Equilibrium partial pressures and mean activity and osmotic

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5369

J . Phys. Chem. 1990, 94, 5369-5380

Equilibrium Partial Pressures and Mean Activity and Osmotic Coefficients of 0-1 00 % Nitric Acid as a Function of Temperature Simon L. Clegg* Plymouth Marine Laboratory, Citadel Hill, Plymouth PLl 2PB, U.K.

and Peter Brimblecombe School of Environmental Sciences, University of East Anglia, Norwich NR4 7TJ. U.K. (Received: November 22, 1989)

Rational activity and osmotic coefficients and Henry's law constants KHx (for the reaction HN03(g)= H+(aq)+ given as functions of temperature, enable p(H20) and p(HNO,) to be calculated from -60 to 120 OC, CklOO% HN03. Apparent molal enthalpies, heat capacities, and activity coefficients (derived from electromotive force (emf), freezing point, and partial pressure data) are represented with the use of the species-interaction models of Pitzer. Equations for the partial molal functions are given. The Henry's law constant is evaluated from p(HN0,) data at 298.15 K, tabulated heats of formation, and heat capacities. Calculated p(H20) and p(HN0,) over the entire concentration range agree with available data over 5 orders of magnitude of p ( H 2 0 ) and 7 orders of magnitude of p(HN0,).

I. Introduction There is considerable interest in the solubility of volatile strong electrolytes in atmospheric clouds and aerosols, necessary for understanding processes such as elemental cycling and urban aerosol behavior. Recent attention has focused on stratospheric cloud systems, of which droplets of frozen nitric acid are likely to be an important component.'q2 Data for the solubility of HNO, in pure aqueous solutions at 25 OC are readily available, and a number of studies have been carried out on the thermodynamics of this system.3-5 In addition, theoretical advances in multicomponent solution theory (e.g., Pitzer6) have enabled the solubility of HNO, and other gases to be predicted in multicomponent salt s o l ~ t i o n s . ~However, ~~ in most cases the treatments are restricted to temperatures close to 25 OC, due to lack of data. While Clavelin and Mirabe15carried out their analysis at both 0 and 50 OC, it was assumed that, at other temperatures, the logarithms of the partial pressures of both solution components were simply proportional to 1 / T. Hanson and Mauersbergerg have presented measurements of p ( H N 0 J in equilibrium with concentrated aqueous solutions and ices containing HNO, to temperatures of approximately -60 OC, for use in the calculation of stratospheric cloud properties. It would clearly be of benefit, for both atmospheric and industrial applications, if these low-temperature data could be combined with those at higher temperatures and the activities and partial pressures of both H N 0 3 and H 2 0 described using a unified thermodynamic approach. In this work we show that Henry's law constants and activity coefficients of HNO, at 25 "C,when combined with other data for apparent molar enthalpy and heat capacity, may be used to describe measured solubilities of H N 0 3 in aqueous solutions from 0 to 100% acid, -60 to 120 "C. In section I1 the fundamental thermodynamic relations and the activity coefficient model used

are described, and in section I11 activity and osmotic coefficients at 25 "C are determined to high aqueous phase concentration. To calculate solute and solvent activities at other temperatures, enthalpies and heat capacities of aqueous HNO, are required: these are evaluated and the necessary model equations presented in section IV. An equation for the Henry's law constant of HNO, as a function of temperature is also given. In section V, model predictions are compared with activity and osmotic coefficients obtained from partial pressure data at all temperatures. An extension to the basic thermodynamic model is presented and the treatment of activity extended to pure liquid H N 0 3 . 11. Theoretical Background I . Solubility Relations. The equilibrium of strong acids such

as HNO, between aqueous and gas phases may be represented aslo On a molal basis the thermodynamic Henry's law constant K H (mol2 k g 2 atm-I) for HNO, is therefore given by KH ,= m(H+)m(N03-)Y*2/P(HN03)

where m represents molality, yf is the mean molal activity coefficient of HNO, in solution, and p ( H N 0 3 ) (atm) is the equilibrium partial pressure of HN03.10 On the mole fraction (rational) scale the corresponding relation for the Henry's law constant KHx (atm-I) is (3) wheref,* is the mean rational activity coefficient (infinite dilution standard state) and x indicates mole fraction. Complete dissociation in solution is assumed, and mole fractions are calculated on the basis of the total number of components; thus x(H+) = x(NO,-) = n(H+)/(2n(HN03)

