Heat Capacities and Volumes of Dissociation of Phosphoric Acid

Heat capacities and volumes of dissociation of phosphoric acid (kt, 2nd, and 3rd), bicarbonate ion, and bisulfate ion in aqueous solution

Departtnent of Chemistry and Oil Satzds Research Laboratory, University of Le t l~b r id~e , Lethbridge, Alta., Canada T l K 3 M 4 Received January 18, 1982

JOHN W. LARSON, KEVIN G. ZEEB, and LOREN G. HEPLER. Can. J. Chem. 60,2141 (1982). Measurements at 25'C with aflow calorimeter and aflow densimeter have led to heat capacities and densities of aqueous solutions

of H,PO,, NaH2P04, Na2HP04, Na,P04, NaHCO,, Na2C0,, H2S04, and various buffer solutions. Results are presented in terms of apparent molar heat capacities (4,) and apparent molar volumes (4,). Analysis of the experimental 4, values to obtain the desired standard state 4 , O values for single electrolytes requires that allowance be made for hydrolysis and dissociation reactions of certain solutes, along with extrapolation to zero concentration. Analysis of 4, values requires similar considerations of hydrolysis and dissociation reactions, allowance for enthalpy and equilibrium changes ("relaxation") during calorimetric measurements, and extrapolation to zero concentration. Procedures for these calculations are presented. The and 4 , O values that result from all of these measurements and calculations are used to obtain ACpO and AVO values for acid dissociation reactions, which are then related to effects of temperature and pressure on other thermodynamic properties.

JOHN W. LARSON, KEVIN G. ZEEB et LOREN G. HEPLER. Can. J. Chem. 60,2141 (1982). Des mesures a 2SC, l'aide d'un calorimttre a Ccoulement et d'un densirnetre a 6coulement, ont conduit aux capacites

calorifiques et aux densites des solutions aqueuses de H3P04, NaH2P04, Na2HP04, Na,P04, NaHCO,, H2S0, et de plusieurs solutions tampons. On prksente les resultats en termes de capacites calorifiques molaires apparentes, (I$,), et de volumes molaires apparents, (4,). Lorsqu'on effectue une analyse des valeurs experimentales de I$,, en vue d'obtenir les valeurs 4 , O de 1'Btat standard d'electrolytes specifiques, on doit tenir compte des reactions d'hydrolyse et de dissociation de certains solutes en plus de faire une extrapolation b dilution infinie. Pour faire une analyse des valeurs I$,, on doit aussi prendre en consideration les reactions d'hydrolyse et de dissociation, on doit tenir compte des variations d'enthalpie et d'tquilibre ("relaxation") pendant les mesures calorimCtriques et faire une extrapolation b concentration zero. On presente les marches b suivre pour ces calculs. On a utilise les valeurs de et de 4 , O qui resultent de toutes ces mesures et de tous ces calculs, pour obtenir les valeurs de AC,,O et de A V0 pour les reactions de dissociation des acides que l'on peut alors relier aux effets de temperature et de pression sur d'autres proprietes thermodynamiques.

[Traduit par le journal]

Introduction The thermodynamic properties of the aqueous

acids listed in the title are important in such fields as boiler water chemistry, interaction of aqueous solutions with minerals, biochemistry, marine science, and analytical chemistry. The research to be described in this paper was undertaken to provide information about changes in heat capacity (ACP0) and volume (AVO) accompanying dissociation of these acids and then to use these values in calculations of the thermodynamics of dissociation over ranges of temperature and pressure. Obtain- ing the desired standard state changes in volume from measured densities of solutions requires appropriate extrapolations to zero concentration and also "corrections" for such chemical reactions as hydrolysis and dissociation, for which we have developed convenient methods. Evaluation of standard state changes in heat capacities involves similar "corrections" and extrapolations and also requires that allowance be made for "relaxation"

'Visiting scientist from Department of Chemistry, Marshall University, Huntington, W. VA. 25701, U.S.A.

effects that are associated with the changes in equilibrium states that occur with changing temperature during the course of calorimetric measurements of heat capacities.

Experimental Our heat capacity measurements have been made with a

Picker flow calorimeter of the type previously described (1). A small systematic error in measurements with this calorimeter has been detected and corrected on the basis of measurements (2-4) on NaCl (aq) solutions. Densities of solutions were measured with a flow densimeter (5). Results of all of these measurements refer to (25.0 + 0.1)"C.

BDH AnalaR grade Na2C0, was dried at 270C. BDH reagent grade NaH2P04 and Na3P04 were recrystallized from water- ethanol and dried at 150C. Fisher certified Na2HP04.7H20 and Baker reagent grade NaHCO, were used as received. The Na2HP0,.7H20 was analyzed for water by heating to constant weight at 150C.

