)
+ ~~(H,co,) + 2n(co2)
(22)
Taking the differential of G' in eq 6 and substituting eq 19 yields
- n'(water)
dp(H20) -
am+>
n'(W (23) At specified pH, dp(H+) = 0, and in dilute aqueous solutions dp(H20) = 0,so that (dG')~,p,,H,(H,O, = ~'(TotcO2)&'(TotCOJ
(24)
Thus, when T, P, and pH are specified in dilute aqueous solutions, TotC02 can be taken as a single component with a transformed chemical potential given by aG'
= p'(TotC0,)
(25)
(an'(Totco2))~,P,~H,,H20~
This is the reason that a sum of amounts of pseudoisomers can be treated as a single reactant in treating equilibria in biochemical reactions with Legendre-transformed thermodynamic quantities. In making derivations with the fundamental equation, chemical potentials and transformed chemical potentials are used, but in making actual calculations, molar Gibbs energies of formation A&i and molar transformed Gibbs energies of formation A&' are used. For ideal solutions of a species, the molar Gibbs energy of formation is given by
+ RT ln([i]/c")
(26) The molar transformed Gibbs energy of formation of a species is given by
(13)
(15)
4-p(H20) &'(water)
PW+>W H )
A F i = A@:
(12)
~ ' ( c o , ~ - ) = p(c0,2-) (14) When T, P,pH, and p(Hz0) are specified at equilibrium, C02, HzC03, HC03-, and co32-are pseudoisomers, and it can be shown6 that at equilibrium the transformed chemical potentials of these species are all equal and can be defined as the transformed chemical potential p'(TotCO2) of the pseudoisomer group. Consequently, eqs 11-14 can be rearranged to obtain the following expressions for the chemical potentials of the species containing C02: p(C02) =p'(T0tCOz) - p(H20)
(dG)T,P= p'(TotC0,) dn'(TotC0,)
(dG')T,p = p'(TotC0,) dn'(TotC0,)
N ~ ~ o ( 1is' )the number of water molecules in a molecule of species i minus 1, and N H ( ~is) the number of hydrogen atoms in a molecule of species of i. The value of NH20(i)for COZis -1, and it is 0 for the species HzC03, and co32-.NH(i) is 2, 2, 1, and 0 for C02 (+H20), H2C03, HC03-, and C032-. Note that n'(water) is used rather than n'(H20) because H2O represents a species, and n'(H) is used because H+ represents a species. The Gibbs energy of the system is given by G = &ni, and when this is substituted in eq 6 with eqs 7 and 8, we obtain
G*=
differential of the Gibbs energy is given by
w:= A@,'" + RT ln([i]/c")
(27) The relationship between A&i and A&' is given by eq 10, which becomes A@? = A@:
- NH2,(i)AP(H20(1)) -
+
NH(i)[AP(H+) RT ln ([H+]/c")] (28) when eqs 26 and 27 are substituted and RT ln([i]/c")is canceled on both sides. The standard Gibbs energy of formation A&"(H+) of hydrogen ion is 0 at zero ionic strength by convention, but it is not 0 at higher ionic strengths. The corresponding equation for the standard transformed enthalpy of formation of species i is
A$:" = A,&:
- NH20(i)AfH"(HzO(1))- NH(i)AP(H+)
(29) In the absence of more detailed information about the composition of the buffer and the corresponding parameters, the standard formation properties of species can be adjusted to the desired ionic strength using the extended Debye-Huckel
Alberty
11030 J. Phys. Chem., Vol. 99, No. 27, 1995
t h e ~ r y . *The ~ ~ standard enthalpy of formation and the standard Gibbs energy of formation of species i at ionic strength Z and 25 "C can be calculated from the standard formation properties at zero ionic strength using A@:(I)
= A@:(Z
A@:(I)
A@:(Z
= 0)
+ 1.4775~,21"~/( 1 + BZL/2) (30)
= 0) - 2.91482zfZ1"/(1
+ BZ'I2)
(3 1)
where the formation properties are expressed in kJ mol-', Zi is the charge on species i, and B = 1.6 L'12 mol-'12. The effect of ionic strength on ArW for a chemical reaction at 25 "C is obtained by substituting eq 30 in ArW(I) = CViAfH?:
+ (1.47751"')cvizf/( 1 + BZ1/2)
A,.W'(Z) = A,.W'(Z=O)
