J . Phys. Chem. 1990, 94, 7821-7830
7821
Apparent and Partial Molar Heat Capacities and Volumes of Cr0,2-(aq), HCr0,-(aq), and Cr,O?-(aq) at 25 'C: Chemical Relaxation and Calculation of Equilibrium Constants for High Temperaturest Jamey K. Hovey*.t and Loren G. Hepler* Department of Chemistry, University of Alberta, Edmonton, Alberta, Canada T6G 2G2 (Received: January 11. 1990)
Calorimetric and densimetric measurements on solutions containing K2Cr04in water and dilute KOH and on solutions of K2Cr207in water and dilute perchloric acid have been made at 25 "C. The resulting apparent molar heat capacities and volumes have been analyzed considering the presence of Cr042-(aq), Cr20?-(aq), and HCr04-(aq) to obtain the standard-state partial molar heat capacities and volumes for these species. This analysis required consideration of the speciation of Cr(V1) in the solutions and evaluation of the 'chemical relaxation" contributions to the heat capacities. The standard-state volumes and heat capacities for Cr042-(aq),Cr2072-(aq),and HCr04-(aq) were found to be 19.9 i 0.2, 72.8 i 0.4, and 45.0 f 0.6 cm3 mol-' and -271 i 4, -131 i 5 , and 2 f 5 J K-l mol-', respectively. The present results for the standard-state partial molar volumes and heat capacities have been combined with calorimetric standard-state enthalpy changes for evaluation of the temperature dependence of equilibrium constants to 175 "C. Comparisons of calculated equilibrium constants with recent measurements provide evidence that such calculations are usefully accurate over the stated temperature range.
Introduction Thermodynamic properties of aqueous chromium(V1) species are useful in many applications, several of which have been mentioned by Palmer et al.' and have been the subject of many previous investigations and a critical review.2 Previous investigations have provided the basis for s e l e c t i ~ n ~ ? ~ of reliable values of Nf?,AGIO, and So for the important species CrOd2-(aq), Cr2072-(aq),and HCrO,-(aq), all at 25 "C. As discussed later, a few prior investigations have also led to apparently reliable values for the partial molar heat capacity and volume of Cr042-(aq). For Cr20:-(aq) and HCrO,-(aq) the only reliable partial molar volumes come from the investigation of Rohwer et al.4 We do not know of any previous calorimetric measurements leading to information about partial molar heat capacities of Cr2072-(aq)and HCrO,-(aq). A principal purpose of the present investigation of aqueous solutions of Cr(V1) has been to obtain values for the partial molar heat capacities of Cr,072-(aq) and HCrO,-(aq). This requires that the relaxation contributions (due to various temperaturedependent equilibria) to the total measured heat capacities be evaluated and accounted for before appropriate properties can be assigned to the various individual ionic species. A secondary purpose has been to try to improve on the accuracy of already available partial molar capacity and volume values for CrOt-(aq). A principal application of partial molar heat capacities involves their use in extrapolating Gibbs free energies (also equilibrium constants and standard electrode potentials), enthalpies, and entropies from 25 "C to much higher temperatures. In this connection we are fortunate to have for comparison the excellent results' of Palmer et al., who have made accurate equilibrium measurements over the range of temperature to 175 "C. The equilibria in aqueous solutions containing Cr(V1) can be represented by several combinations of chemical equations, including the following that we later use: Cr042-(aq) + H2O(l) = HCrO,-(aq)
+ OH-(aq) Cr2072-(aq)+ H20(I) = 2Cr042-(aq) + 2H+(aq) Cr042-(aq) + H+(aq) = HCrO,-(aq)
(1)
(2) (3)
Note that it is not necessary to use both eqs 1 and 3; it is only for convenience that we have used each of these equations in 'This article is dedicated to Professor Kenneth S. Pitzer on the occasion of his 75th birthday. *Present address: Department of Chemistry, Pitzer Research Group, University of California, Berkeley, CA 94720.
0022-3654/90/2094-7821$02.50/0
different calculations. Other representations of equilibria in aqueous Cr(V1) systems are the following: Cr2072-(aq)+ 20H-(aq) = 2Cr042-(aq)
+
+ H20(I)
2HCr04-(aq) = Cr2072-(aq) H20(1)
(4) (5)
Equilibrium constants and other thermodynamic properties associated with the different chemical reaction equations are related to each other by the equilibrium constant and other thermodynamic properties for the ionization of water that is represented here by H20(1) = H+(aq)
+ OH-(aq)
(6) We also mention that there is evidence2 for other Cr(V1) species such as H2Cr04(aq)and HCr207-(aq) that are not expected to be important for the dilute solutions that we have investigated. Our uses of the new partial molar heat capacities and volumes that have resulted from the present investigation have focused on the pressure and especially the temperature dependences of certain equilibrium constants. These calculations have been done with the following thermodynamic equations:
iar), 3 d In K
AH"
=
(7)
(F), = AC,"
(9)
Experimental Section Water used for all solutions was passed through an activated charcoal filter and then through a Milli-R04/Milli-Q reagent grade mixed-bed ion-exchange system to achieve a final resistivity of 16 MQ cm or more. ( 1 ) Palmer, D. A.; Wesolowski, D.; Mesmer, R. E. J . Solution Chem. 1987, 16, 443-463.
(2) Dellien, 1.; Hall, F. M.; Hepler, L. G. Chem. Rev. 1976, 76, 283-310. (3) Wagman, D. D.; Evans, W. H.; Parker, V. B.; Schumm, R. H.; Halow, 1.; Bailey. S. M.; Churney, K. L.; Nuttall, R. L. The NBS tables of chemical thermodynamic properties. Selected values for inorganic and C, and C, organic substances in SI units. J . Phys. Chem. Ref. Datu 1982, I I , Suppl. 2. (4) Rohwer, E. F. C. H.; Brink, J. A,; Cruywagen, J. J. S. Afr. J . Chem. 1978, 31, 19-22.