All compounds mentioned above were titrated with standardized HCI or NaOH and found to contain no appreciable acid-base impurities 'ind had overall purities of not less than 99.7%.

Stock solutions of sulfuric acid and phosphoric acid were prepared from ACS reagent grade concentrated acids and distilled water, and were then analyzed by titration with standardized NaOH solutions.

Buffer solutions of NaHCO,/Na,CO, and NaH2P04/Na2H-

0008-4042/82/162141-10$01.00/0 01982 National Research Council of Canada/Conseil national de recherches du Canada

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TABLE 1. Apparent molar heat capacities and volumes of aqueous NaH2P0,/Na2HP0, buffers at 25.0C

1?1 4c 4 v m 4c 4v mol kg-' J K-' mol-' cm3 mol-I mol kg-' J K-' mol-' cm3 mol-'

F2 = 0.900 F2 = 0.500 0.0440 2.2 27.95 0.0501 -61.2 17.73 0.0820 5.0 28.28 0.0719 -53.1 17.80 0.1235 11.7 28.78 0.1004 -43.9 18.68 0.1558 12.7 28.50 0.1177 -37.9 18.85 0.2206 20.4 28.76 0.1565 -30.2 19.58 0.2394 26.7 29.63 0.2028 - 19.9 20.00 0.3185 30.9 29.35 0.2460 -12.1 20.40 0.3951 41.6 30.48 0.2792 2.5 20.78 0.4823 45.1 30.10 0.3500 6.8 2 1.42 0.6921 60.9 30.99 0.4189 15.9 22.01 0.8286 73.2 32.34 0.493 1 26.9 22.58

F2 = 0.100 0.0366 -116.1 7.27 0.0538 - 112.2 7.63 0.0641 - 106.5 7.96 0.1025 -93.1 8.74 0.1030 -91.3 8.86 0.1450 -76.6 9.63 0.1681 -69.8 9.78 0.2035 -58.0 10.52 0.2410 -49.2 10.81 0.2870 -35.9 11.63 0.3504 -21.7 12.17

PO4 were prepared from known masses of the solids and apparent molar heat capacities and volumes, are distilled water. These buffer solutions were made with mole listed in ~ ~ b l ~ ~ 1, 2, and 3. ~ ~ ~ ~ t ~ ~ ~ t of these ratios equal to 911, 111, and 119. Solutions of Na3P04 were DreDared from known masses of solid and distilled water. is described in the FuAher solutions were prepared by quantitative dilutions of paragraphs. several of these solutions. Our principal interest in this investigation has

Experimental results and analyses of data Results of measurements with the Picker flow

calorimeter are heat capacities per unit volume of solution. Combination of these heat capacities with the densities that we have also measured leads to heat capacities per unit mass of solution. Further combination of these latter heat capacities with the already known (6) heat capacity of water (4.1793 JK-I g-l) leads to apparent molar heat capacities. Similarly, densities of solutions in combination with the already known density of water (0.997044 g ~ m - ~ ) lead to apparent molar volumes. These apparent molar properties are defined according to [ I 1 +Y = Y(so1'n) - n lYlOI Cn,

i > 1

in which Y is the extensive property (heat capacity or volume) of a specified quantity of solution, n1 is the number of moles of water in the specified quantity of solution, Y10 is the molar property of pure water, and ni for i > 1 represents the numbers of moles of solutes. Results of our measurements of heat capacities and densities, expressed in terms of

been to obtain the +,O and +,O values for zero concentration that are identical to the corresponding partial molar quantities that then lead to ACpO and AVO values for various dissociation reactions. Analysis of our apparent molar volumes for this purpose requires that appropriate allowance be made for hydrolysis and dissociation reactions of certain solutes along with extrapolation to zero concentration. Analysis of our apparent molar heat capacities requires similar considerations of hydrolysis and dissociation reactions, allowance for enthalpy and equilibrium changes during calorimetric measurements, along with extrapolations to zero concentration. Because of their relative simplicity, we consider volumes first.

