Absolute Potential of the Standard Hydrogen Electrode and the

May 24, 2010 - The absolute potential of the standard hydrogen electrode, SHE, was calculated on the basis of a thermodynamic cycle involving H2(g) at...
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7894

J. Phys. Chem. B 2010, 114, 7894–7899

Absolute Potential of the Standard Hydrogen Electrode and the Problem of Interconversion of Potentials in Different Solvents Abdirisak A. Isse† and Armando Gennaro* Department of Chemical Sciences, UniVersity of PadoVa, Via Marzolo 1, 35131 PadoVa, Italy ReceiVed: January 15, 2010; ReVised Manuscript ReceiVed: April 23, 2010

The absolute potential of the standard hydrogen electrode, SHE, was calculated on the basis of a thermodynamic cycle involving H2(g) atomization, ionization of H•(g) to H+(g), and hydration of H+. The most up-to-date literature values on the free energies of these reactions have been selected and, when necessary, adjusted to the electron convention Fermi-Dirac statistics since both e- and H+ are fermions. As a reference state for the electron, we have chosen the electron at 0 K, which is the one used in computational chemistry. Unlike almost all previous estimations of SHE, ∆GQaq(H+) was used instead of the real potential, Raq(H+). This choice was made to obtain a SHE value based on the chemical potential, which is the appropriate reference to be used in theoretical computations of standard reduction potentials. The result of this new estimation is a value of 4.281 V for the absolute potential of SHE. The problem of conversion of standard reduction potentials (SRPs) measured or estimated in water to the corresponding values in nonaqeuous solvents has also been addressed. In fact, thermochemical cycles are often used to calculate SRPs in water versus SHE, and it is extremely important to have conversion factors enabling estimation of SRPs in nonaqueous solvents. A general equation relating EQ of a generic redox couple in water versus the SHE to the value of EQ in an organic solvent versus the aqueous saturated calomel electrode is reported. Introduction The definition and possible determination of an absolute standard reduction potential, SRP, is an intriguing subject, which continuously attracts the interest of researchers both from theoretical and experimental points of view.1 Experimental values of half-cell reduction potentials are generally anchored to the standard hydrogen electrode, SHE, in water, to which the conventional value of exactly 0 V has been assigned, or to the aqueous saturated calomel electrode (E ) 0.241 V versus SHE), which is more practical than SHE. Thus, the experimentally measured values are relative SRPs, and these are adequate for many applications of standard potentials. However, the growing development of computational efforts and the need to make comparisons between theoretical and experimental values set the question of absolute SRPs into relevant actuality. This is particularly true in the case of organic radicals, which are very important intermediates in many processes. In fact, it is very often difficult to experimentally determine SRPs of reactive intermediates such as free radicals, whereas theoretically computing them has recently become a quite easy task.2 Radicals are fundamental intermediates in polymerization processes, particularly in controlled/living radical polymerizations such as atom-transfer radical polymerization, which is an important synthetic technique enabling the preparation of many advanced materials with well-defined architectures.3-5 Radicals are also involved in the development of modern electron-transfer theories, particularly in the case of dissociative electron transfer, whereby the injection of one electron is accompanied by the fragmentation of a σ bond (O-O, C-X, S-S, etc.).6 Computational chemistry is increasingly of interest for radicals, especially their redox properties, thanks to the develop* To whom correspondence should be addressed. E-mail: armando. [email protected]. † E-mail: [email protected].