0 1990 American Chemical Society
Hovey and Hepler
7822 The Journal of Physical Chemistry, Vol, 94, No. 20, 1990 Standard solutions of NaCl were prepared by mass from the salt (Fisher certified, ACS) after drying at 1 IO "C for 24 h. Potassium hydroxide (1 M, certified solution) and perchloric acid (60R, certified ACS) were both obtained from Fisher and used without further purification. Three stock solutions of potassium chromate were labeled C I , C2, and C3. The first stock solution (CI) was prepared by mass from K2Cr04(Fisher certified ACS, dried at 150 "C for several m KOH. Some of the same sample of K2Cr04 hours) and from Fisher was recrystallized, it was dried at 150 "C, and then a known mass was dissolved in a known mass of purified water to yield the stock solution C2. The third stock solution (C3) was prepared by mass from purified water and another sample of K2Cr04(BDH certified ACS, dried at 1 I O OC to constant mass). Other solutions of K,Cr04 were prepared by diluting CI with m KOH and by diluting C2 and C3 with purified water: all such dilutions were done by mass. Potassium dichromate (Fisher certified ACS, dried at 150 "C m HCIO, to yield stock to constant mass) was dissolved in solution D1. Another stock solution (D2) was prepared by dissolving some of the same sample of K2Cr207(freshly dried at 150 "C) in purified water. Other solutions were prepared by diluting m perchloric acid and stock solution stock solution DI with D2 with purified water. All of these preparations and dilutions were done by mass. Heat capacities were measured at 25.00 f 0.01 "C with a Picker flow micro~alorimeter.~ The specific heat capacity of pure water was taken from Ke1I6 and had the value 4.1793 J K-' g-l. Corrections were made for a very small unsymmetrical heat loss, based on our measurements of the heat capacity of a standard solution of NaCl and reference data from Desnoyers et al.' The resulting heat loss correction factor was found to be f = 1.005 f 0.001. Densities of all solutions were measured relative to water at 25.00 f 0.01 "C with a Sodev 0 3 D vibrating tube densimeter,* which had been calibrated with purified water and standard solutions of NaCI. The density of pure water was taken from Ke1I9 and had the value 0.997 047 g cm-): the densities of standard NaCl solutions were taken from Picker et al.* Temperatures of the calorimeter and densimeter were controlled by separate Sodev CT-L circulating baths, which were maintained constant to f O . O O 1 "C. Temperatures were measured with calibrated thermistors mounted in the circulating tubes connecting the baths and calorimeter or densimeter. Results Calorimetric heat capacities and the densities of solutions can be related to the desired partial molar heat capacities and volumes of solutes by way of apparent molar heat capacities and volumes. ) defined by Apparent molar properties of a solute ( @ Y 2are Y - n, Y," @Y2= (10) n2
in which Y is the extensive property (heat capacity or volume) of a specified mass of solution, YIo is the same extensive property of 1 mol of pure solvent, and n , and n2 are the amounts (moles) of solvent and solute, respectively. For a "simple" solute such as NaCl for which there are no speciation complications, it is appropriate to express the composition dependence of @Y2in terms of a Debye-Huckel equation of the form "2
= my2" + A * I ' I 2
+ ByI
(11)
in which @Y2"is the apparent molar property at infinite dilution (5) Picker, P.; Leduc, P. A,; Philip, P. R.; Desnoyers, J . E. J . Chem. Thermodyn. 1971, 3, 631-642. (61 Kell. G. S. Thermodvnamic and transcart DroDerties of fluid water. In Wate-A Comprehensioe'Treorise; Frank;. F.: Ed.; Plenum: New York. 1972; Vol. I , pp 363-412. (7) Desnoyers, J. E.; deviser, C.: Perron, G.:Picker, P. J . Solution Chem.
(equal to the desired standard state partial molar property), AY is the Debye-Huckel limiting slope,lO,llw is a "valence factor", I is the ionic strength, and B y is an adjustable parameter. The ionic strength, valence factor, and molality are related by I = um =
t/zCmizi2
The heat capacities and volumes of present interest cannot be analyzed in the simple fashion outlined above because of complications associated with mixed electrolytes and also because of relaxation contributions (due to temperature-dependent chemical equilibria) to the measured heat capacities. The treatment that we have used proceeds from the simple treatment outlined above as follows. The starting point for reporting the experimental results for our more complicated solutions and then for analyzing these results is provided by using an "experimental" apparent molar property (@PXP) that is defined according to
in which n2,irepresents the stoichiometric or initial amount (moles) of solute that was dissolved in the specified amount of water. The superscript exp is used to emphasize that these quantities are based on stoichiometric compositions and the direct results of calorimetric and densimetric measurements; no chemical equilibrium calculations have been involved in obtaining these quantities. We also emphasize that nz in eq IO refers to the number of moles of solute in the solution, while nz,i in eq 13 refers to the number of moles of solid solute added to the specified amount of water to form the solution. For "simple" solutes such as NaCI, n, = n2,i. The situation is more complicated for solutions of Cr(V1) because of the various equilibrium reactions; hence n2 # n2,,. Equations that relate these defined experimental apparent molar quantities to heat capacities and densities and to stoichiometric molalities (m2,,)are 1 4p-P
=
+ m2jM2 - - 1 d dl " m2.1
(14)
and
Here cp, d,cp0,,and dI0represent the heat capacities and densities of solution and pure water and M z is the molar mass of the solute (substance 2). Some of our results in the form of stoichiometric molalities, - I ] and (d - dlo), directly measured values of [cpd/(cpo,dlo) and the defined experimental apparent molar heat capacities and volumes are given in Table I. Note that stock solutions labeled C2, C3, and D2 and solutions made by dilutions of these stock solutions contain no electrolyte other than those in the added compound of Cr(V1) and other ions formed by chemical reactions such as those represented by eqs 1-5. Solutions of K2Cr04 labeled C I and of K2Cr207labeled DI were prepared from the specified compound of Cr(V1) and added potassium hydroxide in C 1 and added perchloric acid in DI. In Table I1 we report our experimental results for these solutions, making use of "mean" apparent molar heat capacities and volumes (@Pean) that are defined according to
where n3 is the amount (moles) of added KOH or HCIO,.
1976. 5. 605-616.
(8) Picker, P.; Tremblay, E.; Jolicoeur, C. J . Solution Chem. 1974, 3. 377-384. (9) Kell, G . S. J . Chem. Eng. Datu 1967. 12. 66-69.
(IO) Bradley, D. J.; Pitzer, K. S.J . Phys. Chem. 1979, 83, 1599-1603. ( I I ) Ananthaswamy. J.: Atkinson, G.J . Chem. Eng. Dora 1984, 29, 81-87.