Apparent molar volumes of reasonably dilute solutions of electrolytes are accurately represented by equations of the form [2] +, = +,O + Av(d 1)11Z(o)31Z(m) l I z + Bvom in which AV and o come from the Debye-Hiickel theory and Bv is an adjustable parameter for each solute. We have used

AV(dlO)llZ = 1.865 cm3 molk3I2 kg1I2

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TABLE 2. Apparent molar heat capacities and volumes of aqueous NaHC031Na,C03 buffers at 25.O"C

171 4c(4csp) 4 v m 4c(4cW) 4 v mol kg-I J K-I mol-I cm3 mol-I mol kg-I J K-' mol-I cm3 mol-I

F, = 0.900 F, = 0.500 0.0660 -25.2(-25.7) 20.85 0.0568 -82.7(-87.7) 10.30 0.0788 - 15.9(- 16.9) 2 1.60 0.0844 -82.2(-85.3) 9.95 0.1418 - 12.3(- 12.5) 2 1.44 0.1021 -74.8(-77.3) 10.39 0.1621 -7.3(-7.5) 21.93 0.1518 -68.5(-70.1) 10.72 0.2435 -2.4(-2.5) 21.96 0.1847 -61.2(-62.5) 11.14 0.2837 3.6(3.5) 22.50 0.2376 -57.1(-58.1) 11.39 0.3887 9.7(9.6) 22.53 0.2903 -45.5(-46.3) 11.95 0.4556 15.7(15.6) 22.86 0.3172 -43.2(-44.0) 12.04 0.5127 17.7(17.7) 22.96 0.3872 -31.9(-32.5) 12.56 0.6023 25.5(25.5) 23.37 0.4265 -27.5(-28.0) 12.69 0.7080 32.6(32.6) 23.54 0.5227 - 16.5(- 17.0) 13.3 1 0.8369 39.2(39.2) 24.03

F2 = 0.100 0.0326 - 121.0(171.1) -2.60 0.0584 - 127.5(- 158.1) - 1.44 0.0815 -123.1(-145.6) - 1.01 0.0958 - 123.6(- 142.8) -0.60 0.1293 -116.6(- 131.1) -0.19 0.1526 - 115.9(- 128.2) 0.24 0.2063 - 101.9(- 11 1.1) 0.79 0.2150 -99.5(- 108.4) 1.06 0.2904 -83.2(-90.0) 1.79 0.3048 -81.7(-88.2) 2.02 0.4148 -60.6(-65.7) 2.96

TABLE 3. Apparent molar heat capacities and volumes of aqueous acids and salts at 25.O"C

m 4c(4cSP) 4 v m 4c(4cW) 4 v mol kg-I J K-I mol-' ~ m ~ m o l - ~ mol kg-' J K-I mol-I cm3 mol-'

H3PO4 Na3P0, 70.4(61.4) 43.78 0.0203 -3.3(-341.7) - 18.19 80.1(74.2) 45.32 0.0405 -39.q-301.8) - 17.22 84.5(79.8) 45.89 0.0617 -56.6(-270.8) -16.29 89.3(85.3) 46.18 0.0825 -65.1(-249.5) -15.42 90.0(86.4) 46.39 0.1028 -70.4(-235.1) - 14.68 93.8(90.6) 46.50 0.1295 -70.3(-217.2) - 13.88

0.1746 -66.7(- 195.2) - 12.84 H2S04 Na3P04 in 0.020171 NaOH

47.6(-53.5) 29.73 51.9(-14.9) 32.81 0.0150 35.5(-315.5) -16.73 55.3(-3.2) 33.15 0.0234 -7.4(-309.0) - 18.27 52.9(0.4) 33.97 0.0499 -42.q-260.9) - 15.96 55.5(9.7) 33.93 0.0766 -55.3(-234.3) - 14.82 53.q12.6) 34.50 0.1035 -61.7(-217.3) - 13.90 56.q19.1) 34.39 0.1310 -60.5(-200.8) -13.03 55.5(22.3) 34.92 0.1671 -56.q- 183.9) -12.02 56.3(28.9) 35.21

from Millero (7) and the "valence factor" o as molality. For non-electrolytes and undissociated commonly defined (8) is weak electrolytes such as H,PO,(aq) we have [3] o = Z/m = (0.5 Evizi2mi)/m [2'] instead of [2]: in which Z is the mold ionic strength. m; is the [2'1 4 v = 4v0 + Bvm molality of a particular solute, and > is the total Many of our measurements have been made on

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2144 CAN. J. CHEM. VOL. 60, 1982

buffer solutions that are described as containing component solutes in ratios equal to 911, 111, and 119. We now define solute mole fractions F2 and F3 according to

and

in which subscript 2 refers to a 1: 1 electrolyte and subscript 3 refers to a 2: 1 electrolyte. For F2 values of 0.9, 0.5, and 0.1 we have o values equal to 1.2, 2.0, and 2.8, respectively. As shown in Tables 1 and 2, we have made measurements on solutions having three different F2 values at several different total molalities.