ment of ab initio and density functional theory calculations.7 Use of theoretical methods for the calculation of SRPs of complex molecules of relevance in many fields of chemistry and biochemistry has also become a common practice. However, when dealing with the computed SRPs, the value of the absolute potential of SHE comes to the forefront. This is because the calculated values, which are related to the Gibbs free energy of the redox reaction O + ne ) R, are absolute standard reduction potentials, EQ ) -∆rGQ/nF. Thus, their conversion to relative SRPs requires a reference system with a kown absolute potential defined in the same manner, that is, involving only chemical potential terms. In addition, often SRPs of redox couples in solvents different from water are of interest since many processes are carried out in organic dipolar aprotic solvents and, more recently, in ionic liquids. In this context, estimation of SRPs relative to aqueous SHE needs to consider the intersolvent potential. It would also be very helpful to be able to convert values of SRPs versus SHE measured or calculated in any solvent to the aqueous SCE scale, which is more frequently used for comparing experimental data. This paper addresses the long-standing issue of the absolute potential of SHE and is divided into two major sections with distinct objectives. In the first section, the problem of the absolute SHE potential is discussed, and calculation of the best up-to-date value is carried out, after clearly choosing a reference state for the electron. The aim of this calculation is to obtain a SHE value based on the Gibbs hydration free energy of the Q (H+), rather than on the real potential of the ion, proton, ∆Gaq + Raq(H ). Such a value is of a fundamental importance to theoretical chemists who use Gibbs solvation free energies when computing reduction potentials. Regrettably, many theoretical SRPs computed with a high level of theory are flawed because SHE values based on Raq(H+) such as the one recommended by IUPAC8 have been used for their conversion. The reason for this error is in part due to the lack of a SHE value based on

10.1021/jp100402x  2010 American Chemical Society Published on Web 05/24/2010

Absolute Potential of the Standard Hydrogen Electrode

J. Phys. Chem. B, Vol. 114, No. 23, 2010 7895

Q ∆Gaq (H+) and perhaps in part because some authors were not fully aware of the implications of these thermodynamic properties. In the second section, a method of conversion of SRPs measured or estimated in water to the corresponding values in nonaqeuous solvents is described.

SCHEME 1

Background The absolute standard potential, EQ(H+/H2)abs, of the halfcell reaction + H(aq)

+

e(g)

1 a H2(g) 2

(1)

has been the subject of several papers. In particular, a IUPAC recommendation indicated a value of 4.44 V three decades ago;8 this value has recently been updated to EQ(H+/H2)abs ) 4.42 V.9 Other recent approaches to determine EQ(H+/H2)abs include theoretical computational methods affording a value of 4.24 V10 and a gas-phase ion nanocalorimetry technique by which values of 4.05, 4.11, and 4.2 V have been obtained in different methods.1 There are several reasons for revising the value of EQ(H+/ H2)abs. First, the reference state for computing the energy of the electron was the object of different choices, and this has contributed much to the differences reported above. Second, the value recommended by IUPAC does not refer to a pure chemical potential of the proton. In fact, since it was considered not possible to know the standard hydration free energy of the Q (H+), the real potential, Raq(H+), was used instead proton, ∆Gaq + Q of ∆Gaq(H ). The real potential represents the work done to take an ion from a vacuum into a solution through the gas/ liquid interface. Therefore, it differs from the Gibbs solvation free energy by an electrostatic term representing the contribution due to the electric potential difference at the vacuum/water interface. In the case of H+ hydration

Raq(H+) ) ∆GQaq(H+) + Fχaq

(2)

where χaq is the surface potential of water. Last, a mixture of different conventions, which affects the final result, has been used in almost all previous estimations of EQ(H+/H2)abs. As pointed out by Trasatti,11 the value estimated for EQ(H+/ H2)abs primarily depends on the reference state chosen for the electron. Three different reference states are possible, (i) free electron in the solution, (ii) free electron at rest in vacuum near the surface of the solution, and (iii) free electron at rest in vacuum at infinity. Of course, each of these is a possible choice for the definition of an “absolute” potential. Therefore, with the lack of a universal reference state for the electron, Trasatti11 suggested the use of the term “single” SRP rather than the “absolute” SRP. A second important issue which affects the value of the single SHE potential is the Gibbs free energy of H+ in H2O. Also in this case, different possible choices have been adopted; it appears that also, in a few cases, some inconsistencies have been involved in the calculations. The IUPAC electrochemical scale8 assumes the electron in vacuum close to the solution surface as the zero level for electron energy. This differs from the level at infinity by the outer potential, that is, the Volta potential, ψS, of the solution. This choice has been made because the energy involved in the passage of an electron from this state to the bulk metal, via the electrode/solution interface, is directly comparable with physical

