130
Joseph Williams and Sergio Petrucci
~ r e s s ~ r e Relaxation - ~ u ~ ~ Kinetics of the Complexation of Nickel( I I) Thio yanate in Methanol' h Williams and Sergio Petrucci* Deparbnont of Chemistry, Polytechnic lnstitute of Brooklyn, Brooklyn, New York I1201
(Received May 3, 1972)
Pressure-jump relaxation spectra of the complexation of Ni(NCS)+ and NCS- in methanol in the concentration range 10-3-10-1 M and a t the temperatures 19.7, 25, 30, and 33.5" are reported as the averaged relaxation times. The data are interpreted by a two-step dissociative interchange mechanism. In terms of this mechanism the activation parameters suggest an increase of the activation entropy for s01vent exchange of Ni(MeOH)5NCS+ with respect to Ni(MeOH)e2+. Calculations of the outer-sphere parameters AS0 and AH0 are done in a corrected way by means of the Fuoss theory of association. Alterr,ate calculations of AS0 and AH0 by means of the Bjerrum theory are reported.
Introduction After the development of the techniques of relaxation kinetics and their application to aqueous solutions,2a activity in recent vears has shifted to the investigation of the mechanism of complexation of transition metals in nonaqueous solvents. The general Eigen multistep mechanism,2a reclassified by Langford2b as dissociative interchange, has found support in the work by Ellgen and Pearson3 for Ni2+ in methanol with various nucleophilic ligands. More recently, however, Bennetto and Galdin4 have questioned the general validity of the relation kf = Kohl (where KO is the outer-sphere ion-pair pre-equilibration constant and k1 is the rate constant for solvent exchange). Bennetto and Craldin' argue that values of kf(expt) as low as 1/100 of the requireld product Kokl have been found. It is true that the k e x c h should be multiplied by a statistical factor 1/S, with S a parameter corresponding to the solvation number of the cation in the second coordination shell. However, S ha:, been found in water2b to be equal to 5-10 (by comparing /zp(expt) with K o k e x c h ) , about one order of magnitude too small in order to rationalize the above difference?,. The doubt that soime of the observed discrepancies5 were due to the theoretically calculated KO was clarified when an investigation" of the outer-sphere ion-pair formation proved that the calculated KO'scorrespond in order of magnitude to the experimental ones. The problem of determining the mechanism of complexation becomes even more challenging when the entrance of the sccond ligand is considered. Hammes and Steinfeld7 were Lhe first to notice changes in the rate constants of sub3titution with respect to the entrance of the first ligand i n the case of aqueous Ni2-t and Co2+ with amino acids. A qualitative rationalization in terms of a releasing effect on the rate of the exchange of the solvent by the first ligand was proposed. Later, similar results were found by Hunt, ~t ul , in a series of papers.8 Recently the pressure jump kinetics of NiNCS+ with NCS- in lMeOl3 at 25" have been investigated by Hoffmann, et a1 9 The same rate constant, within experimental error, 1 hat Ellgen and Pearson3 determined for the process T\;i(MeOM)62 d- N C Z Ni(MeOH)5NCS+, namely, kf = 1.5 x 105 M I sec 1, has been found. However, whereas KO = 150 M - I for a 2:1 electrolyte, KO is only 10 M-1
-
The Journalof Physical Che~ifslry,Vol. 77, No I , 1973