( I ) McElroy, M. B.; Salawitch, R. J.; Wofsy, S. C.; Logan, J. A. Geophys. Res. Let?. 1986, 13, 1296-1299. (2) Hamill, P.; Turco, R. P.; Toon, 0. B. J . Atmos. Chem. 1988, 7, 287-315. (3) Davis, W.; DeBruin, H. J. J . Inorg. Nucl. Chem. 1964,26, 1069-1083. (4) Redlich. 0.; Gargrave, W . E.; Krostek, W . D. Ind. Eng. Chem. Fundum. 1968, 7, 21 1-215. (5) Clavelin, J.-L.; Mirabel, P. J . Chim. Phys. 1979, 76, 533-537. (6) Pitzer, K. S . Reo. Mineral. 1987, 17, 97-142. (7) Clegg, S. L.; Brimblecombe, P. Atmos. Enoiron. 1988, 22, 91-100. (8) Clegg, S. L.;Brimblecombe, P. Atmos. Enoiron. 1988, 22, 117-129. (9) Hanson, D.; Mauersberger, K. J . Phys. Chem. 1988, 92,6167-6170.

0022-3654/90/2094-5369$02.50/0

(2)

+ n(H20))

(4)

where n refers to the number of moles of each component present. Note that according to this definition both x(H+) and x(NO 0.36, values were obtained from cubic-spline fit to data.

large deviations. Accordingly, the model was fitted to the lower range of concentration. A few data points, which differed greatly from the original fit, were removed. The Forsythe and Giauque data (here a single concentration) were weighted preferentially, and data from other sources were given equal weights. Note that, for the initial aim of obtaining best estimates of +Cp(and @L)at 298.1 5 K, a higher value of p than that listed in Table I1 was used, as it gave an improved fit to data at the lowest concentrations. To obtain estimates of the functions J , and J2 over the full concentration range, fitted values of W, for 0.5 C xI I1.O are also required. The earlier treatments of Parker32 and Mish~ h e n k o ,agree ~ most closely with the data of Forsythe and Gia ~ q u e . ,Therefore, ~ we have used only these data, together with a few points from Mishchenko and Ponomareva,40to obtain +Cp at high concentration. Because of the lack of a suitable theoretically based expression for *Cp,and difficulties associated with was here estimated by use of a cubic spline, a polynomial fit, @Cp for xI C 0.5. taking care to ensure consistency with the fitted @Cp The results of both fits are shown in the inset to Figure 5 , and values of @Cp at rounded concentrations are listed in Table V. Expressions for the partial molal heat capacities of solute and solvent ( J 2 and J , ) are given by eq 22 and 23, respectively. Although given as functions of molality, these equations are readily expressed in terms of xI and may then be evaluated to xI equal to 1.0 ( m = a). Note that the very small variation of +Cpat high concentration means that, of the two functions, J I is subject to the greatest uncertainty since it consists of a single term in the differential of @Cp with respect to concentration. Values of @C, and a@Cp/aml/z were derived numerically by using the fitted eq 25 and the cubic spline mentioned above, and J , and J 2 were then calculated as functions of x I . For practical calculations it is convenient to represent the functions as in terms of Chebychev polynomials. The polynomial equation is given in Appendix 2, and the coefficients for calculating each of the partial molal functions are given in Tables XIV and XV. Further functions rzand rl (eqs 18a and 19) remain to be evaluated. Knowing the temperature dependence of *C, as a function of concentration, eq 28, that of J2 and J , can be derived from eqs 22 and 23, yielding

r2= 1 .7722xl(x1 - 2) - o.35496x,2.5(5xI - 7) rl = ~ ~ ~ ( 0 . 8 -8 60.8874xI1,’) 1

(29)

(30)

(40) Mishchenko, K . P.; Ponomareva, A. M . Zh. Fiz. Khim. 1952, 26, 998-1006

The Journal of Physical Chemistry, Vol. 94, No. 13, 1990 5375

Thermodynamics of 0-100% Nitric Acid

TABLE VII: Calculated Values of Apparent Molal Enthalpy (QL/J mol-’) at 298.15 K

TABLE VI: Literature Sources of Heat of Dilution Data (for Calculation of Apparent Molal Enthalpies)’

temp/’C

reference Becker and RothS2 Bertholets3 Forsythe and Gia~que’~

XIb

19.8 9.7 25 1 8.02c =I8 20

0.308 0.990 1 .oo 0.0358 0.998 0.1666

NaudB50

Prttats’ Richards and R ~ w e ~ ~

With the exception of the data of Forsythe and Giauque, results are quoted giving heat evolved from the system as positive. bData are usually given for dilution from some concentration xl,, to another x1,*. This column lists upper limit of x,,’. cMean of values ranging from 17.56 to 18.37 “ C .