properties such as the metal work function, ΦM, and as recently underlined, it has now become familiar to measure also the ion work function.9 The more recent theoretical scale used by Truhlar10 and others7 assumes, as zero level for electron energy, the electron in vacuum at 0 K, which is slightly different from the third state above. This appears to be a more universally acceptable state and is the one largely used in computational chemistry. The most relevant difference between the IUPAC recommended value and that theoretically calculated by Truhlar10 is related to the free energy of hydration of H+. In the electrochemical treatment,8,9 the free energy of solvation is defined as the real potential Raq(H+) ) -1088 kJ mol-1,8 which has recently been updated to -1091 kJ mol-1.9 Farrell and McTigue13 measured Raq(H+) and suggested its use as the hydration free energy of the proton primarily on the grounds that while ∆GQaq(H+) could not be directly determined at that time, the real potential was measurable, both in principle and in practice. These authors, however, conceded that the contribution Fχ of the surface potential in Raq(H+) represents a nonchemical component but gave some justifications for that choice, the principal of which was that use of the real potential did not need any extrathermodynamic assumption. On the basis of this choice, Trasatti8 calculated EQ(H+/H2)abs ) 4.44 V in water, without any extrathermodynamic assumption. Nowadays, it is recognized that it is possible to determine the absolute hydration enthalpy and free energy, without making any of the usual extrathermodynamic assumptions, by methods such as the cluster-pair-based approximation14a or the cluster-pair-based common point,14b which give very similar results. Tissandier et Q (H+) ) -1104.5 kJ mol-1 by the clusteral.14a measured ∆Gaq pair-based approximation method, using a solution-phase concentration of 1 mol L-1 as the standard state. This value has recently been recommended as the best estimate of the free energy of hydration of the proton.14c Truhlar and co-workers10 Q used the value of ∆Gaq (H+) reported by Tissandier et al14a to calculate the absolute potential of the normal hydrogen electrode, E*(H+/H2)abs ) 4.24 V. The superscript asterisk is used to underline that the potential refers to a 1 M solution of H+ in equilibrium with H2 at p ) 1 atm. This is slightly different than the potential of the SHE, which is defined as the potential of the following half-cell + Pt|H2(g,p)1 bar) |H(aq,a)1)

Results and Discussion Absolute SHE Potential. Defining the absolute SHE potential requires a reference state for the electron energy. Among the various reference states reported in the literature, we chose the one used in most theoretical works, which assumes as the zero level for electron energy the electron in vacuum at 0 K. The absolute SHE potential can be calculated by using the thermochemical cycle shown in Scheme 1.

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Isse and Gennaro

This involves, as an alternative route to the redox reaction 1, formation of H+ in the gas phase followed by dissolution in water. Accordingly, EQ(H+/H2)abs can be expressed as the sum of the free energy of H+ formation in the gas phase, ∆fGQ(H+(g)), Q and its hydration free energy, ∆Gaq (H+). It is worth stressing that we use the Gibbs free energy of hydration rather than the real potential, Raq(H+), because we are interested in a SHE value based on the pure chemical potential, that is, µaq(H+). Q E(H +/H )abs ) 2

1 + ) + ∆GQaq(H+)] [∆ GQ(H(g) F f

(3)

The most updated value of ∆fGQ(H+(g)) is that calculated by Bartmess, who used the electron convention, EC, for defining the energetics of the electron and Fermi-Dirac statistical mechanics, FD, for both the electron and the proton (they are fermions).12 Bartmess also calculated the thermodynamic functions of the electron and the proton at different temperatures according to the EC-FD convention. According to this treatment, the integrated heat capacity, HQ298 - HQ0 , and entropy, SQ298, of the electron at T ) 298.15 K are 3.145 kJ mol-1 and 22.734 J mol-1K-1, respectively. To obtain ∆fGQ(H+(g)), the gas-phase formation of H+ can be expressed as the combination of two reactions, homolytic dissociation of H2 and ionization of H•.