For our NaH2P04/Na2HP04 and NaHC031 Na2C03 buffer solutions we find that hydrolysis and dissociation corrections are negligible. We have therefore obtained values of +v5 and Bv by least-squares fitting of 4, - Av(d10)112 (o)312(m)112 against om, as suggested by eq. [Z], with results that are given in Table 4. Further linear least- squares fits of +,O values for buffers against corresponding F2 values lead to +,,,O and +,,,O values for single electrolyte components of the buffers, as reported in Table 5. Using

from Millero (7), based on the useful convention that

we then obtain the conventional ionic volumes for H2P04-(aq), HP0,2-(aq), HC03-(aq), and C032-(aq) that are listed in Table 6.

For solutions of phosphoric acid we need consider only the first dissociation as in

for which we have the equilibrium constant K6 from a recent review (9). We calculate the fraction TABLE 4. Parameters for eqs. [2] and [16] for aqueous NaH,PO,/

Na,HPO, and NaHC031Na,C03 buffers at 25.O"C

4c0 Bc 4 v 0 BI, F2 J K-' mol-I J K-I m01-~ kg cm3 mol-I cm3 mol-z kg

TABLE 5. Standard state heat capacities and volumes of aqueous solutes at 25.O"C

4c0 4 1 . O Solute J K-I mol-I cm3 mol-I

H3P04(aq, undiss.) 94.5 48.1 NaH2P04(aq) 8.6 30.1 Na,HPO,(aq) - 168.1 2.8 Na3P04(aq) -367 -22.4 NaHC03(aq) -11.6 23.6 Na2C03(aq) -215.5 -6.8 H+(aq) + HS0,-(aq) 21.7 35.2

dissociated (cr) from K6 and the Davies equation for activity coefficients (y): [7] log y = ( - A Z ~ I " ~ ) / ( ~ + Ill2) + 0.1z21 The apparent molar volumes of solutions of phosphoric acid are expressed as [81 +V = (1 - cr)+v(H3PO,) + @+V(H+ + H2P04-) Substitution of the appropriate version of eq. [2] and [Z'] in [8] leads us to obtain the desired apparent molar volume of undissociated phosphoric acid, +,0(H3P04), by extrapolating bv - crbvO(H+ + H2P04-) - (u)312Av(d10)112(tn)112

(1 - a ) to zero concentration. Our value for +,0(H3P04) is listed in Table 6.

The hydrolysis of solutions of sodium phosphate is represented by [9] PO,'- (aq) + H,O(liq) = HPO,,~-(aq) + OH-(aq) We have calculated cr (fraction hydrolyzed) values from the equilibrium constant (9) and activity coefficients (10). The apparent molar volumes of solutions of sodium phosphate are then expressed as

[lo] +, = (1 - cr)+,(3Na+ + + ~ + ~ ( 2 N a + + HP042-) + c+(Na+ + OH-)

- cr V0(H20) Substitution of appropriate versions of eq. [2] in [lo] and rearrangement of the resulting compli- cated expression has led us to extrapolate

[+V - ~t+~O(2Na+ + HP042-) - E + ~ O ( N ~ + + OH-) + cr V0(H20)

- (6 - 2~t)A~(d,~)~~~(Z)'~~]/(l - a) NaHC0,1Na2C03 to zero ionic strength to obtain the desired

0.900 -32.7 39.1 20.5 1.2 +,O(3Na+ + that is listed in Table 5 and 0.500 -112.1 35.9 8.4 1.1 thence the conventional +v0(P043-) that is listed in 0.100 -195.8 38.7 -3.8 1.2 Table 6.

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TABLE 6. Standard state heat capacities and volumes for aqueous species at 25.O"C,

based on cbYo(H+) = 0 for ions

Species

H3P04(ad H2P04-(aq) HP042-(aq) ~ 0 , 3 - ( 4 HC03-(aq) C032-(aq) HS0,-(aq) Sod2-(ad

4c0 J K- mol-

"From ref. 19. bFrom ref. 7.

Sulfuric acid in our solutions is completely dissociated into H+(aq) and HS0,-(aq) and some of the bisulfate ions are further dissociated as in

We have calculated a (fraction of HSO,- dissociated) values from the equilibrium constant (9) and the Davies equation (10). The apparent molar volumes of sulfuric acid solutions are expressed as

Substitution of the appropriate forms of eq. [2] in [12] and rearrangement lead us to obtain +vO(H+ + HS0,-) by extrapolation of

to zero ionic strength to obtain the +vO(H+ + HS0,-) that is listed in Table 5. The conventional +,O(HSO,-) is listed in Table 6.

