J. Phys. Chem. 1995,99, 13198-13202
13198
Evaluation of the Redox Potential of Agl1(CN)2-/Agl0(CN)2*-in Aqueous Solution S. Remita, P. Archirel, and M. Mostafavi* LPCR (CNRS URA75) BBt. 350, Universitk Paris-Sud, 91405 Orsay Cidex, France Received: January 3, 1995; In Final Form: March 14, 1995@
The redox potential of the A~I'(CN):!-/A~~O(CN)~~couple in aqueous solution is evaluated with the aid of a thermodynamical cycle including gas phase and solvated species. We use the SCF method for the determination of the electronic structure of the gaseous species and the cavity model for the solvation effect. We find the value -2.6 V/NHE, which is lower than the value (-1.8 V) relative to the Ag+/Ago couple and relative to the bulk metal. hence much lower than the value (-I
1. Introduction
2. The Ag+/Ago Couple in the Presence of CN-
The first step of the synthesis of metal aggregates from M+ ions in solution is the reduction of the metal cations into atoms. Thermodynamics of this step is still poorly understood. The electron transfer is controlled by the relative values of the redox potentials of the two systems, metal ion as electron acceptor and reducing agent as electron donor. The hydrated electron eaq- is the most efficient reducing species in solution,'.2 with a very negative redox potential:
It has been showng that the solvated electron readily reacts with the stable complex Ag'(CN)2-. The species formed has not been observed yet and is unknown. First of all, we will discuss the possible products. Let us consider a thermodynamical cycle involving an electron transfer toward Ag'(CN)2-:
e-
+ nH20
+
E" = -2.9 V/NHE
eaq-
(1)
Therefore, the solvated electron is able to reduce most of the metal cations and the rate constants can be obtained from the e!q- decay directly observed by pulsed radiolysis. It was discovered, two decades ago, that the redox potential of the M+/ Mo couple is quite different from that of the M+/M,,, couple and much more n e g a t i ~ e . ~By. ~using a thermodynamical cycle involving the electrochemical potential of the silver bulk electrode and the sublimation energy of metallic Ag (the solvation energy of one single Ago atom was neglected), Hengleid estimated the redox potential of the Ag+/Agocouple: Ag++e--Ago
Eo=-1.8V/NHE
(2)
According to the last value, only reducing agents with a redox potential lower than - 1.8 VINHE can reduce free Ag+ ions in solution. Nevertheless, electron donors with a higher redox potential can reduce Ag+ when it is adsorbed on Ag, clusters, or other particles, or impurities present in the solution, or on the reactor walls. The reason is that the redox potential of metal aggregates Agn+/Agnincreases with the nuclearity n, up to the value corresponding to the bulk electrode. The higher the nuclearity, the easier the reduction of the adsorbed Ag+, and the size dependence is especially marked for the small values of n.637 Actually, the photographic development has been explained with the aid of this phenomenon.8 It is well-known in electrochemistry that the redox potential of the M+/Mmetcouple is shifted toward lower values when the ions are complexed by ligands. The aim of the present work is to investigate the influence of ligands on the redox potential of the M+l(L)/MO(L)monomer couple. We study the case of Ag+ ions complexed by cyanide ligands (L = 2CN-). ~
_
_
Ag1(CN)2-
~ _ _ _
@Abstractpublished in Advance ACS Abstracts, July 15, 1995.