The Journal of Physical Chemistry, Vol. 94, No. 20, 1990 7823
Partial Molar Quantities of Cr(V1) Species TABLE I: Experimental Apparent Molar Properties for K2CrO4(C2, C3) and K2Cr20, (D2)
-44.17 44.22 43.18 43.00 42.34 41.45 40.68 40.19
27.59 27.48 20.01 18.87 14.55 9.707 5.785 3.639
0.366 76 0.366 76 0.253 72 0.237 92 0. I78 36 0.11535 0.066 662 0.041 187
54.037 54.01 8 37.821 37.821 26.825 17.498 10.184 6.319
-137.15 -135.63 -155.04 -1 57.50 -168.95 -182.54 -1 96.06 -204.38
0.496 94 0.496 94 0.377 53 0.377 53 0.295 78 0.237 35 0.18352 0.10588 0.053 230
72.498 72.533 55.779 55.756 44.050 35.573 27.679 16.132 8.177
K2Cr04(C3) 44.76 44.70 43.70 43.76 43.06 42.49 41.90 40.84 39.93
35.56 28.26 28.33 22.95 18.90 14.99 8.987 4.679
-121.10 -137.60 -138.16 -1 50.75 -160.81 -171.23 -187.97 -203.74
0.292 68 0.242 I7 0.205 39 0.167 25 0.147 23 0.11131 0.074 353 0.037 747
56.488 47.058 40.1 I4 32.848 28.996 22.408 14.818 7.567
K2Cr207 (D2) 95.50 95.10 94.79 94.39 94.21 93.71 93.20 92.71
26.22 22.06 18.93 15.62 13.82 10.67 7.037 3.536
11.01 4.71 0.16 -5.15 -7.34 -10.01 -1 2.06 -8.60
Calculations of Partial Molar Quantities A first step toward obtaining the desired partial molar heat
TABLE 11: 'Mean" Apparent Molar Properties for K2Cr04(Cl) and K2Cr& (D1) - 1 0 3 [ ( ~ , d / 4c m a n , i03(d- d I o ) , W - n , m2,i, g mol kn-I cm3 mol-' ~ . ~ , d-,11~ )J K-? mol-] m K2Cr04(CI); KOH = 1.003 X 77.619 45.16 37.65 0.53441 -1 14.76 60.270 44.20 30.26 -1 31.31 0.409 87 43.36 23.93 46.096 -147.02 0.3 10 29 20.14 38.103 42.81 0.254 90 -156.25 30.795 42.19 16.58 0.204 69 -166.29 12.86 23.453 41.55 0.15488 -176.16 16.194 40.83 9.08 I 0.106 18 -187.80 7.654 38.94 4.400 0.049 394 -205.76 0.353 41 0.290 64 0.245 54 0.201 49 0.150 99 0.095 288 0.050 920 0.022 347
K2Cr207(DI); 67.570 56.019 47.6 IO 39.3 I O 29.690 18.886 10.167 4.503
HCIOI = 9.43 X IO" m 96.21 30.92 95.78 26.02 22.36 94.45 95.05 18.53 14.12 94.45 93.90 9.028 93.26 4.820 91.97 2.045
20.40 12.04 5.67 1.81 -5.49 -10.79 -1 1.43 -2.41
Equations used in calculating these defined mean apparent molar properties are 1
+ m2jM.2 + m3M3 _ - 1 d m2,i + m3
@ p a n=
cp( 1
@CPmcan =
dl O
+ m2,iM2 + m3M3) - cp0 I m2,i + m3
(17) (18)
Here m3 and M, are the stoichiometric molality and the molar mass of the added electrolyte (KOH or HCIO,). We have defined the mean apparent molar properties as in eq 16 to be in accord with Young's rule12for analyzing the properties of solutions of mixed electrolytes. For our purposes, Young's rule can be expressed as 9ymcan
=F,@F
+
in which 6 is an excess mixing term, F2 = m2,i/(m2,i m3),F3 = m3/(m2,i + m 3 ) ,and V x P and @Y3refer to the apparent molar properties of solutes 2 and 3 at the total ionic strength of the solution. Because the molalities of added KOH and HC10, were much smaller than the molalities of the chromium solutes, it is justifiedI2 to take 6 = 0. Since we want to use equations of type eq 19 to obtain values of @PxP from the reported @Pan values, we use values of +Y3for aqueous KOH from Roux et al.I3 and for aqueous HCIO, from our previous ~ o r k . ' ~Table , ~ ~111 summarizes the results of our and @Fn for K2Cr04(CI) and calculations leading from @Cpman K2Cr207( D l ) to @Cpexp and @ P X P values for these solutions.
+ F3@Y3+ 6
(19)
capacities and volumes from the tabulated experimental results is to calculate the equilibrium molalities of the species present in our solutions. These calculations have been made by using the equilibrium quotients at specified ionic strengths of NaCl from Palmer et al.' along with auxiliary data from Busey and MesmeP and from Liu and Lindsay.I7 Because the present measurements pertain to solutions at ionic strengths controlled by the Cr(V1) species present and not to added NaCl as do the equilibrium constants, these values may not reflect the "appropriate" values at finite ionic strengths. However, these differences are expected to be small and become negligible upon extrapolation to infinite dilution. The stoichiometric molalities, calculated equilibrium molalities, ionic strengths, and equilibrium quotients1 at the specified ionic strengths are listed in Tables IV and V . It is unnecessary to list speciation of K2Cr04( C l ) solutions in this way because the small molality of KOH in these solutions was sufficient to ensure that molalities of all Cr(V1) species other than Cr042-(aq) were negligible. To analyze the contributions of temperature-dependent chemical equilibria to measured heat capacities, we follow Mains et al.I8 and express the total enthalpy H of a homogeneous system containing k components as
H = ZnkHk k
(20)
in which nk and Hk are the amounts (moles) and partial molar enthalpies of the components. Differentiation of eq 20 leads to an equation for the heat capacity of the system at equilibrium:
The first term on the right is the conventional compositional heat capacity of the system while the second term is the relaxation contribution due to temperature-dependent equilibria, which we call @Cpleland evaluate as @Cple'= AH,,(
g)p
AH,,, and (aa/aT), in this equation represent the enthalpy of the reaction under consideration and the shift in equilibrium composition with changing temperature. It has been shown in several previous investigati~nsl~-*~ that the experimental apparent ~~
(12) Young, T. F.; Smith, M. B. J . Phys. Chem. 1954, 58, 716-724. (13) Roux, A. H.; Perron, G.; Desnoyers, J. E. Can. J . Chem. 1984, 62, 878-885. (14) Hovey, J. K.; Hepler, L. G.; Tremaine, P. R . Thermochim. Acta 1988, 126, 245-253. (15) Hovey, J. K.; Hepler, L. G. Can. J . Chem. 1989, 67, 1503-1509. (16) Busey, R. H.; Mesmer, R. E. J . Chem. Eng. Data 1978,23, 175-176. (17) Liu, C.; Lindsay, W. T. J . Solution Chem. 1972, I , 45-69. (18) Mains, G. J.; Larson, J . W.; Hepler, L. G. J . Phys. Chem. 1984,88, 1257-1261, (19) Woolley, E. M.; Hepler, L. G. Can. J . Chem. 1977, 55, 158-163. (20) Jolicoeur, C.; Lemelin, L. L.; LaPalme, R. J . Phys. Chem. 1979,83, 2806-2807.