Analysis of experimental apparent molar heat capacities for reactive systems to obtain the desired +,O values of specified solute species requires that we allow for the temperature dependent state of equilibrium and related thermal effects. These "relaxation" effects have been considered previously for other systems by Woolley and Hepler (11) and by Jolicoeur, Lemelin, and Lapalme (12). As these earlier investigations have shown, experimental apparent molar heat capacities (+,) can be regarded as a sum of two kinds of contributions. One contribution is from the solute species present in the solution of interest and is designated +,"; it is this contribution that we will relate to the desired standard state heat capacities of various solute species. The other contribution to the total

experimental apparent molar heat capacity is from the thermal effects associated with changes in chemical equilibrium ("relaxation") and is designated We therefore have

The contribution of "relaxation" to the apparent molar heat capacity is given (1 1, 12) by

Thus the desired contributions due to the aqueous solute species can be obtained as

Procedures for calculating (daldr) from equilibrium constants, enthalpies, and activity coefficients for systems of various charge types follow those previously described (11, 12) and recently extended (13). The required equilibrium constants and enthalpies have been taken from a recent review (9).

The "relaxation" effects on heat capacities of the NaH2P04/Na2HP04 buffer systems are very small compared with the experimental uncertainties. We have therefore taken +, = +cSP for these solutions and used the +, values in Table 1 for our subsequent calculations for this system. For all other systems we have calculated +," values by way of experimental +, values and eq. [15], which has involved calculation of AH(da/dT) as described in detail elsewhere (11-13), with results presented in parentheses in Tables 2 and 3.

The next step has been to analyse the +cSP values in Tables 2 and 3 and the +, (= +,") values in Table 1 on the basis of the equation

in which Ac is derived from the Debye-Hiickel theory and Bc is an adjustable parameter. We have used

from Desnoyers (14,15). In this connection we note that there is recent evidence (16, 17) for a slightly larger value of AC(dl0)li2, but the difference is not enough to be significant for our present purposes.

We have used our +," results with eq. [16] in carrying out calculations and extrapolations to zero concentration in ways that are equivalent to those already described in detail for our +, values. The resulting +cO values are listed in Tables 4 and 5. These +cO values have been combined with the conventional +cO(H+) = 0 and +,O(Na+) = 43 J K-I mol-I and +C0(S042-) = -278 J K-I mol-I from earlier work (18, 19) to obtain the conventional +,O values listed in Table 6.

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TABLE 7. Standard state changes in heat capacity and volume for acid dissociation reactions at 25.0C, from

+,O and values in Table 6

AC, O AVO Reaction J K-I mol-I cm3 mol-l

We have used the conventional standard state heat capacities and volumes listed in Table 6 for calculation of ACpO and AVO for the acid dissociation reactions listed in Table 7. It should be emphasized that the arbitrariness in setting (PyO(H+)= 0 to obtain the conventional values in Table 6 cancels out of the calculations leading to Table 7 so that the reported ACpO and AVO values are independent of the conventional values chosen for +,O(H+).

We estimate that uncertainties in most of our +,O and 4 , O values are about f0.6cm3 mol-I and about f 4 J K-I mol-I, respectively; exceptions are Na3P04(aq), P043-(aq), and HS0,-(aq) for which uncertainties are larger than for the other solutes. Our further estimates for uncertainties in 4,O are about f 3 cm3 mol-I for Na3P04(aq) and P043-(aq) and about f 0.8 cm3 mol-I for HS0,-(aq). We also estimate that uncertainties in are about f 20 J K-I mol-I for Na3P04(aq) and P043-(aq) and about f 8 J K-I mol-I for HS0,-(ad. As shown in the next section, results of several earlier investigations provide evidence that most of our 4,O values (Tables 5 and 6) and related AVO values (Table7) are reliable to within the estimated uncertainties; the only exceptions are Na3P04(aq) and P043-(aq) and the related AVO for dissociation of HP042-(aq), for which the true uncertainties may be larger than our estimates above. Uncertain- ties in most earlier ACpO values are so large that they provide little evidence for or against our ACpO values and our estimates of their uncertainties.

Comparisons of results There have been many investigations of partial

molar volumes and also dissociation constants at various pressures that have yielded volumes to be compared with our values in Tables 6 and 7. Nearly all such results obtained before 1973 have been cited and summarized in Hamann's excellent review (20). Now we also have some useful results from more recent investigations.