0022-365419512099-13198$09.00/0
-
+ eAg+
Ago
+ 2CN-
+ 2CN- + e-
This cycle shows that the redox potential involving free silver metal atoms is more negative than the redox potential of the solvated electron, E"(H201eaq-): Eo(Ag'(CN),-/Ago
+
2 C N 7 = E"(Ag+/Ago) - 0.059 log K (3) = -3.02 V/NHE at 298 K
(4)
(with K = 1020,7:complexation constant of Ag' by 2CN- in aqueous solution). The following reaction is therefore thermodynamically impossible: Ag'(CN),- f eaq-
-
Ago
+ 2CN- + nH20
(5)
Henglein et a1.,I0 using a similar cycle, recently evaluated the potential of the couple Au'(CN)2-/Auo 2CN- and found -3.8 V: this value is much more negative than E"(H20/eaq-) and therefore is in disagreement with the experimentally observed reduction of Au'(CN)2- by eaq-. This value would mean, on the contrary, a spontaneous oxidation of Auo by water. We thus have to consider that reaction 5 yields a product other than free Ago. The simplest hypothesis is to assume a charge addition:
+
Ag'(CN),-
+ eaq-
-
Ago(CN),2-
+nH20
(6)
Note that the following hypothetic reaction, Ag0(CN):-
-
Ago
+ 2CN-
(7)
obtained by the difference between reactions 5 and 6, is also thermodynamically forbidden. We thus have to evaluate the redox potential of Ag'(CN)2-/Ago(CN)22-. For that purpose we 0 1995 American Chemical Society
Redox Potential of Agl '(CN)~-/A~IO(CN)~~-
-67.6
J. Phys. Chem., Vol. 99, No. 35, 1995 13199
I
-67.7 -67.8 A
-67.9 -
Figure 1. Potential curves of Ag1(CN)2-(SCF) and Ag0(CN)z2-(fictitious state) in the linear (symmetrical) geometry and with respect to the R A ~ - c distance.
consider the following cycle: Ago(CN)zz-
(11)
I
A@4
Ago(CN)zz-
-
+ e-
-
+ e-
A@!
A@,
Agl(CN)z-
Agl(CN)B-
in solution
+ +
in the gas phase
where AGO1 is the ionization potential of Ago(CN)22- in solution, AGO2 = IPgasis the ionization potential of Ago(CN)22- in the gas phase (we assume here that the entropic term is negligible: this should be reasonable in the gas phase), and AGO3 = AGosolv(Ag1(CN)2-)is the standard solvation free enthalpy of Ag'(CN)2-. AGO4 = AGo,,~,(Ago(CN)22-) is the standard solvation free enthlapy of Ago(CN)22-. The ionization potential of a species in solution is related to the redox potential Eo according to
AGO, = eEo(V/NHE)
+ 4.5
(8)
where 4.5 eV is the Fermi potential of the normal hydrogen electrode (NHE), with respect to the vacuum.'' This energy shift makes the comparison between gas phase and solution data possible. We thus have
+
AGO, = eEo(Ag1(CN),-/Ago(CN),2-) 4.5 = AGO,
curve is shown in Figure 1. The binding energy, De, amounts to 9.5 eV; the equilibrium value of R is Re, = 4.2 au. A more sophisticated CASSCF calculation (including the 5 s orbital of Ag and the 2s of C) yields almost the same values. Note that the binding energy refers to the Ag+ 2CN- asymptote, which means that the charge transfer AgCN CN- asymptote is not taken into account. Unlike the previous system, the study of the complex Ag0(CN)z2-is problematic because we found it to be autoionizing at the Re, distance. This was proved by the behavior of the SCF solution with respect to the extension of the basis set: if we add diffuse s Gaussians (with exponents tending to 0), then the s electron of Ag always occupies the most diffuse Gaussians. The CASSCF calculation (using the same space as before) did not succeed in binding this electron. We thus define some fictitious state of Ag0(CN)z2-: this state is the result of the crude substitution of Agf by Ago inside the Ag'(CN)2- system at its equilibriumgeometry, with modification of only the overall electrostatics. This amounts to adding 2/Re, (electrostatic attraction between Agf and two CN-) from the energy of the Ag1(CN)2- system. The potential curve for this fictitious state is shown in Figure 1. Note that this fictitious state is nonstationary in two respects: firstly it is autoionizing, and secondly it is dissociative. The simple expression of the vertical ionization potential of Ago(CN)22-,
+ (AGO, - AGO,)
(9)
AGO, = IPg(Ago) - [E(Ago(CN),2-) - E(Ag'(CN),-)] (1 1)
(10)
We now have to evaluate the three terms of eq 10.