for a 1:l electrolyte. Therefore, for the reaction studied by Ellgen and Pearson (assuming the interchange mechanism) k l = 1 x IO3 secc1 which is in excellent accord with the rate constant for solvent exchange. On the contrary, for the reaction studies by Woffmann, et d , 9 hl = 15 x lo3 sec-1, that is, 15 times as high. The doubt that a circular argument had been used, namely, that the differences were due to the assumption of the mechanism rather than to differences in rates of exrhange of NiS$-+ with respect to NiSGNCS+ (S = MeOH), was present. One way to try to ascertain the reality of a difference between the two reactions was to determine the parameters of activation for the second reaction. For this reason it was decided to reinvestigate Ni(NCS)Z in methanol by the available pressure-jump relaxation technique extending the work to various temperatures. Experimental Section The pressure-jump instrumentation has been described eisewhere.5.10 For the present work the 361-66 Tektronix carrier amplifier connected to the 564 oscilloscope was used. The temperature of the solution was maintained within k0.05" by a Forma Junior circulating bath. The temperature of the solutions in the pressure-jump cell was measured by a Pt--Pt-10% Rh thermocouple assembly within f0.1". (1) This work is part of the thesis of J. Wiliiams in partial fulfillment to the requirements of the Ph.D. degree, Polytechnic lnstitute of Brooklyn. Support by the IBM Corporation in the form of a graduate fellowship to J.W. is acknowledged. (2) (a) M. Eigen and L. DeMaeyer, "Techniques of Organic Chemistry," Vol. V i l l , Part 1 1 , 2nd ed, S. L. Friess, E. S. Lewis, and A . Weissberger, Ed., Interscience, New York, N. Y., 1963, p 895; (b) C. H. Langford and H. Gray, "Ligand Substitution Reactions," W. A. Benjamin, New York, N. Y . , 1965; C. H. Langford and T. Stengle, Annu. Rev. Phys. Chem., 19, 193 (1968); C. H. Langford, "Ionic Interaction," Vol. / I , S. Petrucci, Ed., Academic Press. New York, N. Y., 1971 Chapter i. R. G. Pearson and P. Ellgen, lnorg. Ghem., 6, 1379 (1967). H. P. Bennettoand E. F. Caldin, J. Chem. SOC.,2191, 2198 (1971). G. Macri and S. Petrucci, lnorg. Chem., 9,1005 (1970) A. Fanelli and S . Petrucci, J. Phys. Chem., 75, 2649 (1971). G. G. Hammes and J. G. Steinfeld, J . Amer. Chem. Soc,, 84, 4639 (1962). M. Grant, H. W. Dodgen, and J. P. Hunt, J. Amer. Chem. Soc.. 92, 2321 (1970); A. G. Desai, H. W. Dodgen, and J. P. Hunt {bid., 91, 5001 (1969);92, 798 (1970). F. Dickert, H. Hoffmann, and W. Jaenicke, Ber: Bunsenges. Phys. Chem., 74,500 (1970). D. Saar, G. Macri, and S. Petrucci, J. lnorg. Nucl. Chem., 33, 4227 (1971).
Complexation of NilMCS):! in Methanol
131
PJi(N@S)z (K & K Labs.) was dried a t 105" and used without further purification. Methanol (Matheson Coleman) (ACS) was distilled over aluminum amalgam in a 5-ft all-Pyrex Vigraux column collecting only the central portion (about 1/31. The boiling point was 64.5'. Solutions were prepared by weighing Ni(NCS)z on an analytical balance (Mettler 6 H) and dissolving the salt into a volumetric flask. Dilutions from this stock solution were prepared by pipetting a desired amount into a volumetric flask and diiuting to mark with distilled methanol. The comparison solution for the pressure-jump cell was prepared by matching the impedance of the Ni(NCS)Z solution to be examined with a methanolic solution of Bu4NBr . Two stock solutions were prepared during the present work. Dilutions from both solutions gave the same results within experimenlai error. Filling of the pressure-jump cell took less than 60 sec. Longer period exposure to atmosphere gave no apparent deviations in press#ure-jumprelaxation spectra. After having filled the cell, the system was left to reach temperature equilibrium; generally, 1 hr was sufficient. This was chccked by the pressure-jump recording which was repeat.ed a t 15-min intervals until constancy of relaxation times wit,hin experimental error was reached. About five-seven photographs of the relaxation spectra were recorded for each solul:ion. esults Table I reports the concentrations and averaged relaxation times