(21) Allred, G. C.; Larson, J. W.; Hepler, L. G. Can. J . Chem. 1981, 59, 1068-1073.
7824
Hovey and Hepler
The Journal of Physical Chemistry, Vol. 94, No. 20, 1990
TABLE 111: Young's Rule Analysis of 'Mean" Apparent Molar Properties of K2Cd4 (CI) and K2Cr20, ( D l ) [(m2,,+ m 3 ) / m 2 , i 1 + ~ " , t m d ~ , i ) ' V 3 , 'pXp3 [(m2,,+ ~ d / m 2 , , 1 ~ C ~ ' " ~ "( .M ~ / M ~ , ~ ) ' C ~ ,'C ~ ,-P m2.u mol kg-' cm3 mol-' cm3 mol-' cm3 mol-' J K-' mol-' J K-' mol-' J K'Pm&' 0.53441 0.409 87 0.31029 0.254 90 0.204 69 0.154 88 0.106 18 0.049 394
45.25 44.30 43.50 42.98 42.39 41.82 41.22 39.730
0.02 0.02 0.03 0.03 0.03 0.04 0.06 0.12
K2Cr04( C I ) 45.23 44.29 43.47 42.95 42.36 41.78 41.16 39.62
-115.0 -131.6 -147.5 -1 56.9 -167.1 -177.3 -1 89.6 -209.9
0.353 41 0.290 64 0.245 54 0.201 49 0.150 99 0.095 288 0.050 920 0.022 347
96.21 95.79 95.45 95.06 94.45 93.91 93.28 92.08
0.00 1 0.001 0.002 0.002 0.003 0.004 0.008 0.019
K2Cr207 t D 1) 96.21 95.79 95.45 95.06 94.45 93.91 93.27 91.99
20.4 12.0 5.7 1.8 -5.5 -10.8 -1 I .4 -2.4
-0.13 -0.19 -0.27 -0.35 -0.46 -0.65 -1.00 -2.34
-1 14.8
-131.4 -147.2 -156.5 -1 66.6 -176.7 -1 86.6 -207.6
2 x 10-4 2 x 10-4
20.4 12.0 57 I .8 -5.5 -10.8 -1 1.4 -2.4
1 x 10-4
I -2 -8 -2 -7
x IO" x 10-4 x 10-4
x lo-' x 10-3
Not used in least-squares fits.
TABLE IV: Speciation of K2Cr0, Solutions (C2, C3)O m2(2KC+Cr042-), m i ,mol ke-' mol ka-l
1O h 3 ( K++HCrO,J, mol kg-'
0.366 76 0.253 72 0.237 92 0.17836 0.11535 0.066 662 0.041 187
0.366 70 0.253 67 0.237 87 0.17831 0.1 15 31 0.066 628 0.041 159
K2Cr04(C2) 5.78 5.21 5.1 1 4.67 4.05 3.35 2 80
0.496 94 0.377 53 0.295 78 0.237 35 0.18352 0.10588 0.053 230
0.496 88 0.377 47 0.295 7 2 0.237 30 0.18347 0.105 84 0.053 199
K2Cr0, (C3) 6.25 5.83 5.44 5.10 4.7 1 3.94 3.09
I , mol kg-'
log
QI
1.1003 0.7612 0.7 138 0.5351 0.3461 0.2000 0.1236
-8.034 -7.966 -7.955 -7.908 -7.843 -7.772 -7.717
1.4908 1.1326 0.8873 0.7121 0.5506 0.3176 0. I597
-8.098 -8.040 -7.994 -7.955 -7.913 -7.831 -7.745
log
QH
-14.01 -1 3.90 -1 3.89
-13.82 -1 3.74 -13.65 -13.59 -14.11 -14.02 -13.94 -13.89 -13.83 -1 3.72 -13.62
"log QI and log QJ9 refer to reactions represented by eqs 1 and 39. respectively TABLE V: Speciation of K2Cr1O7Solutions (Dl, D2)' r r 1 ~ ( 2 K + + C r ~ O ~ ~ - ) ,m3(K++HCr04-), m2,i,mol kg-' mol kg-l mol kg-'
104m4(K++H++Cr042-), mol kg"
I , mol kg-l
log QI
log
Q39
K2Cr20, ( D I ) 0.35341 0.290 64 0.245 54 0.201 49 0.150 99 0.095 29 0.050 92 0.022 35
0.32647 0.265 84 0.222 43 0.180 22 0.132 13 0.079 79 0.039 21 0.014 57
0.053 55 0.049 29 0.045 92 0.042 28 0.037 48 0.030 8 I 0.023 26 0.0 1 5 46
0.292 68 0.242 17 0.205 39 0.16725 0. I47 23 0.113 13 0.074 35 0.037 15
0.267 8 I 0.21920 0.183 94 0.147 57 0. I28 57 0.096 44 0.06045 0.027 59
0.049 44 0.045 66 0.042 62 0.039 1 2 0.037 08 0.033 17 0.027 63 0.020 18
'log Q , and log
Q39
K2Cr207 (D2)
3.28 3.06 2.87 2.66 2.36 1.95 I .48 1.03
1.0339 0.8477 0.7141 0.5837 0.4346 0.2708 0.1413 0.05947
-8.022 -0.7985 -7.955 -7.922 -7.876 -7.810 -7.732 -7.648
-13.99 -13.93 -13.89 -1 3.84 -1 3.78 -1 3.70 -1 3.60 -13.51
3.06 2.85 2.68 2.46 2.34 2.09
0.8538 0.7041 0.5953 0.4826 0.4235 0.3231 0.2095 0.1033
-7.987 -7.953 -7.925 -7.892 -7.872 -7.834 -7.177 -7.699
-13.93 -13.88 -13.84 -13.80 -13.77 -13.72 -13.66 -13.57
I .75
1.30
refer to reactions represented by eqs 1 and 39, respectively.
molar heat capacity of the solute can be expressed as the apparent molar heat capacities of the solute species plus OCple': 4cpexp = Qcspecies + 4~ rei (23) P
It is therefore possible to obtain the desired QCpSpicsfrom our experimental results by evaluating QCTtaccording to eq 22 and
P
(22) Peiper, J. C.; Pitzer, K.S. J . Chem. Thermodyn. 1982. 14. 613-638.