8ool

j

I

.-.

/

i

I

#I

3-

0

3

.. 2 5 . 3 l

J n

1

lIM.$”’ 0

0.2

0.6

0.L

0.8

1.0

XI

Figure 6. Apparent molal enthalpies of HN03(84 Symbols: small square, NaudE? large square, Richards and Rowe;q‘cross(X), PrBtat;5’ plus (+), Becker and Roth;s2 diamond, Berth~let;~’ lines, fitted values. Note scale change (to J mol-’) in inset which shows results at low con-

centration. Taking the apparent molal heat capacities obtained above, heat of dilution data are next used to derive apparent molal enthalpies of aqueous nitric acid. 3. Apparent Molal Enthalpies (OL)of HN03(,, (0 5 xI I 1.0). Sources of experimental data (heats of dilution) leading to @Lare listed in Table VI. The heat of dilution AHD for a solution containing 1 mol of solute from concentration x,,] to xI,2is related to the apparent molal enthalpy by

AHD(XI,I-*XI,Z)= ‘L2 - “1

(31)

The data listed in Table VI are for the temperature range 10-25 O C . Heats of dilution show a marked dependence upon temperature, given by (aQL/aqp,,y,=

+ep - #ep”

(32)

Accordingly, the experimental heats of dilution were first corrected to 25 O C using the apparent molal heat capacities already obtained, and assuming @Cpto be invariant with temperature over the small range encountered here. It is not possible to plot heats of dilution directly (for the sake of comparison) because of the wide range ~. the data were initially fitted as (@L2- #Ll) using of x ~ , Therefore eq 24 to a maximum concentration ( x ~ ,of ~ )0.2, covering the full range of x , , ~in the experimental data. The fitted model, which was in excellent agreement with the data, was then used to calculate *L2 for each experimental measurement and so obtain the apparent molal enthalpy QL;see Figure 6. Test fits of eq 24 to the data in the figure (weighted equally) showed that a maximum concentration of 0.24 mole fraction gave optimum results, with higher concentrations leading to a decrease in accuracy for small XI. For mole fractions greater than 0.24 the data were fitted by a polynomial in x11/2.Results of the fit are shown in Figure 6

XI

0.0 0.0002 0.0004 0.0010 0.002 0.004 0.008 0.012 0.0 16 0.02 0.04 0.06 0.08 0.10 0.12 0.14 0.16 0.18 0.20

*L 0.0 130.2 175.2 251.7 320.9 393.8 457.4 481.9 489.9 489.4 451.7 43 1.3 46 1.9 555.0 715.8 947.2 1251 1629 2080

XI

QL

0.24 0.28 0.32 0.36 0.40 0.44 0.48 0.52 0.56 0.60 0.64 0.68 0.72 0.76 0.80 0.84 0.88 0.96 1.oo