3. Calculation of the Gas Phase IP of Ag0(CN)z2The study of the system Ag'(CN)2- is easy because the main components of the binding energy (the electrostatic interactions between the ions and the short-range repulsions) are clearly taken into account by the simple SCF method. We have done the SCF calculation with core pseudopotentials (simulating the inner shells of the atoms) and Gaussian basis sets.I2 We have frozen the CN distance, using the experimental value rCN = 2.21 au,I3 and assumed linear geometry and central symmetry of the complex. The simplest electrostatic model shows that this configuration is the most stable. The only geometric parameter is thus R, the Ag-C distance. The SCF potential
(where E is the cohesion energy of the species and the zeropoint vibration energy has been neglected), now may be written as AGO2 = IP,(Ago) - 2/Re,
(12)
= -5.39 eV
(13)
where we have used the experimental value (7.57 eV) of the IP of Ago l4 and the above given value of Re,. The advantage of the present method is that the IP of the fictitious state only slightly depends on the quantum method chosen, through the value of Re,, The influence of the error on Re, on the solvation energies will be estimated in the next section. The IP of eq 13 is negative, which is normal for an autoionizing state: this reflects the fact that putting the electron on the Agf ion inside
13200 J. Phys. Chem., Vol. 99, No. 35, 1995 TABLE 1: Values of the Multipole Moments of the Two
Complexes 1
Ag(CN12- M,O(au)
0
2 4 6
-1.00 -35.3
-622.3 -10 978.1
Ag0(CN)2*-M,O(au)
-2.00 -35.3 -622.3 -10 978.1
the complex requires work, for fighting its repulsion by the two CN-. Cycle I1 is now considered to be written with this fictitious state, %andAGO4 is thus its solvation energy.
4. Calculation of the Solvation Energies of Agi(CN)2and A$(CN)z24.1. Principle of the Method. We use the cavity model of Rinaldi et al.I5 In this model the solvent is replaced by a continuum medium and the solute is located in the middle of some convenient elliptic cavity. The corresponding solvation free energy, namely, the electrostatic interaction between the solute and the polarized solvent, is given by
where the f l are the spherical multipole moments of the solute and t h e r ' are the reaction field (RF) factors. The RF factors depend only on the shape of the cavity and on the dielectric constant of the solvent. The multipole moments are calculated according to the formulaI6
where qi and ri are the value and the location of the different charges and the origin is located on the Ag atom. The linear and central symmetry of the complex yields nonzero moments for m = 0 and 1 even only. The values of the moments are given in Table 1. Since we assume the same geometry, the two complexes Ag1(CN)2- and Ag0(CN)z2- have the same multipole values, except the total charge M",. The shape of the cavity is deduced from the location of the different atoms and from their van der Waals (rVdW) radii." The problem is that these parameters are not known for C-, Ag, and Ag+. Moreover the method has been mainly developed for neutral molecules, which weakly perturb the solvent, and may yield significant errors when used on ionic species in its standard form. We have thus decided that the radii are adjustable parameters, which may be optimized in order to reproduce the experimental values of AGosolvof Ag+ and CN-. In addition, we have considered that rc- = r ~ this : assumption is supported by the fact that the CN- ion is quite stable and therefore, concentrated and that C- has the same electron configuration (s2p3)as N. 'Moreover, we have considered that r A g + = rAgo, owing to the definition of the fictitious state of Ago(CN)22- (section 3). We have found the optimum values rc- = r~ = 1.38 A and ' A g + = rAgO = 1.27 8. Note that these values are rather small (for instance the value of the van der Waals radius of N is 1.55 A). Of course we have carefully examined the consequences of the variation of the parameters in the vicinity of their optimum value: the results may be seen in Tables 2 (variation of rN with r A g = 1.27 A) and 3 (variation of r A g with r~ = 1.38 A). These tables will be examined in detail in the next section.