with their standard deviations at the various temperatures investigated. The relaxation times were calculated from the oscillograph recordings as l / e of the ordinate corresponding to the maximum displacements of the bridge potential. In the Concentration range investigated (10-1-10-3 M ) the concen.tration of NiZ+ is negligible (Table 11). Hoffmann, et al.,9 have shown that the average ligand number ii determined t h o u g h potentiometric titrations can be quantitatively reproduced by considering only the species NiN@S+ and. N ~ ( I V I C Swithout )~ the necessity of accountLE I: C I D ~erntrations I: of Ni(NCS)Z, Averaged Relaxation Times, and 'Their Standard Deviations at the Various Temperatures 1 nvestigated
M
7,
rnsec
C, M
t = 19.7" 6.6!5 :k1.33 4.70 :* 0.50
0.002
0,005 0.010
3.76:i.0.39 2.37:& 0.11 .1.56 :k0.10 1.21 :Ir 0.15
0.025 0.05 0.10 t
300
0.05 0.10
0.50 r t
0.01 0 0.015
0.05
0.10
_.I
3.41 :k 2.82 :$ 2.17 :t 1.45 :k 0.93:tr 0.72 .t
0.001 0.002 0.005
0.001 0.002 0.005 0.010 8.025
0.21 0.10 0.06
0.001 0.002 0.005
0.07
0.010 0.025
0.09 0.03 0.04
0.05 0.10
jM"2)
for
ing for higher aggregates such as Ni(NCS)s - and Ni(NCS)2-. The two thermodynamic formation constants for the two former species areg K1 = 4.9 .X 105 M - l and KZ = 2000 M-1 at 25". Because of the above, a t 25" the system Ni(NCS)Z in MeOH in the concentration range in question (Table 11) is in effect a 1 : l electrolyte, that is, NiNCS+ and NCSpartially associated to Ni(NCS)Z, all the other species being negligible. This being the case, a plot of 7-l us .\/cl with C the total concentration of electrolyte, should be approximatively a straight line. Indeed, according to EigenZa for a symmetrical electrolyte one may write 7-l
= k,@
+ k,,
(1)
where
~ a ~ c ~ ~ a ~ i o ~ ~
C.
Figure 1. Plot of 7 - l (sec-l) vs. d[Ni(NCS)2] the temperatures investigated
Ni(NCS)Z at
7,
t = 25" 4.08 3.74 2.63 1.84 1.31 0.88 0.67
msec
f 0.05
f 0.10 rt 0.17 f 0.24 f 0.10 f 0.03 f 0.04
t = 33.5O 2.70 f O . 1 1 2.09 f 0.12 1.50 f 0.20 1.20 f 0.09 0.775f 0.039 0.571 f 0.036 0.386 f 0.01 0
8 zz 2aCy*Z
and
K =kf/h for large K one has a = 1/v& and taking y=2 = 1 as a first approximation
Plots of 7 - l us. Zre for Ni(NCS)Z in MeOH a t 19.7, 25, 30, and 33.5" are shown in Figure 1. The solid straight lines have been calculated by least-squares analysis. This analysis gives a t 25" a slope = 4523.9 M - 1 / 2 sec-1 and an intercept = 78.80 sec-I. Ratio of slope over intercept gives K = 824 M-I which is of the same order of magnitude as the thermodynamic9 Kz = 2000 M - I . Then kf = 6.49 x 104M-1 sec-1 and k R = 78.8 sec -1. In the same figure the crosses represent the average data of Hoffmann, et ~ l . a ,t 25". ~ The agreement with the present data is excellent. The straight linea reproduce the trend of the data without systematic deviations within experimental error. At 25" it would be enough to use the value of Kz = 2000 M - - l to compute 8. Since, however, no association data are available at the other temperatures, a successive approximation method has been tested to see whether kinetic data would suffice to provide Kz and therefore 8. With the value of Mz determined from the r - 1 us. v'C plot, a The Journal of Physical Chemisfry, l,/ol. 77, No. 1, 1973
Joseph Williams arid Sergio Petrucci
132
TABLE II: Concentration of Various Species Present for Ni(NCS);!in MeOH in the Concentration Range 10-3-10-'Ma [Ni(NCS)p],
Mb
[Niz+],
[NiNCS+], M
M
[NCS-1, M
[Ni(NCS)d M
-__I_-
0.001
3.00 X 4.00X 4.60X l o W 6 0.050 7.06X 0.10 8.9 x 10-6 * Calculations has& on K, = 490,000 M-' and K P = 2000 M - l . 0.005 0.010
5.72 x 1.55x 2.48 x 7.52 x 1.15 X
10-4 5.80 x 10-4 1.57x 10-3 10-3 2.50 x 10-3 10-3 7.53 x 10-3 10-3 1.15 X lo-;! lo-' b Stoichiometric concentration of the salt.