(32) where vi is the stoichiometricnumber of species i in the reaction, CvizF is the change in z: in the reaction, and reaction properties are expressed in kJ mol-'. The effect of ionic strength on the equilibrium constant K for a chemical reaction at 25 "C is obtained by substituting eq 3 1 in ArG"= CviArGiO = -RT 1nK: log K(I) = log K(Z=O)
+ ( 0 . 5 1 0 6 5 Z 1 ~ 2 ~ v i z ~+) /BZ112) (1 (33)
Since C02, H2CO3, HC03-, and C032- are pseudoisomers at a specified pH in dilute aqueous solution, the equation that is used to calculate the standard Gibbs energy of formation of a group of isomers at chemical equilibrium can be used to calculate the standard transformed Gibbs energy of formation of a group of pseudoisomers at specified pH.",'' The standard transformed Gibbs energy of formation of TotCO2 under the conditions discussed here is given by
+
A,.G'"(TotCO,) = -RT ln{e~p[-A@'~(C0~)/RT] exp[-AfG'"(H2C03)/Rfl
TABLE 1: Standard Formation Properties from the NBS Tables at 298.15 K, 1 bar, and I = 0 A&P/kJ mol-l AG0Mmol-' COz(g) -393.509 -394.359 -385.98 COdao) -413.80 -623.1 1" HzCOdao) -699.63" HCOy-(ao) -69 1.99 -586.77 -527.81 COP(ao) -677.14 -285.830 -237.129 HzO(1) " These values were adjusted from the values -699.65 and -623.08 in the NBS tables to make them agree with the convention of these tables that these values are equivalent to COZ(ao) + H20(1). TABLE 2: Standard Formation Properties of Species in Aqueous Solution at 298.15 K, 1 bar, and I = 0 AfPM mol-l A&OM mol-'
c02(sP)
-4 13.81 -694.91 -691.99 -677.14
HzCOdsp) HC03-(sp) co32-(sP)
-385.97 -608.33 -586.77 -527.81
are based on the convention that ArG" = ArW = 0 for H20(1)
+ CO,(ao) = H2C03(ao)
(38)
This convention is used when a species exists as a hydrated and unhydrated form in aqueous solution and there is no way to distinguish between these forms experimentally, as is currently the case for a dozen similar reactions in the NBS tables. In the case of carbon dioxide, it is possible to distinguish between the species C02 and H2CO3 because the conversion of one into the other is slow enough that the rate constants have been determined. The history of these measurements is described by Kern.I3 In order to distinguish between the use of the molecular formulas CO2 and H2CO3 in the NBS tables and in this paper, the reaction forming the species H2CO3 from the species C02 is written as
+ exp[-A@'"(HC03-)/Rfl + e~p[-A@'"(CO~~-)/Rfll)(34)
with the equilibrium constant for the hydration reaction written as
where the sum is a kind of partition function. The equilibrium mole fraction ri of any one of the pseudoisomers i in the pseudoisomer group is given by
ri = exp{[A@'"(TotCO,)
- A,.G'"(i)]/RT)
(35)
which is an analog of the Boltzmann distribution. The standard transformed enthalpy of formation of the pseudoisomer group is a mole fraction weighted average and is given by
The standard transformed entropy of formation of the pseudoisomer group is given by A$'"(TotCO,)
= [A$P(TotCO2) - A@'"(To~CO~)]/T
(37) Calculation of the Standard Formation Properties of Species of Aqueous Carbon Dioxide The entries in the NBS tablesI2 involved in the present calculations are given in Table 1. The values for COz(a0) and HzC03(ao) in this table are not for species because the values for COz(a0) are for the sum of unhydrated and hydrated species. The values for H2CO3(ao) are not for the species because they