Rtmita et al. In addition, the value of the solvation energy may be affected by a variation of the parameter Re,, through a variation of both the multipole moments and the RF factors. We show in Table 4 (obtained with the optimum radii) that a variation of Re, of 0.1 au between 4.15 and 4.25 au leaves the solvation energy of Ag0(CN)2*- unaffected and induces a variation of the solvation energy of Ag1(CN)2- of only 1%. We eventually discuss the convergence of the calculation: we give in Table 5 the RF factors and the cumulated contributions of the calculation of AGosolvwith the optimum radii. It may be seen that the convergence of the multipole series seems to be achieved at 1 = 6. 4.2. Solvation Energy of Agi(CN)2- and A$(CNh2-. The results of Table 2 call for the following comments: 1. The order of magnitude is roughly AGosolv= -3 eV for the ions of charge - 1 (Ag1(CN)2- and CN-) and - 10. eV for the ion of charge -2 (Ago(CN)22-). We thus verify that the solvation energy is roughly monitored by the square of the charge, in agreement with the simple Bom model. 2. It may be seen that a variation of r~ of 0.15 A induces significant variations of the solvation energy of the different species: CN- (0.38 eV), Ag1(CN)2- (0.48 eV), and Ag0(CNh2(0.70 eV). In every case reducing the radius makes the solvation energy more negative. Again this is consistent with the Born model. 3. The species Ag0(CN)z2- displays a variation which roughly is the double of the variation displayed by CN-. This shows that the two ligands of the complex are active with respect to solvation. 4. The species Ag1(CN)2- displays a much smoother variation. This shows that the solvation of this complex is monitored not only by the CN- but also by Ag+. 5. We call the "optimized value" of the solvation energy the value yielded by the optimized radii. For Ag1(CN)2- this value is -3.34 eV, which is smaller (in absolute value) than the experimental valueI8 by 0.16 eV (5%). We attribute this error to the cavity model, best suited to neutral species. We shall not improve it within the present work. 6. The "optimized value" for Ago(CN)22- is -10.62 eV. Since the experimental value is not available, we can only estimate the error. Since the present complex is an ion, it is clear that the above value must be too small in absolute value, and moreover since its charge is double, the error must be larger than 5%, the error on Ag1(CN)2-. The results of Table 3 call for the following comments: 7. A variation of r A , of 0.3 induces a variation of the solvation energy of Ag+ of 1.07 eV, which is quite large. 8. Unlike Ag', the complexes Ag1(CN)2- and Ago(CN)22display smooth variations (of 0.13 and 0.32 eV, respectively) of their solvation energy. We conclude that the Ag+ ion is (at least within the present model) largely imbedded in the comblex: of course it influences the solvation through its charge, but not through its volume. We have verified that setting r A g = 0 still yields an acceptable value (-3.42 eV) for the solvation energy of Ag1(CN)2-.
5. Redox Potential of A ~ ~ * ( C N ) Z - / A ~ ~ O ( C N ) ~ ~ 5.1. Results and Discussion. The redox potential of A ~ I ' ( C N ) ~ - / A ~ ~ O ( CisN evaluated )~~from eq 9 and 10. Its values according to the variation of the radii are given in Tables 2 and 3. The following may be seen: 1. The variations of r~ and r A , induce variations of E" which are much smoother than the corresponding variations of the solvation energies: 0.2 V across Table I1 and Table 111.