Hamann (20) has listed AVO = - 16.6 and

-16.2cm3 mol-I for the first dissociation of phosphoric acid. More recently, Lo Surdo, Bern- strom, Jonsson, and Millero (21) have reported AVO = - 16.26 cm3 mol-' for this dissociation. All of these results are in good agreement with our AVO = - 16.8cm3 mol-I.

Hamann (20) has listed AVO = -24.0 and -24.1 cm3 mol-' for the second dissociation of phosphoric acid, for which Lo Surdo et al. (21) have recently reported AVO = -25.85 cm3 mol-I. Our AVO = -26.1 cm3 mol-I for this dissociation provides support for the value from Lo Surdo et al. (21) in preference to the earlier values.

Lo Surdo et al. (21) have also reported AVO = -35.96cm3 mol-' for the third dissociation of phosphoric acid, for which we found AVO = -24.0 cm3 mol-I. There is no obvious explanation for the difference in these values.

We also note that the cPVO values reported by Lo Surdo et al. (21) for H3P04(aq), NaH2P04(aq), and Na2HP04(aq) are all in good agreement with our corresponding values in Table 5. On the other hand, their 4,O for Na3P04(aq) is in poor agreement with our value.

Previous reviews by Millero (7) and by Hamann (20) have led to a choice of AVO = -27.7 cm3 mol-I for dissociation of HC03-(aq). The more recently reported partial molar volumes of Perron, Des- noyers, and Millero (8) lead to AVO = -28.4cm3 mol-I. As shown in Table 7, our measurements lead to AVO = -29.2cm3 mol-I for this dissociation. We suggest that both of the latter two values are more accurate than the first value quoted above.

Both Millero (7) and Hamann (20) chose AVO = -21.7 cm3 mol-I for dissociation of HS0,-(aq). More recent volumetric measurements by Rohwer, Brink, and Cruywagen (22) lead to AVO = -20.0 f 1.0 cm3 mol-I. These earlier values are in good agreement with our AVO = -21.2cm3 mol-I for this dissociation.

We now turn to some comparisons of our heat capacities with earlier results.

Bates (23) has evaluated the first dissociation constant of H3P04(aq) at 5" intervals from 0 to 60C. An equation of type log K = f(T) has been fitted to the results; two differentiations with respect to T have led to a reported ACpO = - 154 f 5 J K-I mol-I at 25C. More recently, Mes- mer and Baes (24) have investigated the first dissociation of H3P04(aq) at 25" intervals from 0 to 300C. Their results have led to ACiO = 34 f 29 J K-I mol-I for the neutralization reaction

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Combination of this result with ACpO = -215 J K-I mol-1 for dissociation of water from a recent review (9) leads to ACpO = - 186 f 30 J K-' mol-' for the first dissociation of H,PO,(aq). Because of the well-known magnification of uncertainties and errors associated with double differentiation and also because of the difficulties in evaluating accurately the relatively small pK of H3P04(aq) as reviewed recently by Pitzer and Silvester (23, we believe that our ACpO = - 128 J K-I mol-I is the best available for the first dissociation of H,PO,(aq) -

For the second dissociation of phosphoric acid we have four separate values of ACpO to compare with our value. Bates and Acree (26) have determined this dissociation constant at 5" intervals from 0 to 60C; two differentiations of the derived log K = f(T) have led to ACpO = -226 J K-I mole'. Bates and Acree (26) also calculated ACpO = - 189 J K-I mol-I from the earlier dissociation constants of Nims (27), determined over the temperature range 20 to 50C. Grzybowski (28) has determined dissociation constants at 5C intervals from 5 to 50C and has similarly calculated ACpO = -200 f 9.5 J K-I mol-I for the second dissociation. Finally, Mesmer and Baes (23) have determined equilibrium constants for

at 25" intervals from 0 to 300C and have calculated ACP0 = 32 + 25 J K-I mol-I for this neutralization reaction. Combination of this result with ACpO for dissociation of water (9) leads to ACpO = - 183 f 26 J K-I mol-I for the second dissociation of phosphoric acid. We suggest that our ACpO = -220 J K-' mol-I is the best available value. Although we are generally skeptical of ACpO values based on double differentiation of log K = f(T), we note that the equilibrium constants of Bates and Acree (26) and Grzybowski (28) appear to have been determined with considerable skill and accuracy; it is therefore gratifying to note that their (26, 28) ACpO values are in good agreement with our value.

We do not know of any earlier results that permit calculation of the heat capacity change for the third dissociation of phosphoric acid to compare with our ACpO = -242 J K-I mol-I.