1 • H a H(g) 2 2(g)

(4)

• + H(g) a H(g) + e(g)

(5)

The standard enthalpy and Gibbs free energy of formation • Q • Q • are ∆fH298 (H(g) ) ) 217.965 kJ mol-1 and ∆fG298 (H(g) )) of H(g) -1 12 • 203.246 kJ mol . For the ionization of H(g) (eq 5), the reaction enthalpy, entropy, and free energy at T ) 298.15 K are given by12

∆HTQ(IE) ) E0 + (HTQ - HQ0 )(H+) + (HTQ - HQ0 )(e-) (HTQ - HQ0 )(H•) ) 1315.134 kJ mol-1

+ • ∆fGTQ(H(g) ) ) ∆GTQ(IE) - ∆fGTQ(e(g) ) + ∆fGTQ(H(g) ))

1513.321 kJ mol-1 With the reference state chosen for the electron, these are the most up-to-date values for the enthalpy and free energy of formation of the proton in the gas phase. Note that in the above calculations, according to the EC convention,12 the electron is considered to be equivalent to an element, which implies that both its standard enthalpy, ∆fHQT (e-(g)), and free energy, ∆fGQT (e-(g)), of formation are zero at all temperatures. Instead, both the integrated heat capacity and entropy have nonzero values, which have been accounted for in the calculation of ∆HTQ(IE) and Q + (H(g) ) is about ∆GTQ(IE). The above computed value of ∆fG298 3.6 kJ mol-1 smaller than that adopted by Trasatti, who utilized Q (IE) ) 1313.82 kJ mol-1 from JANAF Tables,15 which ∆G298 are based on the EC-B convention (electron conventionBoltzmann statistical mechanics), for which the integrated heat capacity of the electron is 5/2RT, that is, 6.197 instead of 3.145 kJ mol-1 of the EC-FD convention. In fact, calculation of the enthalpy and Gibbs free energy of formation of (H+(g)) according Q + (H(g) )EC-B ) 1536.212 kJ to the EC-B convention gives ∆fH298 Q + (H(g) )EC-B ) 1516.956 kJ mol-1.12 This value mol-1 and ∆fG298 Q + (H(g) )EC-B has been used also by Fawcett in his recent of ∆fG298 updating of the IUPAC recommended EQ(H+/H2)abs value as 4.42 V.9 Regarding the second term of the right-hand side of eq 3, the absolute enthalpy and free energy of hydration of the proton have recently been determined by a cluster-pair-based approximation method14 without invoking extrathermodynamic assumptions. This method, however, requires knowledge of the enthalpy and Gibbs free energy of formation of H+ in the gas Q + (H(g) ) and phase. Therefore, using the values of ∆fH298 Q + (H(g) ) calculated by Bartmess according to the EC-B ∆fG298 convention, Coe and co-workers14 obtained the absolute values Q Q (H+) ) -1150.1 ( 0.9 kJ mol-1 and ∆Gaq (H+) ) of ∆Haq -1 -1104.5 ( 0.3 kJ mol by extrapolation to an infinite number of solvating water molecules (i.e., the bulk of solution). These values should be corrected by using ∆fHQ298(H+(g)) and ∆fGQ298(H+(g)) values calculated according to the EC-FD convention. This can be done by positively shifting the EC-B based values of Q Q (H+) and ∆Gaq (H+) by 3.0524 and 3.577 kJ mol-1, ∆Haq Q (H+) ) respectively, which leads to the following values: ∆Haq -1 + Q -1147.05 ( 0.9 kJ mol and ∆Gaq(H ) ) -1100.9 ( 0.3 kJ mol-1. On the other hand, if we consider that

+ • ∆STQ(IE) ) STQ(H(g) ) + STQ(e(g) ) - STQ(H(g) ))

16.968 J mol-1 K-1 ∆GTQ(IE) ) ∆HTQ(IE) - T∆STQ(IE) ) 1310.075 kJ mol-1 where E0 is the ionization energy at 0 K. The integrated heat • , considered as a monatomic ideal gas, is 5/2RT capacity of H(g) ) 6.197 kJ mol-1 at T ) 298.15 K. With these values in hand, the standard enthalpy and Gibbs free energy of formation of H+ in the gas phase can now be calculated at T ) 298.15 K as + • ∆fHTQ(H(g) ) ) ∆HTQ(IE) - ∆fHTQ(e(g) ) + ∆fHTQ(H(g) ))