J. Phys. Chem., Vol. 99, No. 35, 1995 13201
Redox Potential of Agl'(CN)2-/Aglo(CN)2*-
TABLE 2: Values of the Solvation Energy of Different Species and of the Redox Potential for Different Values of the Parameter rc- = rN calculated values experimental values1* 1.40 1.45 1.30 1.35 1.38 rc- = rN (A) -3.60 -3.48 -3.64 -3.64 -3.72 AGos0dCN-) (eV) -3.86 -3.28 -3.13 -3.50 -3.44 -3.34 AG",,I~(A~(CN)~-) (eV) -3.61 - 10.53 -10.31 - 10.62 - 10.76 AG0sn~v(Ago(CN)2*-) (eV) -11.01 -2.71 -2.49 -2.57 -2.61 -2.64 E" (VI TABLE 3: Values of the Solvation Energy of Different Species and of the Redox Potential for Different Values of the Parameter r ~ =~~ + A ~ O calculated values experimental values'* 1.40 1.27 1.3 1.10 1.20 rAgt = rA$" (A) -4.31 -4.00 -4.40 -4.41 -5.07 -4.66 AGosodAg+)(eV) -3.26 -3.50 -3.34 -3.32 -3.31 AG0sndAg(CN)2-) (eV) -3.39 - 10.44 -10.62 - 10.59 - 10.62 AGosoIv(Ago(CN)22-)(eV) - 10.76 -2.71 -2.61 -2.62 -2.52 -2.58 Eo (VI TABLE 4: Values of the Solvation Energy of the Two Complexes and of the Redox Potential for Different Values of the Parameter Re, Re, ( a 4 4.15 4.2 4.25 AG"sndAg(CN)2-) (ev) -3.35 -3.34 -3.33 AGo,,lv(Ago(CN)22-)(eV) -10.63 - 10.62 -10.62 Eo (V) -2.61 -2.61 -2.60
EO
(V/NHE)
t &+/Ago
-2
TABLE 5: Reaction Field Factors and Cumulated Values of the Solvation Energy of the Two Complexes 1,l'
0,o
22 032 4,4 0,4 2,4 66 0,6 2,6 4.6
f,P(au) 0.26 0.31 x 10-3 -0.41 x 10-2 0.78 10-7 0.38 x 10-4 -0.44 x 10-5 0.17 x lo-" -1.2
10-7
0.17 x 10-7 -0.35 x 10-9
AGO,,I, (eV) Ag(CN)zAgo(CN)12-
-3.55 -8.81 -4.83 -5.24 -5.88
-3.25 -3.26 -3.22 -3.40 -3.34
-14.21 -19.47 -11.51 -11.92 -13.21 -10.58
-10.58 - 10.5I - 10.69 - 10.62
Obviously this is due to a partial cancellation of errors: both complexes are poorly described by the cavity model, but roughly in the same way. 2. Our "optimized value" is E" = -2.6 V. It is difficult to evalute the error on this value: the error on the solvation energy difference is likely to be small, but the arbitrariness of the fictitious state of Ago(CN)z2- makes the discussion difficult. We have tried another definition of this fictitious state, namely the extrapolation of the asymptotic curve (displaying a 1/2R behavior), and got very similar values. In addition, the'present result is consistent with recent experiments.21 This "optimized value" is much lower than the redox potential of Ag+/Ago with no ligands (-1.8 V). This shows the strong complexation effect of the cyanide. In the absence of nuclei or impurities, only extremely strong electron donors, like the solvated electron, can reduce Ag1(CN)2- to Ag0(CN)2*-. In other terms reaction 6 is thermodynamicallypossible, in contrast with reaction 5, which is not. This is summarized in Figure 2. Moreover, this very low value explains the noticeable stability of Ag1(CN)2- in aqueous solution toward reduction. This stability is due to the protection of the Ag+ by the two CN-. The ionization potential of the free silver atom is strongly decreased by the solvent effectI9 and even more by the complexation, as shown in Figure 3. The ligand thus reinforces the solvent effect. 5.2. Complexation Constant of Ago with Two CN-. The redox potential can be used to evaluate the complexation
Figure 2. Relative positions of the redox potentials of Ag'/Ago with and without CN- ligands with respect to the solvated electron. IP(eV)
'I I 4
In Gas Phase In Aqueous Solution Figure 3. Effect of the hydration and of the complexation on the ionization potential of the silver atom.
constant (K') of Ago by 2CN-. This is done through the cycle (in water solution) Ag+ + 2CN-
(111)
tiI t
+ e-
-
-1 .E
v
4.059log K
Ag'(CN)P-
+
Ago 2CN-
tIIt
4.059 log IC
+ e- 5AgO(CN)22-
which gives the following equation:
E"(Ag'(CN)2-/Ago(CN)22-) = Eo(Ag+/Ago) - 0.059 log K -I-0.059 log K' (16) This yields K' = lo7 at 298 K. This value is quite large. This implies that the complexation equilibrium is strongly shifted toward the formation of the complex and that Ago(CN)22- is predicted to be quite stable in solution, unlike the gas phase, where we have proved it to be autoionizing and dissociative.