Edsall14has reviewed the thermodynamic and kinetic measurements that have been made on this reaction and recommends A,Go = 14.77 kJ mol-' and ArW = 4.73 kJ mol-' at 298.15 K, where molar, rather than molal, concentrations are used. Thus, Kh (298.15K) 2.584 x Note that R = 8.3143 J K-' mol-' is used in the calculations here because the NBS tables is based on this value. These values also used by Edsall and Wyman.15 The value of Kh is from conductance measurements by Wissbrun, French, and PattersonI6 at high electric fields (Wein effect), and ArW is from measurements by Roughton." The standard properties for reaction 39 can be used with the standard formation properties in Table 1 to calculate the standard formation properties of COz(sp) and HzCOdsp), as described below. The values calculated here are given in Table 2. The formation values for HC03-(sp) and C032-(sp) are the same as in the NBS tables. The standard formation properties of COl(sp) can be calculated by considering the Henry's law constant from the NBS tables. The phase distribution can be written as a reaction:
Standard Transformed Formation Properties of C02
J. Phys. Chem., Vol. 99, No. 27, 1995 11031
with ArW = 20.29 W mol-' and ArG" = -8.38 kJ mol-' at 298.15 K. The Henry's law constant is
where Po = 1 bar is the standard state pressure and mo = 1 mol kg-' is the standard state molality. Now we need the value of a Henry's law constant K Hin~terms ~ of the species COz(sp), which is the equilibrium constant for the reaction
so that the equilibrium constant expression is
Now consider the dissociation of H2CO-,(sp): H2C03(sp)= Hf(ao)
+ HC0,-(sp)
(51)
The equilibrium constant is CO,(SP) = CO,(g)
(43)
The Henry's law constant in terms of species is defined by Note that where co = 1 M is the standard state molarity. These two Henry's law constants are related by (45) Thus, K H =~29.466 ~ at 298.15 K. The standard Gibbs energy of reaction for reaction 43 is given by ArGo = A&O(co,(g)>
-~Q(CO~(SP))
Gibbs energy of formation of CoZ(sp): AG"(CO,&p)) = A&"(C02(g)) RT In 29.466 = -394.36 8.39 = -385.97 kJ mol-'. Note that this value is not very different from the NBS value for A&"(COz(ao)) because of the small equilibrium concentration of HzCO3(sp). If AfHO is independent of T near 298.15 K for KHand Kh, as assumed here, ArW for reaction 43 will depend on temperature, as indicated by eq 45. However, since Kh is small, this effect is very small. ArW for reaction 43 can be calculated by differentiation of eq 45, but since the effect is small, it is easier to make a numerical calculation over a small temperature range = 20.30 kJ mol-' and near 298.15 K. This yields A,HO(KH~~) hfW(C02(sp)) = -413.81 W mol-'. Note that this value is not very different from the NBS values for AfW(COz(ao)). The standard formation properties of H~C03(sp)can be calculated from a consideration of the first acid dissociation constant K I calculated using the NBS tables. Probably the best way to calculate K I from the NBS tables is to write the chemical equation as
+
H20(1)
+ CO,(ao) = Hf(ao) + HC03-(ao)
(47)
since the standard formation properties of CO2(ao) are known from the Henry's law constant. Thus,
K, =
[Hf (ao)][HCO
- ao (
[CO,(ao)lm"
= 4.2992 x lo-'
(48)
since H20 is treated as the solvent and has unit activity in the limit of zero concentration of C02. The standard reaction Gibbs energy and standard reaction enthalpy for reaction 47 and A,G" = 36.34 kJ mol-' and A,W = 7.64 M mol-'. An altemative way to write the chemical equation and the equilibrium constant for the first acid dissociation is H,CO,(ao) = H+(ao)
+ HCO,-(ao)
The value of KH,CO,at 298.15 K is readily calculated and is 1.6681 x so that A,G" = 21.56 M mol-' for reaction 51, and A&"(HzCOs(sp)) is -608.33 kJ mol-', according to
(46)
This makes it possible to calculate the value of the standard
+
(53)
(49)
The value of AfW(HzC03(sp))can be calculated by computing KH,CO,with eq 53 at two temperatures close to 298.15 K. This yields ArG" = 2.92 W mol-'. Thus, ApH"(H2C03(sp))= A,H(HCO,-(sp))
- 2.92 kJ mol-'
=
-694.91 kJ mol-' (55) Now we have the standard formation properties for the species shown in Table 2.