4.30 X 3.45 x 10-3 7.50 x 10-3 4.25 X IO-' 8.85 X
TABLE Ill: Results of 81f @Visec-I), k R (sec-'), Their
Standard Deviation, and Average Ratio ( M - l ) for Ni(NCS)' in MeOH at the Temperatures Investigated -
--____-__-_
i,"c 19.7 25 30 33 5
kf,
Rz, ( M - 1 ) 10-3
k K , see-'
( ~ - sec-')10--5 1
0 86 3~ 0 04 1 40 A 0 05 221 f 0 0 8 3 32 & 0 44
27 f 27 73 f 25 72 33 91 k 54
1.9 31 37
*
first set of values of 0 has been computed as follows. A two equation system has been solved K
I-a
- ___ a'Cy*2
I
(3)
where S and A are the Debye-Huckel values. The parameter a has been set a = 6 X 10-8 cm, as done by Ellgen and P e a r ~ o n .Starting ~ with the above value, within two cycles of approximations, one obtains kf = 1.40 X lo5 M - l sec--l and hi3 = 73.17 see-I, their ratio being Kz = 1915 M'' in good accord with the thermodynamic Kz = 2000 M - l . The above figures have been calculated by least-squares analysis. Considering the last figures approximate enough t o I)e taken as final, statistical analysis of standard error on slope and intercept1' give kf = (1.40 & 0.05) 105 &-:I sec-:;, K R = (73 16 25) sec-1. .and average
uz= 1.9x 1 0 3 ~ 4 - 1 .
Similar calculations at 30 and 33.5" are reported in Figure 2 and Table III together with the data a t 25". For the data at 19.7", since htt is practically zero, the value of Kz has been obtained as extrapolation of a plot of log Kz us. l/Tgiving Kz = 1.2 Y. IO3 M - I . In order to ccirreiate the data of kf a t various temperatures, the Eyring expression has been used
ki
~
kT h
-.
eSSr#!R
e-AHiSIRT
(4)
a plot of log ( h f , l T )us. 1/T gives an intercept = 17.46 f 0.90 and a slopa? .=: -4422 f 269, both being calculated by least-squares analysis (Figure 2 , insert). Then because n tercept = log dope
one obtains ASt kcal/mol.
h
# + 0.43 ASi R
AH^+ = - -2.30 R
== 32.6 f 4.1 eu and
AHff = 20.2 f 1.2
Discussion By retaining the interchange mechanismzb for the present reaction The Journal of Physi(:al Chemistry, Vol. 77,
No. I, 1973
0x IOJ-
Figure 2. Plot of T-' (sec-I) vs. 0 = 2acyi* (MI for Ni(NCS)Z at the temperatures investigated. Insert shows plot of log ( k f / T ) ( M - I sec-I) vs. ( 1 / T ) (deg-l) for Ni(NCS)2in MeOH.
(SCN)S4NiS2+
+ NCS-
KO
(SCN)S,NiL+S,NCS-
f? , /L,
S,Ni(NCS)?