13202 J. Phys. Chem., Vol. 99, No. 35, 1995
Rkmita et al.
6. Conclusion
References and Notes
We have evaluated the redox potential of the Ag11(CN)2-/ Agln(CN)z2- couple with the aid of a thermodynamical cycle, including a fictitious state of the complex Ag0(CN)z2- in the gas phase. We have used the SCF method for the electronic structure of the species and the cavity model for the solvation effects. The redox potential of the Ag+/Agn couple in aqueous solution (- 1.8 V) is significantly decreased by the cyanide ligands down to E" = -2.6 V. It is clear that the cavity model is poorly suited to the solvation of ions. We have circumvented this problem by considering that the radii of the atoms are adjustable parameters. Of course this method is disputable, but it preserves the simplicity of the model. In this respect the present results are only preliminary. Improving the model would imply the explicit treatment of a first solvation layer, namely, the quantum chemical study of the two complexes, solvated by a few water molecules. This work is in progress. Note that the present model is strengthened by the fact that both complexes display comparable errors, which partially cancel out when only the redox potential is considered. Other nuclearities should be addressed. Actually it seems that the redox potential of aggregates Ag,+/Ag, is decreased by ligands whatever their nuclearity, and all the more when the ligand is strongly bound: this has been experimentally shown on very large nuclearities.2n In addition, it would be interesting to extend the study to other complexed systems.
(1) Swallow, A. J. Radiation Chemistry, an Introduction; (Wiley: New York, 1973). (2) Schwarz, H. A. J . Chem. Educ. 1981, 59, 101. (3) Delcourt, M. 0.;Belloni, J. Radiochem. Radioanal. Lett. 1973, 13, 329. (4) Basco, N.; Vidyarthi. S. K.; Walker, D. C. Can. J . Chem. 1973, 59, 2309. (5) Henglein, A. Ber. Bunsen-Ges. Phys. Chem. 1977, 81, 556. (6) Henglein, A.; Tauch-Treml, R. J. Colloid I n t e ~ a c eSci. 1981, 80, 84. (7) Mostafavi, M.; Marignier, J. L.; Amblard, J.; Belloni, J. Radiat. Phys. Chem. 1989, 34, 605. (8) Belloni, J.; Mostafavi, M.; Marignier, J. L.; Amblard, J. J . Imaging Sci. 1991, 35, 68. (9) Anbar, M.; Hart, E. J. J . Phys. Chem. 1965, 69, 271. (10) Mulvaney, P.; Giersig, M.; Henglein, A. J . Phys. Chem. 1993, 97, 7062. (11) Reiss, H. J . Phys. Chem. 1985, 89, 4207. (12) Barthelat, J. C. Mol. Phys. 1977, 33, 159. (13) Huber, K. P.; Herzberg, G. Constants ojDiatomic Molecules; (Van Nostrand Reinhold Co.: New York, 1979). (14) Moore, C. Atomic Energy Levels; NBS circular 467, 1949. (15) Rinaldi, D.; Ruiz-Lopez, M. F.; Rivail, J. L. J . Chem. Phys. 1983, 78, 834. (16) Buckingham, A. D. Adv. Chem. Phys. 1967, 12, 107. (17) Rinaldi, D.; Rivail, J. L.; Rguini, N. J . Comput. Chem. 1992, 13, 675. (18) Krestov, G . A. Thermodynamics ofSolvation; Ellis Honvood Series in Physical Chemistry; University of Wanvick, 1991; p 122. (19) Khatouri, J.; Mostafavi, M.; Amblard, J.; Belloni, J. Z. Phys. D: At. Mol. Clusters 1993, 26, 82. (20) Remita, S.; Mostafavi, M.; Delcourt, M. 0. New. J . Chem. 1994, 18, 581. (21) Texier, I.; Mostafavi, M. To be published.
Acknowledgment. We thank the Laboratoire de Chimie ThCorique in Nancy for providing us with the cavity program and for helpful comments, and J. Belloni and M. 0. Delcourt for helpful discussions.
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