Calculation of Standard Transformed Gibbs Energies of Formation of Species of TotCOz at Various pH Values The standard transformed properties of the species calculated using eqs 28 and 29 are given in Table 3 at five pH values at two ionic strengths. Table 3 shows that the most stable pseudoisomer at pH 5-6 is C02 and that the most stable pseudoisomer at pH 7-9 is HC03-. The standard transformed enthalpies of formation of species are not a function of pH. The standard transformed Gibbs energies of formation of TotCO2 were calculated using eq 34. Note that they are necessarily more negative than the A&? of the most stable pseudoisomer. The standard transformed enthalpies of formation of TotCO2 were calculated using eqs 35 and 36; they are functions of pH because of the change in the relative amounts of the four species with pH. The equilibrium mole fractions of the species within the pseudoisomer group at these five pH values and two ionic strength values, calculated from their standard transformed Gibbs energies of formation using eq 35, are given in Table 4.
Apparent Henry's Law Constant for Carbon Dioxide at Specified pH
In an earlier section the Henry's law constant KHof 29.39 at 298.15 K was calculated from the NBS tables, and the Henry's law constant in terms of species K Hat ~298.15 ~ K was found to be 29.466 using Kh. The solubility of c02(g) in a buffer will increase with the pH. Now we want to consider the apparent
11032 J. Phys. Chem., Vol. 99, No. 27, 1995
Alberty
TABLE 3: Standard Transformed Gibbs Energies of Formation A@'' and Standard Transformed Enthalpies of Formation ArH'' of Species and of TotCOl in kJ mol-' at 298.15 K, 1 bar, Five pH Values, and Two Ionic Strengths?
Note that ArH:'
6 -554.61 -552.99 -539.84 -538.22 -552.52 -552.52 -527.81 -531.05 -555.33 -554.49 -699.3 3 -696.62
5
Ihf
-566.02 -564.4 -55 1.25 -549.63 -558.23 -558.23 -527.81 -53 1.05 -566.13 -564.61 -699.31 -699.8 does not depend on pH.
TABLE 4: Equilibrium Mole Fractions ri of Species within the Pseudoisomer Group at 298.15 K, 1 bar, Five pH Values, and Two Ionic Strengths IN
coz
0
H2C03
0
0.25 0.25
HCO3-
0
0.25 co32-
0
0.25
5
0.9563 0.9212 0.0025 0.0024 0.0413 0.0764 O.oo00 O.oo00
6 0.6972 0.5458 0.0018 0.0014 0.3010 0.4527 O.oo00 O.oo00
PH 7 0.1879 0.1074 0.0005 0.0003 0.8112 0.8908 O.OOO4
0.0015
8 0.0225 0.0117
9 0.0022 0.0010
o.oo00 o.oo00
o.oo00 o.oo00
0.9728 0.9715 0.0046 0.0168
0.9532 0.8517 0.0446 0.1473
Henry's law constant KH' that is a function of pH:
KHf=
P(C0,)C"
PH 7 -543.19 -541.57 -528.42 -515.38 -546.82 -541.11 -527.81 -53 1.05 -547.33 -547.10 -693.42 -692.88
8 -531.77 -530.15 -517.00 -515.38 -541.10 -541.11 -527.81 -53 1.05 -541.18 -541.18 -692.1 -691.81
9 -520.36 -518.74 -505.59 -503.97 -535.40 -535.4 -527.81 -531.05 -535.52 -535.52 -69 1.34 -689.57
AfHfo
-699.64 -700.46 -694.91 -695.73 -691.99 -691.99 -677.14 -615.5
constants written in terms of P(C02) to apparent equilibrium constants written in terms of [TotCOz], and vice versa.