1533.099 kJ mol-1

Raq(H+) ) ∆GQaq(H+) + Fχaq

(2)

and that χaq has been estimated to be about 0.14 V9 and Raq(H+) ) -1088 kJ mol-1,8 we obtain ∆GQaq(H+) ) -1101.5 kJ mol-1. This value of ∆GQaq(H+) is in excellent agreement with the value determined by the cluster-pair-based approximation method, which we corrected according to the EC-FD convention. The significance of the hydration free energies determined by Coe and co-workers14 has been questioned by some authors.1,16 In particular, Asthagiri et al.16a have argued that the values determined by methods based on solvated cluster pairs include the surface potential of water, χaq, and should therefore be considered to be Raq(H+). In support of their interpretation, these authors cited a paper by Klots17 who explained that for large clusters, where the surface layer takes on the characteristics of pure water, the surface potential will develop. However, as

Absolute Potential of the Standard Hydrogen Electrode

J. Phys. Chem. B, Vol. 114, No. 23, 2010 7897

explained by Coe, who also cites the same paper, the clusters used for the determination of the hydration free energy of H+ by the cluster-pair-based methods are much too small to have reached the point of surface potential development. On the other hand, the difference between the value of the hydration free energy obtained by Coe14b and the real potential determined by Farrell and McTigue13 is in good agreement with the value of χaq recently obtained by Fawcett.9 Therefore, we consider that the values obtained by Coe, by the cluster-pair-based method, utilizing clusters with n e 6, are the absolute enthalpy, Q Q (H+), and the Gibbs free energy of hydration, ∆Gaq (H+), ∆Haq of the proton, without any interference by the surface potential. All parameters in eq 3 being known, we can now calculate the absolute potential of the standard hydrogen electrode. Note, however, that the Gibbs free energies refer to a solution-phase concentration of 1 M, whereas standard potentials are defined for redox species at unit activity.18 In fact, according to the Nernst equation, the potential of the hydrogen electrode is given by

E(H+/H2)abs ) EQ(H+/H2)abs +

(

aH+ RT ln F (pH2 /pQ)1/2

)

(6)

Therefore, the above-reported free energies provide the absolute potential of the normal hydrogen electrode, whose use, however, has long been abandoned.19 + E*NHE,abs ) (∆fGQ298(H(g) ) + ∆GQaq(H+))/F ) (1513.321 1100.9)/F ) 4.275 V

From this value we can calculate the absolute standard potential for the hydrogen electrode in water, EQ(H+/H2)abs, by taking into account the activity coefficient for a 1 mol L-1 solution of H+ in water, which has been estimated to be 0.8.19

EQ(H+/H2)abs ) E*NHE,abs - RT/F ln γH+ ) 4.275 + 0.006 ) 4.281 V This value is in good agreement with the value of 4.42 V estimated by Fawcett9 if we consider that the hydration free energy used in this paper contains only a “pure chemical” contribution. In fact, the difference between the two values is simply the surface potential χaq, which has been quantified as 0.14 V by the same author. In conclusion, using values of pertinent thermodynamic functions based on (i) the EC-FD convention for the electron and the proton in the gas phase and (ii) the cluster-pair-based approximation or cluster-pair-based common-point method for hydration of the proton, we can determine an absolute SHE potential based only on chemical potentials. Thermodynamic data of the proton together with the calculated values of the absolute potential of the normal and standard hydrogen electrodes are reported in Table 1. The value of EQ(H+/H2)abs reported herein can be considered an absolute one, particularly suitable as a reference value for SRP estimations that do not include χaq, since (i) the reference state chosen for the electron is a universally acceptable one, (ii) the EC-FD convention is coherently adopted both for the electron and the proton, (iii) no extrathermodynamic assumption

TABLE 1: Thermochemistry of the Proton in the Gas Phase and in Water at 298.15 K According to the EC-FD Convention + ∆fHQ(H(g) ) + SQ(H(g) ) + ∆fGQ(H(g) ) + Q ∆Haq(H ) ∆GQaq(H+) E*NHE,abs EQ(H+/H2)abs