(5)
with S a solvent molecule, the rate constants for the overall process (SCN)S,Ni2+
+ NCS-
k,
e S,N#NCS), ki,
(6)
are related to the above ones by the expressioiis kf = Kokl and k R = k - 1 . One may retain the Fuosdz association constant for KO and for a 1:1 electrolyte in methanol one has
for u = 6 X 10-8 cm. Then.at 25" k l = 14.7 X lo3 sec-I, about 15 times the rate of solvent exchange from the first coordination shell of NiS$+, as already observed.g Using the same mechanism, the two relations between enthalpies and entropies, respectively, may be derived
AH^+ AS{'
+ AH,+ = AS" + .AS,'
=
AH,
(8)
By retaining the values of AH0 and AS0 calculated through the Fuoss theory (see Appendix), the calculated activation parameters for the second step of the mechanism of association of NiNCS+ and NCS- in methanol are (11) Y. Beer, "Introduction to the Theory 5 Error," 2nd ed, AddisonWesley, Reading, Mass., 1962, Chapter V i .
Complexation of NI(NCS)2 in Methanol
133
AMI* = (20.2 1.4) f 1.2 = 18.8 f 1.2 kcal/mol 0.25X l o 3 = %-exp(kT L 15>800 e x p ( T ASexciIi1I ) RT As,* := (32.6 - 9.1) f 4.1= 23.5 &= 4.1 eu one may calculate ASexch*'' = +5.6 eu in accord with It may be ser'n that although the reported inconsistencies Bennett0 and Caldin.4 The same result could be obtained (see Appendix) for AS0 and AH0 are theoretically signififrom the relation cant, they are numerically small and contained within the standard errors of AHr# and A S f # . N ASexcbiI1 = ASexciiiil,+ R In -$ Comparison of Activation Parameters for Solvent S u b [S, stitutzon and Exchange The activation parameters for It may then be seen that the second-order entropy of solvent exchange from the first coordination shell of Nisolvent exchange from NiSG2+ is only 5.6 eu activation for (Me0H)s2t are bNex,hi" = 15.8 kcal/mol and ASexcp,+ = compared to the determined A S f * = 32.6 f 4.1 eu for sol8.0 eu.I3 It may be seen that the enthalpy of activation for vent substitution from (NiNCS)+S5. This seems to prove the entrance of the second ligand NCS- is comparable that there is a substantial increase for the entropic pawith one for solt-ent exchange. The difference in the rates of solvent irukstitution from NiSs2 i- and NiS5NCS+ seems rameter of activation for solvent ejection from NiSs2' to NiNCS+S5 (unless one invokes the unlikely possibility of due mainly to an increase in the entropy of activation for a ligand participation in the free energy of the activated solvent substitution of the second species. In other words, rate, namely a S N or~ associative activation mode of subthe obtained results suggest that2b the "dissociative event" stitutionzb for NiNCS+S5). of ejecting one solvent molecule from the activated comOne should point out, however, in evaluating the above plex is entropically easier for NiSsNCS+ than for NiSs2+. result, that criticism has appeared14 concerning formula 9 A parallel effect (of a slight increase in the enthalpy of and the conversion of kexchl to kexchII using the bulk solactivation and strong increase in the entropy of activation vent concentration. for solvent exchange has been found by Hunt, et aL,8 for Ni(HzQ)$ +, Ni(H20)4 bipy2+, and N i ( H 2 0 ) ~bipy22 t the Appendix values being LHexclT*= 12.1 f 0.5, 12.6 f 0.5, and 13.7 Calculation of AH0 and AS0 for the Outer-Sphere Pref 0.5 kcal/mol and i!