Equilibrium Constant for the Formate Dehydrogenase Reaction
This reaction was introduced in eqs 1 and 3 because the standard thermodynamic properties of the species of formate are knownI2from nonbiological experiments and the differences in the standard thermodynamicproperties of NAD,, and NAD,d are known.I8 Goldberg et al.I9 have summarized measurements of A,Gfo and ArHfofor the oxidoreductases and have summarized data on a number of reactions producing carbon dioxide. Ruschig et al.,O have studied the formate dehydrogenasereaction from both directions and have shown that the same equilibrium constant is obtained. In their calculation of an apparent equilibrium constant for this enzyme-catalyzed reaction, they refer to [COz(sp)] [H2C03(sp)] as [CO2] and write the equilibrium constant expression for the first acid dissociation constant as
+
[TotCO,] Po
This is the equilibrium constant for TotCO,(aq) = CO,(g)
(57)
(cf. eqs 41 and 42). The standard transformed reaction Gibbs energy for this phase distribution at specified pH is given by
4G'" = -RT In KH) = A@O(CO,(g)) - A~'"(TotC0,)
This can be compared with eq 48. They give their experimental value for the apparent equilibrium constant for the formate dehydrogenase reaction as
(58)
Note that since we are dealing with transformed thermodynamic properties, the standard formation properties of CO2(g) have to be transformed to the same pH as the standard transformed formation properties of TotCO2. Thus at zero ionic strength and pH 7, eq 28 indicates that
+
r =EC0,I [NADH1 = 420 ECHO,-] [NAD]
at 283.15 K, pH 6.2, and I = 0.038 M. When [COz] is eliminated between eqs 61 and 62, the following equilibrium expression is obtained:
A+3'o(COz(g)) = A+30(COz(g)) A+3O(H,O(l)) - 2RTln[H+] = -394.36 - 237.13 2(39.95)
+
= -55 1.59 kJ mol-'
(59)
At zero ionic strength and pH 7, eq 58 yields
A,Gfo= -551.59
+ 547.34 = -4.25
K=
[H+][HCO,-][NADH] [CHO,-] [NAD]co
= 1.75 x
(63)
This is their experimental value for the following chemical reference reaction:
kJ mol-'
(60)
so that KH' = 5.55. Dividing by [COz(sp)]/[TotCO2] = 0.1879 (from Table 4) yields KH = 29.5, as expected. Apparent Henry's law constants are shown as a function of pH and ionic strength in Table 5 . These values approach 29.5 at low pH, as expected. They can be used to convert apparent equilibrium
H,O
+ CHO,- + NAD- = HCO,- + NADH~-+ H+ (64)
which balances charge and atoms. The standard thermodynamic properties for this reference reaction at 298.15 K, 1 bar, and I = 0 are readily calculated from standard formation values given in Table 6.
J. Phys. Chem., Vol. 99, No. 27, 1995 11033
Standard Transformed Formation Properties of C02
TABLE 5: Calculation of Apparent Henry's Law Constants at 298.15 K, 1 bar, Five pH Values, and Two Ionic Strengths PH
IIM 0
A&'"(CO2(g))M mol-]
0.25 A&'"(TotCO2)/kJ mol-'
0
0.25 A,G'"/kJ mol-'
0
0.25
KH'
0
0.25
5
-547.41 -572.79 -566.13 -564.61 -8.28 -8.18 28.24 27.11
6 -563.00 -561.37 -555.33 -554.49 -7.67 -6.88 22.07 16.05
TABLE 6: Standard Formation Properties of Species at 298.15 K, 1 bar, and Z = 0" A$PM mol-' A&"Mmol-' CHOZ-(ao) -425.55 -35 1.O HCO3-(ao) -691.99 -586.77 -237.13 HzO(1) -285.83 NAD-(ao) NADH2-(ao)
0
0
-3 1.94
22.65
" T h e NBS tablesI2 is the source for the first three species, and Alberty'* is the source for the last two species.