3200 4416 5750 7163 8637 10156 11709 13288 14886 16497 181 I7 19742 21369 22996 24620 26239 27852 3 1054 32641

(and inset) and values of @Lat round concentrations are listed in Table VII. Using eqs 2 0 and 21, partial molal enthalpies of both solute ( L 2 ) and solvent (15,)were next calculated over the entire concentration range, in a manner similar to the corresponding functions for heat capacity. For practical calculations these functions are again represented in terms of Chebychev polynomials, whose coefficients are given in Tables XI1 and XIII. Equations 18 and 19, together with partial molal functions as given in Appendix 2, may be used to correct solute and solvent activity coefficients from the reference temperature (298.15 K) to another temperature T. These calculations may be carried out over the entire concentration range, from infinitely dilute solution to pure liquid H N 0 3 , although we note that activity coefficients based upon data at 298.15 K remain uncertain for xI> 0.60. For many applications at relatively low concentration, it would be more convenient to calculate activity coefficients directly from eqs 9 and 13. To do this, it is necessary to obtain model parameters U I , H N 0 3 and WIJNO as functions of temperature. 4. Activity Coehcient Model Parameters from 273.1 5 to 393.15 K. Equation 25 gives apparent molal heat capacities at temperature T in terms of the first and second differentials of Ci],HNO, and W ] , H N 0 3 at the experimental temperature. In the absence of primary heat capacity data over a wide range of temperature, eq 28 was used together with the corrected 298.15 K data (0 I xI I 0.5) to estimate #Cpat 10 “Cintervals from 273.15 to 393.15 K. Model parameters for the Debye-Huckel terms in eq 25 are listed in Table I11 at these temperatures. The unknown parameters to be fitted occur as the pairs ( U ” i , H N 0 3 + ( 2 / T ) U’I,HN03) and (W”I:HNO, + ( 2 / T ) W ’ I , H N 0 3 ) . To obtain the first and second differentials separately, a number of possible functional forms for further differentials were tested. This involved determining analytic expressions for U“l,HNO, and U’I,HNO, (or W“l,HNo, and W’l,HN03) by integration and fitting the heat capacity data set with coefficients in the equations for U”I,HN03, U’l,HNO,, W”1,HN03, and W’I,HN03 as unknowns. The following functional forms were adopted, as they gave a sum of least squares only slightly greater than that obtained by “free” fits to the data at each temperature individually: U”~,HNO,, W ” ~ , H N O ~ =P I

+ p2T In

(T)

+ p3T3 + p4T

(33)

U‘I,HNO~, W‘~,HNO, = 41 + ~ 2 ( In p ( T ) - T,Z In (TI)) + 43(P - Tr? + 4 4 P - T,2) + 9 d T - Tr) (34) = ri + r2(T3 In ( T ) - T? In (TI)) + r3(rS - TIs) + r4(T3- TI3)+ r5(T2 - T,2) + r6(T - TI)(35)

U~,HNO,*Wi.HN0,

where coefficients ~ ( 1 - 4 ) . 4(1-5), and r(!-6)are to be determined. A number of constraints can be applied when fitting the heat capacity data. First, QCpoat 298.15 K was fixed at -72 J mol-’

5376 The Journal of Physical Chemistry, Vol. 94, No. 13, 1990

Clegg and Brimblecombe

TABLE VIII: Coefficients for Calculation of Pitzer and Simohpon Model Parameters (Eqs 33-35) W”I,HNO, -4.987398 X IO-’ -1.174630 X 0.0 8.232756 X

~”LHNO,

PI

-0.029 637 12 -9.972 267 0 X 8.807007 X IO-” 6.567 790 X

P2 P3 P4

uI.HNO~

r2 r3

r4 r5 r6

’

2.5F--------

0

008

016

U‘I.HNO~ 0.068 640 38 -4.986133 X 2.201 752 X IO-” 3.533202 X IOd -0.029 637 12

wl,HNO~

0.232 5640 -1.662045 X 4.403503 X IO-’* 1.233 135 X IOd -0.01481856 2.576 840

rl

ql q2 q3 q4 45

-3.542378 -1.957717 X IO“ 0.0 1.535 269 X -2.493699 X IO-’ 0.598 373 50

‘4

0.24

XI

Figure 7. Calculated values off** (eq 13) from 273 to 373 K. Model parameters are listed in Tables 111 and VIII.

K-’ and d W p o / a Tat that temperature was 1.0284 (eq 28). A significant reduction in the sum of least squares was obtained where amCpo/dTwas also a function of temperature; therefore, a (constant) second differential with respect to temperature was assumed, leading to an equation of the form @Cpo = s, + s2T + s,p (36) Second, values of U‘i,HNO,and W’i,HNO,must agree with those obtained by fitting to molal enthalpy data at 298.15 K. This was done by using eq 24 and the value of p given in Table I1 (14.942) to ensure model consistency. Thus, although there are 12 coefficients in eqs 3 3 , 34, and 36, there are only 8 unknowns. These were determined by fitting the (generated) +CP values (273.15-393.15 K) and weighting the original 298.15 K data by a factor of 5. The results were used to calculate the coefficients in eqs 33-36, which are listed in Table VIII. Note that the variation of Wpowith temperature, while consistent with the heat capacities at higher concentrations, has been estimated by extrapolation and is very dependent upon the accuracy of eq 28, determined from the data of M i ~ h c h e n k o . ~ ) Figure 7 shows calculated mean activity coefficients of HNO, at a number of temperatures from 273.15 to 373.15 K. While the fitting of @Cp data could have extended below the freezing point of water, this would involve assumptions about the behavior of model parameters associated with the Debye-Huckel term, even though solutions would be relatively concentrated (and thus long-range interactions have only a small effect). Comparisons with activity coefficients calculated using 298.1 5 K values and eq 18 showed that, at temperatures far from 298.1 5 K, the Pitzer and Simonson model calculations are reliable only to the maximum concentration of fit of the enthalpy data, about 0.24 mole fraction. However, for concentrations below this, agreement between the two methods is excellent even at the extremes of the temperature range. Thus, while the Pitzer and