1533.099 kJ mol-1 108.947 J mol-1 K-1 1513.321 kJ mol-1 -1147.05 kJ mol-1 -1100.9 kJ mol-1 4.275 V 4.281 V

has been used, and (iv) only pure chemical contributions have been considered for the half-cell reaction

1 + H(aq) + e(g) a H2(g) 2

(1)

Electrochemists refer to the above half-cell reaction as a reduced absolute potential,8 but we think that the term “absolute” is more appropriate for the ∆rGQ of reaction 1, when the reference state for the electron is the vacuum at 0 K. Conversion of SRP to the Aqueous Saturated Calomel Electrode Scale. When comparing computed SRPs with experimentally measured values, which are generally referenced to the aqueous saturated calomel electrode, SCE, we must consider that for a given redox reaction O(S) + e(g) h R(S)

(7)

Q,SCE,aq in any solvent S, EO/R,S is the electromotive force (∆Erev) of the following galvanic cell

M|Hg|Hg2Cl2(solid) |Cl(aq,sat) |O(S,a)1),R(S,a)1) |M

Q,SCE,aq The value of EO/R,S measured in this manner contains an interliquid (intersolvent) potential EL, which depends on the nature of the solvent S as well as on the composition of the Q,abs is the absolute potential of the O/R couple in solution. If EO/R,S abs the solvent S and ESCE, aq is the absolute potential of aqueous SCE, we have

Q,SCE,aq Q,abs abs ) EO/R,S - ESCE,aq + EL EO/R,S

(8)

abs Q where ESCE,aq ) Eabs (H+/H2) + 0.241 V ) 4.522 V. Potentials in nonaqueous solvents such as acetonitrile, MeCN, and dimethylformamide (DMF) are often measured in the presence of tetraalkylammonium salts (typically 0.1 M). The interliquid potentials of several solvents containing 0.1 M tetraethylammonium picrate, which may be considered to be representative of tetraalkylammonium salts, have been measured by Diggle and Parker.20 The values of EL found for MeCN and DMF are 0.093 and 0.172 V, respectively. A slightly different value can be estimated for MeCN starting from the potential of Pleskov’s electrode21 in the same solvent, measured versus SCE (see Supporting Information). Thus, the standard potential of the redox couple O/R in MeCN and DMF can be expressed as

Q,SCE Q,abs EO/R,MeCN ) EO/R,MeCN - 4.429 V

(9)

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J. Phys. Chem. B, Vol. 114, No. 23, 2010 Q,SCE Q,abs EO/R,DMF ) EO/R,DMF - 4.350 V

Isse and Gennaro

(10)

A widely used method of estimation of SRPs is to incorporate the redox reaction in a thermochemical cycle and calculate its free energy in water from available thermodynamic data. This method yields standard reduction potentials in water referenced to the SHE. It may be desirable to convert standard reduction potentials in water to the corresponding values in a solvent S, referenced to the aqueous SCE. This involves a change of solvent for the redox process and a change of the reference system from SHE to SCE. This problem has previously been addressed by Save´ant22 in the framework of a concerted dissociative ET to an organic halide, RX, in DMF. We will adopt the same procedure to develop a more general method, which can be applied to any redox couple in any solvent S. The standard reduction potential of a generic redox couple O/R is defined, according to the SHE convention, as the opposite of the free energy of the following reaction

1 + O(aq) + H2(g) h H(aq) + R(aq) 2

(12)

and the SRP of the O/R couple versus the standard silver electrode is given by +

Q, Ag /Ag,aq EO/R,aq )

+/Ag,S

Q, Ag EO/R,S

+

Q, Ag /Ag,aq ) EO/R,aq -

-∆rGQ12 Q,abs Q,abs Q,SHE,aq - ESHE,aq - EAg ) EO/R,aq +/Ag,aq ) F Q,abs Q,abs - ESHE,aq - 0.799 V (13) EO/R,aq