(23) Larson, J. W.; Zeeb, K . G.; Hepler, L. G. Can. J . Chem. 1982, 60, 2141-2150.
Partial Molar Quantities of Cr(V1) Species then subtracting this quantity from values of Wpmcan. These calculations can become quite complex when several equilibria are present. In these cases there may be some advantage in using a numerical method of evaluating the relaxation contributions instead of an analytical method such as the one used here.22 Our calculations of +C?' according to eq 22 are based on the a values calculated from the equilibrium quotients (at specified ionic strengths) from Palmer et al.' that were used in the speciation calculations that have already been summarized. It appears that the best AH' values for reactions represented by eqs 1-5 and thence the best AHr' values for Cr04z-(aq), HCr04-(aq), and Cr20:-(aq) are those cited by Dellien et aL2and Wagman et ale3 The more recent AHo and AH (referring to specified ionic strengths) values calculated by Palmer et al. from the temperature dependences of their equilibrium constants and equilibrium quotients (referring to specified ionic strengths) appear to be somewhat less accurate than those citedzv3above that were based primarily on results of calorimetric measurements. However, the AH values from Palmer et al.' have the advantage of referring to specified ionic strengths appropriate to the solutions of present concern. We have therefore based our choices of enthalpies to be used in eq 22 on estimates of the accuracies of the various AHo and AH values and the estimated enthalpies of dilution that might be used to obtain AH from A H o , as specified in the following discussions. We now turn to analysis of the heat capacities and volumes of the solutions that we have investigated, beginning with the relaxation contributions to the heat capacities of the solutions labeled C2 and C3 (K2Cr04,no added KOH). Consideration of the equilibrium quotients ( Q 1and Q4) for reactions 1 and 4 shows that the extent of reaction 1 is sufficient to warrant our attention, but the extent of reaction 4 is entirely negligible. Letting a represent the fraction of Cr042-(aq) that reacts to form HCr04-(aq), we have Q , = a2m2,i/(1 - a ) and thence Q , (1 - a ) = a2m2,i. Differentiation of this last equation with respect to temperature (constant p and m 2 , i )leads to
The Journal of Physical Chemistry, Vol. 94, No. 20, 1990 7825 TABLE VI: Relaxation Contributions to Heat Capacities of K2Cr04 (C2. C3) Solutions
~
K2Cr04(C2) 0.366 76 0.253 12 0.237 92 0.17836 0.1 15 35 0.066 662 0.041 187
1.577 2.052 2.147 2.620 3.513 5.023 6.808
0.496 94 0.377 53 0.295 78 0.237 35 0.18352 0.10588 0.053 230
1.258 1.544 1.841 2. I50 2.569 3.718 5.796
5.723 7.432 7.771 9.464 12.67 18.06 24.47
0.30 0.39 0.41 0.50 0.67 0.96 I .29
K2CrO4(C3) 4.572 5.600 6.663 7.776 9.272 13.41 20.85
0.24 0.30 0.35 0.41 0.49 0.71 1.10
Now we let a and /3 represent the fractions of Cr2072-(aq)and Cr042-(aq) that react as indicated by eqs 2 and 3 . Equilibrium molalities of the various species are therefore as follows: [CrzO$-] = (1 - a)mz,i,[CrOd2-]= 2 ( a and [HCr04-] = [H+] = 2 ( a - /3)mz,i. We begin by considering solutions ( D 2 ) for which m3 = 0. Equilibrium quotients for eqs 2 and 3 can be expressed as
16(a - P)4mz,i3 Qz =
(27)
1-a
and
The next step is to multiply eq 27 by (1 - a),differentiate the resulting equation with respect to T (constant p and m2,i),and then combine this equation with the van't Hoff equation to obtain Next we write the van't Hoff equation in the form
( 1 - a)Q2AH2 RP
64m2,?(a - @I3[
and combine with eq 24 to obtain
(g)p- (
(29)
Equation 29 is rearranged to Before describing our use of eq 26, we emphasize that Q , and AH1 represent the equilibrium quotient (expressed in terms of molalities) and enthalpy of the reaction represented by eq l at the total ionic strength of the solution. We have used values of Q , from Palmer et a1.l in eq 26. For A H , we can use either the A H , values for the ionic strengths of our solutions from Palmer et al. or the AH,' values from Dellien et aL2 that are probably more accurate but do not refer specifically to the ionic strengths of our solutions. The results of calculations with eq 26 (combined with eqs 22 and 23) that are summarized in Table VI have come from our use of AH,' values, but we note that use of AH, values would not have caused a significant change in the final standard-state partial molar heat capacity for CrO&aq) that will be calculated and reported later in this paper. We now turn to consideration of the relaxation contributions to the heat capacities of solutions of K2CrZ07.Consideration of the pertinent equilibrium constants shows that two coupled equilibria must be considered. These calculations might be done in terms of any of several combinations of equilibrium quotients and AH values for reactions represented by eqs 1-5; we have chosen to use eqs 2 and 3 . (24) Barbero, J . A.; Hepler, L. G.;McCurdy, K.G.;Tremaine, P.R.Can. J . Chem. 1983, 61. 2509-2519.