For the dissociation of HC0,-(aq) we have several ACpO values to compare with our value. Harned and Scholes (29) have determined the equilibrium constant at 5" intervals from 0 to 50C and by double differentiation of the resulting equation of type log K = f(T) have obtained ACpO = -276 JK-I mol-I. Nakayama (30) has treated the

experimental results of Harned and Scholes (29) slightly differently to obtain ACpO = -282JK-I mol-I for this dissociation. More recently, Plum- mer and Busenberg (3 1) have analyzed equilibrium data over a wider range of temperature; double differentiation of their log K = f(T) now leads us to ACpO = -291 JK-I mol-I at 298.15K. Plummer (31) has also found that different selection and treatment of the equilibrium constants leads to less negative ACpO values. Perron, Desnoyers, and Millero (8) have measured heat capacities of aqueous solutions of NaHCO, and Na2C03. We have used their reported c$cO values in calculating ACpO = -214 J K-I mol-I for the dissociation of HC0,-- (aq). We know, however, that their reported c$,O values for these salts and this calculated ACpO value are incorrect because of neglect of the chemical relaxation effect ( 1 1-13), which had not yet been recognized (1 1) at the time Perron, Desnoyers, and Millero (8) wrote their paper. We have therefore made use of eqs. [13]-[15] with their experimental results to obtain a corrected ACpO = -249 JK-I mol-I for the dissociation of HC0,-(aq). We also note that Peiper and Pitzer (32) have independently and by a slightly different procedure (also using slightly different auxiliary data) made "relaxation corrections" to the results of Perron. Desnovers.

d ,

and Millero (8) to obtain ACpO = -260 J K-1 mol-I for the dissociation of HC0,-(aq). These "corrected" ACpO values from the measurements of Perron, Desnoyers, and Millero (8) are in good agreement with our ACpO = -247 J K-I mol-I for this dissociation.

For dissociation of HS0,-(aq) we have several values for ACpO as follows. Lietzke, Stoughton, and Young (33) have determined equilibrium constants at temperatures from 25 to 225C and have obtained an equation of type In K = f(T) from which we now calculate ACpO = -209 J K-I mol-I for the dissociation of HS0,-(aq) at 298.15 K. Marshall and Jones (34) have determined equilibrium constants from 25 to 350C and have calculated ACpO = -238 J K-I mol-I. Young, Singleterry, and Klotz (35) have reported results of their determinations of equilibrium constants at 10" intervals from 5 to 55C and have calculated two values for the heat capacity change associated with dissociation of HS0,-(aq): ACpO = - 192 and -238 J K-I mol-'. Readnour and Cobble (36) have measured enthalpies of solution of Na2S04(c) in water and in dilute hydrochloric acid at several temperatures and have calculated ACpO = -356 J K-I mol-I for dissociation of HS0,-(aq) at 25C. More recently, Criss (37) has used the experimental results of Readnour and Cobble (36) in some improved calculations and has

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CAN. I. CHEM. VOL. 60. 1982

TABLE 8. Selected standard state thermodynamic properties for acid dissociation reactions in aqueous solution at 298.15 K and 1 atm

L W O AS0 ACP0 AVO

Reaction Ka J mol-I J K-I mol-I J K-I mol-I cm3 mol-I

"All K values here are based on compositions of solutions expres one molal standard slate so that activity coefficients approach unit)

obtained ACpO = -280 J K-I mol-' for dissociation of HS0,-(aq) at 25C. We believe that our ACpO = -300 J K-I mol-I, supported by the value from Criss (37), is the best value available for the dissociation of HS0,-(aq) at 25C.

Thermodynamic calculations A principal purpose of this research has been to

obtain AVO and especially ACpO values for acid dissociation reactions with a view toward using these values in calculating the dependences of other thermodynamic properties on pressure and temperature. For these calculations we want the "best7' available values for the equilibrium constant K, AH0, AS0, ACpO, and AVO for dissociation reactions at 298.15 K and 1 atm. Our selected values for these quantities are given in Table 8; sources of these values are as follows.

Equilibrium constants for the first and second dissociations of phosphoric acid have already been cited (23-28). For the third dissociation we select the K value from Quist and Vanderzee (38). En- thalpies for these dissociations have been selected from values calculated with dln K/dT = AH0/RT2 (23, 24, 26-28) and from results of calorimetric measurements (39-43). The selected ACpO values are all based on our calorimetric results as reported in this paper. Selected AVO values are based on our results and on other values already cited (20, 21).

We take the equilibrium constant at 298.15 K for dissociation of HC0,-(aq) from Harned and Scholes (29) and the enthalpy of dissociation from Berg and Vanderzee (44). Our selected ACpO is based on the present calorimetric results and the "corrected" results from Perron, Desnoyers, and Millero (8). The selected AVO is a weighted average of the values cited earlier in this paper.