+/Ag,S

Q, Ag EO/R,S

)-

∆rGQ14 Q,SHE,aq Q,SHE,aq - EAg ) EO/R,S +/Ag,S ) F Q,SCE,aq Q,SCE,aq - EAg EO/R,S +/Ag,S

(14)

(15)

In the above equation, SHE, SCE, or any other reference system can be used to express the difference between the SRPs of O/R and Ag+/Ag. Rearranging eq 15 gives

1 (∆ GQ,aqfS + F tr Ag+ - ∆trGOQ,aqfS) (17)

Q,SHE,aq Q,SHE,aq ) EO/R,aq - EAg +/Ag,aq -

∆trGRQ,aqfS

Q,SHE,aq + Combining eqs 16 and 17 and recalling that EAg /Ag,aq ) 0.799 V leads to

1 (∆ GQ,aqfS + F tr Ag+ - ∆trGOQ,aqfS) - 0.799 V (18)

Q,SCE,aq Q,SHE,aq Q,SCE,aq EO/R,S ) EO/R,aq + EAg +/Ag,S+ -

∆trGRQ,aqfS

Q,SCE,aq SCE,aq + where EAg - (RT/F) ln(0.01γAg+)S. This is the /Ag,S ) EPl,S equation that we sought because it relates the SRP of O/R in any solvent S versus aqueous SCE to its value in water versus Q,SHE to aqueous SHE. To be able to do conversion of EO/R,aq Q,SCE,aq , one needs several parameters. The standard potential EO/R,S of Ag+/Ag, if not already available in the literature,23 can be experimentally measured. It can also be calculated from the potential of Pleskov’s electrode by correcting the latter for the activity term. Also, the free energies of transfer of a large number of ions are available in the literature.24 Rearranging eq 18 to separate it into two terms, one pertaining to the O/R redox couple and another related to the Ag+/Ag reference system, gives

1 (∆ GQ,aqfS - ∆trGOQ,aqfS) ) F tr R Q,aqfS ∆trGAg + Q,SCE,aq EAg - 0.799 V (19) +/Ag,S F

Q,SCE,aq Q,SHE,aq EO/R,S - EO/R,aq +

Considering the same redox reaction in a solvent S and using Pleskov’s reference electrode in the same solvent, we obtain + O(S) + Ag(solid) h Ag(S,0.01 M) + R(S)

1 (∆ GQ,aqfS + ∆trGRQ,aqfS F tr Ag+

∆trGOQ,aqfS)

(11)

If we wish to reference the SRP to Pleskov’s electrode,21 the redox reaction to be considered is + O(aq) + Ag(solid) h Ag(aq,0.01 M) + R(aq)

in the same solvent. This potential can be related to the SRP of +/Ag,aq the O/R couple in water, EQ,Ag , and the Gibbs free energies O/R,aq of transfer of the reagents and products of reaction 14 from water to the solvent S, ∆trGQ,aqfS. Comparing reactions 12 and 14 gives

The term in the right-hand side of eq 19 depends only on thermodynamic properties of Ag+ in water and in the solvent S. This term is therefore constant for each solvent S, regardless of the nature of the O/R redox couple. This term mainly stems from the change of the reference system from SHE to SCE; therefore, we define it as the potential-scale conversion constant, Q,S . ∆ESHEfSCE

Q,S ∆ESHEfSCE

)

Q,SCE,aq EAg +/Ag,S

Q,aqfS ∆trGAg + - 0.799 V F

(20) Q,SCE,aq EO/R,S

)

Q,Ag+/Ag,S EO/R,S

SCE,aq EPl,S

Q,Ag+/Ag,S EO/R,S

+ ) + RT Q,Pl,S SCE,aq + EPl,S ln(0.01γAg+)S ) EO/R,S F Q,SCE,aq EAg +/Ag,S

(16)

is the potential of Pleskov’s electrode in the solvent where ESCE,aq Pl,S SCE,aq is known, eq 16 can S with respect to aqueous SCE. If EPl,S Q,SCE,aq from the SRP measured for the be used to calculate EO/R,S O/R couple in the solvent S with respect to the standard silver Q,Ag+/Ag,S Q,Pl,S , or from EO/R,S , which electrode in the same solvent, EO/R,S is the SRP measured in S with respect to Pleskov’s electrode