(30) Returning to eq 28, we differentiate with respect to temperature and combine with the van't Hoff equation to obtain
I
2m2.i 2 ( a - P )
X
Rearrangement of this equation gives
Substitution of the right-hand side of eq 30 for ($3/87''), in eq
The Journal of Physical Chemistry, Vol. 94, No. 20, 1990
7826
Hovey and Hepler
TABLE VII: Relaxation Contributions to Heat Capacities of K2Cr20, Solutions
0.35341 0.290 64 0.245 54 0.201 49 0. I50 99 0.095 29 0.05092 0.022 35
7.577 8.533 9.4 I O 10.56 12.49 16.27 22.99 34.82
7.623 8.480 9.352 10.49 12.41 16.17 22.84 34.59
9.66 I 10.84 11.87 13.22 15.49 19.70 26.52 36.49
0.292 68 0 242 I 7 0 205 39 0 16725 0 14723 011313 0 074 35 0 03: 7.5
8.498 9.486 10.44 1 1.77 12.67 14.75 18.70 26.90
8.446 9.427 10.38 1 1.69 12.59 14.66 18.58 26.73
10.79 11.96 13.12 14.58 15.74 17.95 22.21 30.12
K2Cr207 (Dl )
K2Cr207
9.681 10.82 11.85 13.19 15.45 19.66 26.46 36.41
13.36 14.99 16.42 18.29 21.42 27.24 36.68 50.47
2.84 3.17 3.47 3.87 4.53 5.76 7.75 10.67
16.20 18.16 19.89 22.16 25.95 33.00 44.43 61.14
(D2) 10.77 1 1.93 13.09 14.55 15.71 17.91 22.16 30.05
14.92 16.54 18.14 20.16 21.77 24.83 30.72 41.65
3.15 3.50 3.83 4.26 4.60 5.25 6.49 8.81
18.07 20.04 21.97 24.42 26.37 30.08 37.21 50.46
TABLE VIII: Standard-State (Infinite Dilution) Partial Molar Heat Capacities, Volumes, and By Parameters at 25.0 OC' cm3 mol-'
(2K+ + Cr042-)( C I ) (2K+ + Cr042-)(C2) (2K+ + Cr042-)(C3) (2K+ + Cr2072-) (K* + HCr04-)
37.91 f 0.02 38.15 f 0.02 37.64 f 0.03 91.1 f 0.3 54.6 f 0.5
QCp02.
B V.
QVZO?
solute
cm3 kg
BC,
J K-I mol-' -245.9 f 0.2 -241.8 f 0.6 -245.2 f 0.4 -105.0 f 1.4 14.7 f 2
0.120 f 0.03 0.131 f 0.04 0.134 f 0.03 -0.55 f 0.3
J ke, K-I mol-2 4.13 f 0.3 1.42 f 0.8 2.06 f 0.5 1.9 f 1.5 (30 f I O )
(Of 1) "The f values are the statistical uncertainties (standard deviations) associated with the fit of eq 1 1 to the experimental results. Total uncertainties are larger than these f values, as discussed in the text. Parentheses around BV and B , values indicate estimates of these quantities as discussed in the texi
32, followed by considerable algebraic manipulation, leads to the following equation that permits calculation of (aa/aT),:
i") =(I[ ~
PRF
T
Q2AH2( 1 - a ) [ 1 f 4m2.1Q3(.- PI1 6 4 ~ 2 . , ~-(PI3 .
11
2m2.1(. - P12Q3AH3
[
[Q2
I
I
c
+
X
+ 64m2,,3(. - PI3l[1 + %,,Q3(a
t -
o
1
PI1
A Olofsson et a / , 1978 0 Stock ( C l )
64m2.,3(a- b13 '%,&%a - P )
Stock (C2)
I
0 Stock
-230 3~
c
(33)
Values of (aa/aT), for each M ~calculated , ~ with eq 33 are then used in eq 30 to obtain corresponding values of (ap/aT),. Because the amount of HC104 added to form solutions labeled DI was very small, algebraic checks have confirmed that setting m3 = 0 as above is appropriate for calculations of (aa/aT), and (ap/aT), as described here. As done earlier for less complicated calculations for the relaxation contribution to heat capacities of solutions of K2Cr04, we have used Q2.Q3,AH2,and AH3 from Palmer et al.' and from Dellien et al.2 Results of all of these calculations for solutions of K2Cr0, are summarized in Table VII. Now that we have applied Young's rule to some of our data and have worked out the relaxation contributions to heat capacities, we are ready to apply eq 1 1 to obtain the desired values for the standard-state partial molar heat capacities and volumes of the Cr(V1) ions as follows. The solutions labeled C I were prepared by dissolving solid potassium chromate in dilute potassium hydroxide so that Cr0,2-(aq) would be the only significant species of Cr(V1). We have already applied Young's rule to our reported (Table 11) heat capacities and volumes for these solutions containing K2Cr0, KOH to obtain the quantities designated o P x P and @CpexP for
+
i
(C3)
*a -250
c 1
0
4
I
1
05
1
I
10
1
15
I / mol kg''
Figure 1. Plots of +Y2 minus the Debye-Huckel limiting law term (DHLL) for K2Cr04solutions after correction for speciation and relaxation.
aqueous (2K+ + that are listed in Table 111. Fitting equations of type eq 11 (using values of A , from Ananthaswamy leads to the and Atkinson") to these values of @ P X P and @Cpexp desired @Y2"(equal to corresponding standard-state partial molar quantities) and B y that are listed in Table VI11 for K2Cr04(Cl). Values of QY2minus corresponding Debye-Huckel limiting law terms are plotted against ionic strength in Figure 1, where similar plots are presented for solutions labeled C2 and C3, which we will discuss later.
Partial Molar Quantities of Cr(V1) Species
The Journal of Physical Chemistry, Vol. 94, No. 20, 1990 7827
L
TABLE IX: Conventional [@Yo(H+,aq)3 01 Ionic Partial Molar Volumes, Heat Capacities, and Standard-State Enthalpies for Aqueous Ions and Liquid Water at 25 OC0 0Cp" = q o ,
@p = p, cm3 mol-'
kJ mo1-l
0 12.8 -140 -27 1 -131 2 75.29
0 9.02 -4.05 19.9 72.8 45.0 18.07
0 -252.38 -229.994 -88 1 . I 5 -1490.3 -878.22 -285.83
J K-l mol-'
ion H+
K+ OHCr02-
Cr2072HCr O i H20
AH?,
"Sources for all volume and heat capacity values are discussed in the text except the volume of OH-, which was derived from MilleroZ8and from Roux et a1.I3 from the volumes of NaOH, and the heat capacity, which was derived from values of NaCI, HCI, and NaOH from Desnoyers et al.,' Tremaine et aI.,j5 and Roux et aI.l3 All enthalpy values were derived from Dellien et aL2 and the NBS table^.^
It is now possible to use the results summarized above to complete our analysis of the heat capacities and volumes of solutions of potassium dichromate. To begin this analysis we obtain properties of the hypothetical electrolyte (K+ + H+ Cr042-) at specified molalities from equations of the type
53t I
+
@Y(K++H++Cr042-) = @Y(2K++Cr042-)- @Y(K++CI-) @Y(H++CI-)(34)
+
Values of @Yfor aqueous KCI and HCI were selected from Desnoyers et Fortier et aLzsand Allred and Woolley.26 For aqueous (2K+ CrO,Z-) we have used @Yvalues calculated from the properties of K2Cr04( C l ) solutions as listed in Table VIII. The relationships between the defined experimental apparent and apparent molar heat capacities (@CpeXP) molar volumes (@Pxp) and the same properties of single electrolytes are as follows: @ P a p = (1 - C Y ) Q V ~ ( ~ K + + C ~ 2P@V3(K++HCr04-) ~O~~-) 2(a - P)@V4(K++H'+CrO4*-) - a P ( H 2 0 ) (35)
1
0
I
I
0.5
1
1 .o
1
I
1.5
I /mol kg-I Figure 2. Experimental apparent molar volume data for aqueous solutions of K2Cr207plotted so that intercepts give @Pvalues and slopes give BV values.