TABLE 7: Standard Transformed Formation Properties of Reactants at 298.15 K, 1 bar, pH 7, and Z = 0.25 M AtH'OM mol-] AfG'"M mol-' Formate -425.55 -311.05 TotCOz -692.88 -547.10 -286.65 -155.66 HzO 1059.10 NAD,, -10.26 -41.38 1120.09 NADred ~~
The standard thermodynamic properties for reaction 64 calculated from Table 6 are A,Go = 24.0 kJ mol-' and ArW = -12.55 kJ mol-'. The value of ArHo can be compared with the calorimetric value obtained by Rekharsky et al.21,22 in the pH range 6.02-8.01 in 0.05 M phosphate buffers; they obtained values close to -15 kJ mol-', uncorrected for heat of protonation of the buffer.23 The standard Gibbs energy of reaction 64 calculated using Table 6 corresponds with an equilibrium constant of 6.24 x at 298.15 K and I = 0. This value can be adjusted to I = 0.038 M by use of eq 33 to obtain 1.05 x It can be further adjusted to 10 "C by use of A,W = -12.55 kJ mol-'. This yields a theoretical K for reaction 64 of 1.37 x compared with the experimental value of 1.75 x by Ruschig et al. Under the circumstances, this is considered to be good agreement. The values of A,G'O and ArH0 for the biochemical reaction (eq 1) at 298.15 K, 1 bar, pH 7, and I = 0.25 M can be calculated using the values in Table 7. This yields A,G'" = -19.3 kJ mol-', which corresponds with K' = 2.4 x lo3 and A,Ho = -11.8 kI mol-'. This shows why Ruschig et al. chose to measure K' at pH 6.2. Goldberg et al.I9 have summarized experimental data on apparent equilibrium constants for other reactions involving COz, which may be used to calculate standard transformed thermodynamic properties of reactants.
Discussion These calculations have been made with standard thermodynamic properties of species obtained with additional information about Kh for the hydration of carbon dioxide. This has the advantage that the equilibrium mole fractions of the species COz, H2C03, HC03-, and C032- can be calculated as a function of pH and ionic strength, as shown in Table 4. However, this raises the question as to whether AG'"(TotCO2) and AfH'O(TotCO2) could have been calculated with values from the NBS tables
7 -557.59 -549.97 -547.33 -547.10 -4.25 -2.87 5.55 3.18
8 -540.16 -538.55 -541.18 -541.18 1.02 2.63 6.63 x lo-' 3.46 x lo-'
9 -528.75 ~~. -527.13 -535.52 -535.52 6.77 8.39 6.52 x 3.39 x 10-2 ~
alone. The answer is that they can be, and the same values of AG'"(TotC02) and AfH"(TotCO2) are obtained. This is so because the values of A@(COz(ao)) and AfW(C02(ao)) in the NBS tables are for all of the forms of C02, hydrated and not hydrated, in dilute aqueous solution. The NBS values can be used to calculatethe equilibrium mole fractions of CO?(hydrated and unhydrated), HCO3-, and H2CO3, and so they give less information about the composition of the solution. Correct thermodynamic calculations can be made without information on Kh,but in connection with the interpretation of kinetics it is of interest to know the concentrations of COz(sp) and HzCO3(sp). The explanation in the NBS tables that A,G" = A,Ho = 0 for the hydration of COZ(reaction 38) may make it seem that this assumption is built into the tables, but this assumption really only affects the values for HzCOs(ao). Actually, the values for HnCO3(ao) in the NBS tables are not needed, as shown by the discussion of eqs 47-50, and since they are redundant, perhaps they should not be included in future tables. The same remarks apply to about a dozen other species that are hydrated to an unknown extent; this list includes NH3, SOZ, and SO3, which are also of interest in biochemistry. There is a distinct advantage in using the apparent equilibrium constant K' in discussing equilibrium in biochemical reactions at specified pH. It is the reactants (sums of species) to which the stoichiometry of the biochemical reaction applies. For example, in the formate dehydrogenase reaction, it is the increase in TotCOz that is equal to the decrease in the concentration of NAD,,. Although the relaxation time for a solution of C02, HzCO3, HC03-, and C032- is a couple of seconds at pH 7 and 298.15 K, these species come to equilibrium in the determination of the apparent equilibrium constant. This assumes there is no diffusion into a gas phase, which is much slower. The equilibrium mixture of species of carbon dioxide at specified T,P, pH, and I can be treated as an entity with its own standard transformed formation properties. There has been discussion in the literature as to whether or not C02 is the true substrate of enzyme-catalyzed reactions involving C02, HzCO3, HC03-, and co32-,but this is irrelevant for measurements of equilibrium constants over periods of minutes because the species come to equilibrium. The pH-dependent Henry's law constant that relates [TotCOz] with P(COz(g)) can be used to express K' for a biochemical reaction in terms of [TotCOz] or P(COz(g)).