Q,S Table 2 reports values of ∆ESHEfSCE calculated for some solvents from literature data. The standard potential of Ag+/Ag in acetonitrile was calculated from the potential of Pleskov’s SCE,aq Q,SCE,aq SCE,aq + ) EAg electrode, EPl,MeCN /Ag,S + (RT/F) ln(0.01γAg+)S. EPl,MeCN 25 ) 0.290 V, whereas the activity term (RT/F) ln(0.01γAg+) is reported to be -0.130 V for a 0.01 M solution of Ag+ in Q,S CH3CN.26 Therefore, EQ,SCE,aq Ag+/Ag,MeCN ) 0.420 V. Besides ∆ESHEfSCE, the difference between the transfer free energies of the reagents Q,SCE,aq to and products, ∆∆trGQ,aqfS, is required to convert EO/R,S Q,SHE,aq EO/R,aq and vice versa. This term depends on the nature, and

Absolute Potential of the Standard Hydrogen Electrode TABLE 2: Potential-Scale Conversion Constants for Some Polar Aprotic Solventsa solvent

Q,SCE,aq + EAg /Ag,S (V)

Q,aqfS + ∆trGAg (kJ mol-1)

Q,S ∆ESHEfSCE (V)

N,N-dimethyformamide dimethyl sulfoxide propylene carbonate acetonitrile

0.538 0.372 0.813 0.420b

-15.6 -14.3 22.8 -23

-0.099 -0.279 -0.222 -0.141

a

Q,aqfS Q,SCE,aq + EAg /Ag,S taken from ref 23a, unless otherwise stated; ∆trGAg+ b Calculated from the potential of Pleskov’s taken from ref 24b. electrode.

particularly on the charge density, of O and R and is to be evaluated differently from case to case. For example, the free energy of transfer of many ions is known, whereas ∆trGQ,aqfS of neutral species may be evaluated from the solvation energies in water and in the solvent S if available or somehow assessable, or it may be simply neglected. Conclusions The absolute potential of the standard hydrogen electrode in water has been calculated from literature data by using an appropriate thermodynamic cycle. The calculation is based on the following set of choices: (i) a free electron in vacuum at 0 K as the reference state for the electron, (ii) electron convention + Fermi-Dirac statistics, EC-FD, for both e(g) and H(g) , (iii) a Q (H+) value based on a cluster-pair-based approximation ∆Gaq or cluster-pair-based common-point method, and (iv) the use of absolute hydration free energy, ∆GQaq(H+), instead of the real potential, Raq(H+). The last choice has been dictated by the necessity for an absolute SHE potential entirely based on pure chemical contributions, which is the appropriate reference in theoretical computations of SRPs where the surface potential of the solvent is not taken into consideration. Using the most up-to-date thermodynamic data based on the above convention, Q (H+/H2)abs of 4.281 V has been obtained without a value of Eabs any extrathermodynamic assumption. This value is in agreement Q (H+/H2) and also with with the most recent estimations of Eabs the value recommended by IUPAC, if it is taken into considQ (H+) eration that the latter is based on Raq(H+) rather than ∆Gaq aq and therefore contains a contribution Fχ from the surface potential. The possibility of converting potentials in a given solvent to the corresponding values in water and vice versa has Q,SCE,aq to been examined. In particular, an equation relating EO/R,S Q,SHE,aq EO/R,aq has been developed. Acknowledgment. This work was supported by the University of Padova through Grant CPDA083370. Supporting Information Available: Estimation of the intersolvent potential between the aqueous calomel electrode and acetonitrile. This material is available free of charge via the Internet at http://pubs.acs.org. References and Notes (1) (a) Donald, W. A.; Leib, R. D.; O’Brien, J. T.; Bush, M. F.; Williams, E. R. J. Am. Chem. Soc. 2008, 130, 3371. (b) Donald, W. A.; Leib, R. D.; O’Brien, J. T.; Williams, E. R. Chem.sEur. J. 2009, 15, 5296,

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