&exch' = 2.6 f 2, 5.1 f 2, and 9.2 f Equilibration Process. The usual procedure reported in 4 eu, respectively. (Clearly, extensive data are needed for textbooks15 is to calculate AGO from a model (so called any generalization to be drawn. double-sphere model) visualizing two charged spheres 2 be In substance, the present data and the previous ones by and Z - e , brought from infinity to the contact distance a. Ellgen3 are consistent with the Eigen dissociative interThis result extended to one mole of ions is change mechonism if one accepts the hypothesis that the rate of solvent, erchorzge from Ni(Me0H)s NCS+ is larger than for Nic MeQH)62+. Comparison of Second-Order Activation Parameters. which may be rewritten, multiplying and dividing by T The objectio~couid be advanced that the conclusion just and recalling that N = R / k , with R the gas constant reached for the activation parameters is based on the assumption of the interchange mechanism, namely, it is viAGO = -RTb (11) tiated by tha> same circularity in argumentation9 that gives k l = 15 X 103 sec-2 from the experimental kf = 1.5 with b the Bjerrum parameter b = I Z + Z _ l e 2 / a l ) k T . Notice that since AGO = -RT In KO, this implies KO = X lo5 M - I t3ec- 1. eb, namely, the Denison-Ramsey16 association constant It is important therefore to be able t p compare the excalculated from a Born cycle. perimental AiYf# and AS,+ with the activation parameThe calculation of AS0 then proceeds as ters for solvent exchange without the drawback of the assumption of a particular mechanism. This may he done, according to Caldin and B e n n e t t ~ , ~ by transforming the activation parameters for the firstand since order rate constant 1of solvent exchange, kexch', to the parameters for the second-order rate constant, kexchII,called total exchange rate.4 The relation between the two constants is4 I -
(9)
where IV, is the coordination number of the cation in the first coordination sphere taken as 6 and [SI is the concentration of solvent, Ai: 25" for methanol p = 0.7866 g/cc, [SI = (1000 X 0.7866/32.04) = 24.551 and kexch" = 0.25 X IO3 sec-I A13sumption of Eyring expressions for kexchI* and hex&' with elimination of the frequency factors k T / h , taking the derivative with respect to T of the logarithmic expressions for kexchI1 and kexch', and neglecting the dependence of [SJ on teinperature give the identity A H e x c k + I ~=
Then from thc expression
nHexchfI
which is identical with the usual formula reported in text books.15
once the transformations indicated between eq 10 and 11 R. M. Fuoss, J. Amer. Chem. Soc., 80,5059 (1958). 2. Luz and S. Meiboom, J. Chem. Phys., 40,2688 (1964). C. H. Langford, J. Chem. Educ., 46, 557 (1969). , K. J. Laidler, "Chemical Kinetics," McGraw-Hill, Mew York, N. Y . , 1965 Chapter 5, p 212; A. A . Frost and R . G. Pearson, "Kinetics and Mechanism," J. Wiley, New York, N. Y.. 2nd ed, 1961, Chapter 7, p 135. (16) J. T. Denison and J. 8. Ramsey, J . Amer. Chem. SOC.. 77, 2615, (1955). ~
The Journal of Physical Chemistry, Vnl. 77, No. 1, 1973
Joseph Williams and Sergio Petrucci
134
are performed. If one writes
KBl =
48 Nu3
1000
b3Q(b) =
with
one obtains
The above is consistent with the use of the Denison-Ramsey16 equation for KO,namely, K D R= e b . If, however, the Fuoss association constant12 K F = KoOeb is retained as is usually the case for KO, an inconsistency occurs in the above calculations of AGO, A&, and AH0 and consequently in AHl# and A&;* in eq 8. In fact if KO= Kooeb
Comparing eq 14 with 11 and 12 we may notice that the and R In KoO, respectively, are missing terms -RT In in the latter equations. In the Fuorss equation the terms KoO = 47rNu3/3O00 represent, approximately the volume fraction due to 1 mol of ions or, in other words, the excluded volume from the solution due to the presence of ions.l7 It is natural then that, because of the entropic nature of Ko0 in the association constant K O (representing the probability of collision between neutral particles increasing with the cross section ~a2),17k(oo had to appear in the AS0 expression. Combination of eq 14 LO calculate AH0 gives the same expression as eq 13 sin612 the terms RT In Ko0 cancel out. Hence, the two theories, the Denison-Ramsey and the Fuoss, give the same result for I&, but not for AS0 and AGO. Unfortunately, many authors have used eq 12 for calculating ASo, and in calculating Ahro have used the Fuoss expression for AGO AH()