For the dissociation of HS0,-(aq) we have equilibrium constants from investigations already cited (33-35) and many others, especially including Covington, Dobson, and Srinivasan (45) and Pit- zer, Roy, and Silvester (46). We select the K,,,.,, value in Table 8 from this last investigation (46).

sed in terms of molalities (mol kg-') of solutes and the hypothetical 1 as m approaches zero.

Pitzer, Roy, and Silvester (46) have also calculated a AH2,,.,,O value for dissociation of HS0,-(aq), which we have considered along with values derived from investigations already cited (33-35) and from several calorimetric investigations (47). Our selected value is given in Table 8. We have also selected ACpO from our results, in good agreement with the recalculated ACpO from Criss, Readnour, and Cobble (36, 37). The selected AVO is based on our results and those cited earlier (7, 20,22).

Afirst approximation to the effect of temperature on thermodynamic properties associated with chemical reactions is to take ACpO = 0. In this approximation both AH0 and AS0 are independent of temperature and the effect of temperature on the equilibrium constant is expressed as follows:

The ACpO values in Table 8 show that ACpO = 0 is a poor approximation for the reactions of interest in this paper; hence eqs. [19]-[21] can be regarded as useful only when T is very close to 298.15K. We therefore go on to the next approximation, a non-zero ACpO that is considered to be independent of temperature.

Integration of (dAHO/dT) = ACpO with ACpO taken to be independent of temperature leads to

and similar integration of (dASO/dT) = ACpO/T leads to

Combinations of eqs. [22] and [23] with

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LARSON ET AL. 2149

or integration of dln KIdT = AH01RT2

with [22] leads to the following:

Effects of pressure on enthalpies and entropies of reaction can be calculated with [28] (dAHO/dP), = AVTO - T(dAVO/dT)p

[24] In KT = In K2,,,15 (AH2,,,,,0/298. 15RT) and x (T - 298.15) - (ACP0/RT) (T -298.15) [29] (dASO/d P), = - (dAVO/dT),

+ (~c-01~) ln (T/298.15) Of more immediate use to us in relation to the data that we have available (AVO, but not usually

[251 I ~ K T = (298.151T) lnK298.,5 + (AS298.1s0/RT) dAVO/dT) is (T - 298.15) - (ACpO/RT) (T - 298.1') [30] (d ln K/dPlT = - AVO/RT + ACpO/R) In (Tl298.15)

Integration of eq. [30] taking AVO to be independent [261 In KT = -(AH298.,s0IRT) + (ASZ~~.I~O/R) of pressure leads to

- (ACpO/RT) (T - 298.15) + (ACpO/R) In (Tl298.15)

Calculations with eqs. [24]-[26] and comparisons with experimental K values for several reactions show (48) that these equations yield very accurate K values over the temperature range 0- 100C (273-373 K) and yield moderately accurate K values at temperatures to 200C (473 K) or even higher.

According to eqs. [19]-[21], In K should be a linear function of 1IT. with no minimum or maxi-

. ,

mum. It is, however, known that In K is not generally a linear function of 1/T and that In K versus 1/T or T often passes through a minimum or maximum. Equations [24]-[26] are substantial improvements over eqs. [19]-[21] in that the equations containing ACpO allow for non-linearity of plots of In K versus 1IT and also permit a minimum or maximum. Consideration of

shows that In K will go through a minimum or maximum at the temperature where AH0 = 0. We therefore set AHT0 = 0 in eq. [22] and solve to obtain [271 Tm = 298.15 - (AH,, ,,,, O/ACp0)

in which K, represents the equilibrium constant at 1 atm and P is the pressure of interest expressed in atm. Use of K and AVO values from Table 8 in eq. [3 11 leads to calculated Kp values that are in good agreement with experimental results (20) over moderate ranges of pressure. Better accuracy over wide ranges of pressure can be obtained by consid- ering AVO to be dependent on pressure (non-zero partial molar compressibility changes) as summarized elsewhere (7, 20, 21).

Acknowledgments We are grateful to the Alberta Oil Sands Tech-

nology and Research Authority and the Natural Sciences and Engineering Research Council of Canada for support of this research. We also thank Dr. Gregory Allred for his helpful advice. Finally, we thank Dr. K. S. Pitzer and Dr. L. N. Plummer for sending us manuscript copies of their papers (31, 32) and Dr. C. M. Criss for sending us the results of his recent calculations.

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VOL. 60. 1982

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Documents

Heat Capacities and Volumes of Dissociation of Phosphoric Acid