+
+
+
+
@CpCXP= (1 - a)Wp,2(2K++Cr2072-) 2P@Cp,3(K++HCr04-) + 2(a - P)@Cp,4(K++H++Cr042-) - a C p 0 ( H 2 0 )+ @Cprei(36) Simultaneous least-squares analysis of eqs 1 1 and 35 leads to values of @V2" and BY for (2K+ Cr2072-)and for (K+ HCr04-). Similar least-squares analysis of eqs 11 and 36 for heat and E , values for these same capacities leads to values of @Cpo electrolytes. Required values for the properties of water (Toand cpo)were taken as listed from Table IX. This kind of analysis in terms of four parameters (two @I"'values and two Byvalues) results in slight statistical overfitting of the experimental results, which has led us to reduce the number of parameters by estimating values of B y for the lowest molality electrolyte (K+ + HCr04-) as follows. The B , and Bc values for some "similar" (+1, -1) electrol y t e ~ ~range , ~ ~from , ~ ~-1.4 to 1.3 cm3 kg mol-2 and from 18 to 39 J kg K-' mol-2, respectively. Using these values as guides, we estimate BY = 0 f 1 cm3 kg mol-2 and Bc = 30 f 10 J kg K-' mol-2 for (K+ + HCr04-). Using these values for BY and Bc, we have repeated the analysis (now the only quantities to be evaluated are two @I"'values and one B y value) described above and have obtained the properties listed in Table VI11 for aqueous (2K+ Cr2072-)and (K+ + HCr04-). Letting the values of BY and Bc for (K+ + HCr04-) vary over the specified f ranges leads to changes of less than 0.2 cm3 mol-l and 2 J K-' mol-' in the @P and @Cpo values for (2K+ + Cr2072-)and to slightly larger changes in these values for (K+ + HCr04-). Graphs in Figures 2 and 3 display the properties of aqueous (2K+ Cr2072-)and (K+ HCr04-); the straight lines in these graphs are based on properties
+
+
+
+
+
(25) Fortier, J.-L.; Leduc, P.-A.; Desnoyers, J. E. J. Solution Chem. 1974, 3, 323-349. (26) Allred, G. C.; Woolley, E. M. J . Chem. Thermodyn. 1981, 13, 147-154. (27) RoUX, A.; Musbally, G. M.; Perron, G.;Desnoyers, J . E.; Singh, p. p.; Woolley, E. M.; Hepler, L. G.Can. J . Chem. 1978, 56, 24-28.
r
1 i
.
r
Y
I
i-
d
I
n
fL
-80
-901
i
K2Cr207 (aq)
i
4
-120/0
1.o
0.5
1.5
I / mol kg-' Figure 3. Experimental apparent molar heat capacity data for aqueous values and slopes solutions of K2Cr207plotted so that intercepts give QC," give B, values.
listed in Table VI11 and equations of type eqs 1 I , 35, and 36. Our first calculations of infinite dilution (standard-state) properties were for solutions of K2Cr04in dilute KOH (labeled C l ) . Allowing for the presence of KOH in these solutions presented only a very minor complication, which was handled by making use of Young's rule. Now we turn to consideration of solutions of K2Cr04 (C2 and C3) in pure water, for which the complication in analysis comes from the equilibrium reaction represented by eq 1. Again we use a to represent the fraction of CrO,'-(aq) ions that have reacted at equilibrium to form HCr04-(aq) ions. Equations that relate the previously defined and tabulated experimental quantities @PxP and @CFpto the desired properties
7828 The Journal of Physical Chemistry, Vol. 94, No. 20, 1990
of (2K+ + Cr0,2-) and the corresponding properties of single electrolytes are as follows: @PXP
+
= ( 1 - c~)@y(2K++CrO,~-)n@V(K++HCrO,-)+ nmV(K++OH-)- tuV'(Hz0) (37)
@CpeXP= ( 1 - ~u)W,,(2K++Cr0~~-) +cu@Cp(Kt+HCrO,-) + CPT,,(K++OH-) -- CYC,"(HzO) + @CFl(38) Properties for (K+ + HCr0,-) are based on Table VI11 and eq 1 1, and properties for KOH solutions were taken from Roux et al.13 The relaxation contributions to heat capacities were calculated as described earlier, with results listed in Table VI. Equations of type eq 1 1 were then fitted to the apparent molar volumes and heat capacities of (2K+ + Cr042-jobtained from eqs 37 and 38. The resulting values for the properties of (2K+ + CrO?-) based on K2Cr04(C2 and C3) are listed in Table VIII. Graphs that illustrate the properties of solutions C2 and C3 are shown in Figure 1 Comparison of Results and Selections of Best Values Millero's useful compilation28lists @F'(2K++CrO,Z-) = 37.12 cm3 mol-', based on the earlier experimental results of Jones and C ~ l v i n .More ~ ~ recent measurements by Olofsson et al.30have led to @F'= 38.17 f 0.06 cm3 mol-' for this electrolyte. Still more recently, Vasilev et have reported @Yo = 37.43 cm3 mol-'. We have applied eq 11 to recalculation of this latter value and have obtained a revised @F'= 37.69 f 0.08 cm3 mol-'. The effects of hydrolysis of CrO:-(aq) as represented by eq 1 are small and have been neglected in obtaining all of these values. Consideration of these values along with our (Table V I I I ) "(2K++Cr042-) = 37.91 f 0.02, 38.15 f 0.02, and 37.64 f 0.03 cm3 mol-' leads us to choose the weighted average (in accord with the inverse square of the estimate of the standard deviation) of 37.92 f 0.2 cm3 mol" as the best value for @V0(2K++Cr042-j. where the f indicates our estimate of the total uncertainty. Here and later it is convenient to use standard-state properties for "whole" electrolytes to obtain conventional standard-state apparent (=partial) molar volumes and heat capacities of single ions, based on @ P ( H + , a q ) 0. For this purpose we use the conventional @ P ( K + , a q )= 9.02 cm3 mol-' from Millero28 to obtain b P ( C r 0 4 2 - , a q )= 19.9 cm3 mol-" as listed in Table IX. We have carried out new calculations with the heat capacities of aqueous solutions containing K2Cr04reported by Olofsson et (@Cpo = -235.0 J K-' mol-') and Vasilev et (@C,," = -244.7 J K-' mol-'). I n these calculations we have used the new value'O,'lfor Ac and allowed for the equilibrium represented by eq 1 (presence of K+, HCr04-and also a relaxation contribution). These new calculations have led to @Cp0(2Kt+Cr042-)= -239.5 f 0.3 and -233.2 f 1.7 J K-l mol-' from the results of Olofsson et aL30 and Vasilev et al.," respectively. Here, as previously. f indicates the standard deviation of fit rather than an estimate of total uncertainty. Our present (Table VIII) values @C," = -245.9 f 0.2, -245.2 f 0.4, and -241.8 f 0.6 J K-' mol-' for (2KC + Cr0,2-) are in better agreement with the result from Olofsson et al.30than with the result from Vasilev et a1.j' We note that the latter)' measurements were made on considerably more concentrated solutions than were the present measurements or those of Olofsson et al.30 Our considerations of uncertainties in each of these values, and our opinion that results obtained from solutions containing dilute KOH are more accurate because of the absence of relaxation contributions, led us to select a weighted average @C,,"(2K++Cr04*-,aq)= -245 f 3 J K-' mol-'. Combination of this selected value with the conventional @C,,"(K+.aq) = 12.8 J K" mol-' from Desnoyers et al.7 and Roux et aL2' leads (28) Millero, F. J. Partial molal volumes of electrolytes in aqueous soluI n Water and Aqueous Solution: Structure, Thermodynamics and Transporr Processes: Horne. R. A.. Ed.; Wiley: New York, 1972; pp 519-564. (29) Jones, G.; Colvin, J . H . J . Am. Chem. Soc. 1940, 62, 338-340. (30) Olofsson. 1. V.; Spitzer. J J.; Hepler, 1. G . Can. J . ('hem. 1978. 56. 1871-1873. (31) Vasilev. V . A.; Lar'kov, A. P.: Kruchina, T. I . Russ. J . Phys. Chem. 1984, 58. 1674- 16??
tions.
Hovey and Hepler to a conventional @Cpo(Cr042-,aq) = -271 J K-I mol-' as listed in Table IX. Measurements by P e ~ c eprovided ~~ data that Fajans and Johnson33used to estimate the standard-state partial molar volume of Cr,072-(aq). Because the measurements were made on concentrated solutions and because HCr0,-(aq) and CrO&aq) were neglected in the calculations, the derivedg3volume of Cr2072-(aq) is quite uncertain and probably contains substantial errors. More recent dilatometric titrations of solutions of K2Cr04with aqueous HCI by Rohwer et aL4 and their sensible data treatment have led them to more reliable values as follows: @V"(2K++Cr20,2-)= 87.81 f 0.9 cm3 mol-' and @VO(K++HCr04-)= 53.15 f 0.18 cm3 mol-'. The first of these values is in poor agreement with our (Table VIII) @VO(2K++Cr2072-)= 91.1 f 0.3 cm3 mol-', and the second value is in fair agreement with our @V"(K++HCrO,-) = 54.6 f 0.5 cm3 mol-'. On the basis of our estimates of total uncertainties in these values, we choose the following weighted averages: @V0(2K++Cr20,2-,aq) = 90.8 cm3 mol-' and @P(K++HCrO,-,aq) = 54.0 cm3 mol-'. Combination of these values with the conventional volume of K+(aq) leads to the conventional volumes for Cr2072-(aq)and HCrO,-(aq) that are listed in Table IX. The only calorimetric results that we know of leading to apparent and partial molar heat capacities of Cr2072-(aq)and HCrO,-(aq) are those reported in this paper. We therefore use the @Cpovalues listed in Table VI11 with the conventional @Cp"( K+,aq) to obtain the conventional standard-state partial molar heat capacities for these ions that are listed in Table IX. Equilibrium Calculations The heat capacities and volumes that we have reported here can be used for many thermodynamic calculations, but in this paper we are concerned only with the relationship of these values to the temperature dependence of equilibrium constants for Cr(V1) species. One principal use of partial molar heat capacities is to combine them with calorimetrically determined enthalpies (usually for 25 "C) and with equilibrium constants (also usually for 25 "C) to calculate new equilibrium constants for other temperatures, especially for temperatures much higher than 25 "C. The largest uncertainty in such calculations is the extent to which it is reasonably accurate to assume that AC,," for a reaction involving aqueous ions is independent of temperature, which then determines the useful range of temperatures for these equilibrium calculations. Palmer et al.' have made measurements leading to equilibrium constants and related thermodynamic properties for the equilibria represented by the following equations: Cr04*-(aq) + H20(I) = HCr0,-(aq) + OH-(aq) (1) 2Cr042-(aq) + H20(I) = Cr207*-(aq)+ 2OH-(aq) (39) They also have made use of information regarding the thermodynamics of ionization of water to obtain equilibrium constants and related thermodynamic properties for the following: Cr042-(aq) + H+(aq) = HCr04-(aq) (3) 2Cr042-(aq) + 2H+(aq) = Cr2072-(aq)+ H,O(I) (40) The equilibrium results have led them to AC," = 0 for the reactions represented by eqs 1 and 39. They have made the observation that AC," is expected to be small for electrically symmetrical reactions such as eqs 1 and 39 and have stated that their data do not provide an accurate estimate of the value of AC," for either of these reactions. Now we use our standard-state partial molar heat capacities (Table IX) to calculate ACp* = 58 f 12 J K-' mol-' for reaction 1 and AC," = 56 f 1 2 J K-' mol-' for reaction 39. The K = f(73 results from Palmer et al.' have led them to AC O = 231 f 6 J K-' for reaction 3 and to AC," = 463 f 13 J I&' mol-' for reaction 40, both at t = 25 "C. The f values are their statistical estimates of errors associated with their least-squares fits. We have used our partial molar heat capacities (Table IX) ( 3 2 ) Pesce, B. Gazz. Chim.Ital. 1935, 65, 448-452. (33) Fajans, K.; Johnson, 0. J . Am. Chem. Sac. 1942, 64, 668-678
Partial Molar Quantities of Cr(V1) Species
The Journal of Physical Chemistry, Vol. 94, No. 20, 1990 7829
TABLE X: Comparison of Experimental' and Calculated Equilibrium Constants log
K
experiment' CrO?-(aq) + H20(I) = HCrO