Acknowledgment. The author is indebted to Robert N. Goldberg for helpful discussion and to John T. Edsall for sending a copy of his 1969 article on carbon dioxide in water. The costs of this research were supported by NIH-1-RO1GM48358-0 1A 1. References and Notes (1) Webb, E. C. Enzyme Nomenclature; Academic Press; San Diego, 1992. (2) Alberty, R. A.; Cornish-Bowden, A.; Gibson, Q. H.; Goldberg, R. N.; Hammes, G. G.; Jencks, W.; Tipton, K. F.; Veech, R.; Westerhoff, H.
11034 J. Phys. Chem., Vol. 99, No. 27, 1995 V.; Webb, E. C. Pure Appl. Chem. 1994, 66, 1641. (3) Alberty, R. A. Biophys. Chem. 1992, 42, 117. (4) Alberty, R. A. Biophys. Chem. 1992, 43, 239. (5) Alberty, R. A. J. Phys. Chem. 1992, 96, 9614. (6) Alberty, R. A. Biochim. Biophys. Acta 1994, 1207, 1. (7) Alberty, R. A. Chem. Rev. 1994, 94, 1457. (8) Clarke, E. C. W.; Glew, D. N. Trans. Faraday SOC. 1966,62,539. (9) Goldberg, R. N.; Tewari, Y. Biophys. Chem. 1991, 40, 241. (10) Smith, W. R.; Missen, R. W. Chemical Reaction Equilibrium Analysis: Theory and Algorithm; Wiley-Interscience: New York, 1982. (1 1) Alberty, R. A. I & EC Fund. 1983, 22, 318. (12) Wagman, D. D.; Evans, W. H.; Parker, V. B.; Schumm, R. H.; Halow, I.; Bailey, S. M.; Chumey, K. L.; Nutall, R. L. The NBS Tables of Chemical Thermodynamic Properties. J. Phys. Chem. ReJ Data 1982,11, Suppl. 2. (13) Kem, D. M. J. Chem. Educ. 1960, 37, 14. (14) Edsall, J. T. C02: Chemical, Biochemical, and Physiological Aspects; NASA SP-188, 1969.
Alberty (15) Edsall, J. T.; Wyman, J. Biophysical Chemistry; Academic Press: New York, 1958; Vol I. (16) Wissbrun, K. F.; French, D. M.; Patterson, A. J. J. Phys. Chem 1954, 58, 693. (17) Roughton, F. J. W. J. Am. Chem. Soc. 1941, 63, 2930. (18) Alberty, R. A. Arch. Biochem. Biophys. 1993, 307, 8. (19) Goldberg, R. N.; Tewari, Y. B.; Bell, D.; Fazio, K. J. Phys. Chem. Ref. Data 1993, 22, 515. (20) Ruschig, U.; Muller, U.; Willnow, P.; Hapner, T. Eur. J. Biochem. 1976, 70, 325. (21) Rekharsky, M. V.; Egorov, A. M.; Gal’chenko, G . L.;Berezin, I. V . Dokl. Akad. Nauk. SSSR 1979,249, 1156. (22) Rekharskv. M. V.; Enorov, A. M.; Gal’chenko, G. L.; Berezin, I. V . Zh. Obsch. KLm. 1980, 56, 2364; J. Gen. Chem. USSR Engl. Transl. 1980.50, 1917. (23) Alberty, R. A.; Goldberg, R. N. Biophys. Chem. 1993, 47, 213. JF9504938