d In D
I=
RT In Kp - RTb d l n T ~
( + ___ dlnT
AHo := - RT 1 n K a -. R T b 1
mixing up t be two theories and introducing inconsistencies such as omitting the R In Ko0 terms for AS0 and improperly introducing --ET In Ko0 into AH,. At 25", for methanol (d log Drd'dt) = -0.264 X 10-2, D = 32.63, a = 6 Y. em, and IZ,Z-J = 1 giving Ko0 = 0.545 and AS0 - 1.20 i10.3 = 9.1 eu. Notice that by retention of the Denison-Ramsey theory for KO,AS0 =: 10 3 eu. Similarly, for AH0 from eq 13 RTb = 1688 cal, RT2b (d In D/dT) = -3071 cal, and AH0 = 1.38 kcal/mol. The =: 358 cal/mol if improperly introduced term -RT ln in the AH0 expression would make AH0 = 1.74 kcal/mol. Exception:; could be raised for having used the Fuoss or the Denison-Ramwy theory instead of the more accurate Bjerrum theory to evaluate the thermodynamic parameters AS0 and AHo. 'lrhe Fuoss statistical theory is superior to the thermodynnniic theory by Denison and Ramsey because it contains the preexponential term Koo which accounts for the excluded volume due to the presence of 1 mol of ions per lite,. of solution. However, it has been experimentally shown18 that the exponential term eb of the Fuoss theory i s an approximation to the Bjerrum term 6 3 Q f b )in the expressior,
The Journal of Physical Chemistry, Vol. 77, No. I , 7973
(In fact, only for low D where b is large h3Q( b ) = e h f b = eb and the two theories give similar numerical values of association constants.) Therefore given In KBl = In 3K0° 4- 3 In b
+ In Q(b)
d In KBI dT
then
butlg
and
also AH,,
= - RT
+ TASO
In K B ,
At 25" in methanol 01 = 6 X cm, b = 2.858, Q(b) = 0.305, K B ~= 11.63 M - l , AS0 = 13.63 eu, and AH0 = 2.6 kcal/mol. It may be seen that although these last two figures are significantly different from the ones calculated by the Fuoss theory, namely, AS0 = 9.1 eu and AH0 = 1.4 kcal/mol, the differences are of the orders of the standard errors of AS,.' and AHf#. In any case the conclusions of the present work remain unaltered regardless of the theory chosen for calculating the outer-sphere association parameters.20 S. Petrucci. "Ionic Interactions," Academic Press, New York, N. Y., Vol I, 1971, p 136. C. DeRossi, B. Sesta, M. Battistini, and S. Petrucci, J. Amer. Chem. SOC.,94, 2961 (1972). A. Bronwell, "Advanced Mathernatics in Physics and Engineering," McGraw-Hill, New York, N. Y., 1953, p 110. Note Added in Proof..In the above calculations of AGO,AH0 and ASo the concentration scale used was the molarity. For thermodynamic calculations it may be useful to modify the derivation to the molality concentration scale. Given mp e, with m, p , and c the molality, solvent density, and molarity, respectively, K m and K, the equilibrium constants in the two units, it follows that K, = Kcp (neglecting differences in the activity coefficierxts). Then d In Km/dT = d In KcldT - CY with 01 (the expansion coeeficient of the solution) = (d In p/dT). I f one identifies Kc with one of the theoretically calculable association constants, say K F ~then ~ eq ~ 14~and , 13 from above become AGO = RTln Ko0 RTb - RTirip ASo = R In Ko0 -RTb d In D/dT R In p RTa AH0 = - R T b ( l f d In D / d In T ) RT2a The additional contribution to AGO, &SO, and AHo in methanol at deg-') are 6AGo = 25" ( p = 0.7866 g/cm3, 01 = 1.16 X 0.14 kcal/mol, 6ASo = -1.2 eu, and 6AHo = -0.21 kcal/mol, bringing the thermodynamic parameters in the molality scale to AS0 = 7.9 eu and AH0 = 1.2 kcal/mol.
=
-
-
+
-
-