W‘l,HNO~

0.056 985 88 -5.873 15 X IO“ 0.0 4.410035 X -4.987398 X IO-’ @Po

SI

s2

s3

-568.03908 2.299061 9 -2.1?091 X IO-’

Simonson model parameters were only determined to 273.15 K, agreement between the two methods of calculation extends down to at least 223 K (although only extremely concentrated aqueous solutions would exist at such a low temperature). The Pitzer and Simonson model may therefore be used to calculate activity coefficients and partial molal properties at temperatures below 273.15 K, where solutions for which x, < 0.24 remain in liquid form. Simple empirical extensions for A,, p, and the other parameters appearing in the Debye-Huckel terms in eqs 9 , 13, 24, and 25, and which were used in the above calculations, are given in Appendix 1. In order to obtain the solubility of H N 0 3 in aqueous solutions over the range of conditions shown in Figure 4, estimates of K H x are required from 223.15 to 393.15 K. 5 . The Henry’s Law Constant of H N 0 3 as a Function of Temperature. Heat capacity data (Cpo/J mo1-I K-I) for HNO,,, were obtained from the J A N A F Tables41 over the temperature range 100-500 K and expressed as a polynomial in T. Combining this with eq 36 to obtain ACpo for the dissolution reaction (eq 1) yields ACpo = -590.339

+ 2.1857287 - (2.09986 X

10-3)T2

(37)

Knowing the value of the Henry’s law constant at 298.15 K (298KHx), its value at temperature T i s obtained from the following expression:

where aHo/aT, the temperature derivative of the enthalpy change for the reaction, is equal to ACpoat the same temperature. Using ~~~~~

~~

(41) Stull, D. R.; Prophet, H. JANAF Thermochemical Tables, NSRDS-NBS-37; U S . Government Printing Office: Washington, DC, 1971. (42) Schufle, J. A,; Venugopalan, M. J . Geophys. Res. 1967, 72, 327 1-3215. (43) Burdick, C. L.; Freed, E. S . J . Am. Chem. Soc. 1921, 43, 518-530. (44) Flatt, R.; Benguerel, F. Helv. Chim. Acta 1962, 45, 1772-1776. (45) Kuster, F. W.; Kremann, R. 2.2.Anorg. Chem. 1904, 41, 1. (46) Pascal, P. Ann. Chim. 1921, 15, 253-290. (47) Thomsen, J. Thermochemische Untersuchungen;Johann Ambrosius Barth: Leipzig, 1882; Vol. I . (48) Marignac, M. C. Ann. Chim. Phys. 1876,8, 410-430. (49) Richards, T.W.; Rowe, A. W. J. Am. Chem. Soc. 1921,43,770-796. (50) NaudB, S. M. Z. Phys. Chem. 1928, 135, 209-236. (51) PrCtat, P.M. Mem. Poudres 1930, 24, 119-136. (52) Becker, G.; Roth, W. A. Z . Phys. Chem. 1935, A174, 104-114. (53) Berthelot, M. Compt. Rend. 1874, 78, 769-777. (54) Potier, A. Ann. Fac. Sci. Uniu. Toulouse Sci. Math. Sci. Phys. 1957, 20, 1-98. ( 5 5 ) Vandoni, M. R.; Laudy, M. J . Chim. Phys. 1952.49, 99-108. (56) Wilson, G. L.; Miles, F. D. Trans. Faraday Soc. 1940,36, 356-363. (57) Berl, V. E.; Samtleben, 0. 2.Angew. Chem. 1922, 35, 201-202. ( 5 8 ) Boublik, T.; Kuchynka, K. Collect. Czech. Chem. Commun. 1960,25, 579-582. (59) Carpenter, C. D.; Babor, J. A. Trans. Am. Inst. Chem. Eng. 1925, 16, I 11-148. (60) Klemenc, V. A,; Nagel, A. Z . Anorg. Allg. Chem. 1915, 155, 257-268. (61) Prosek, J. Collect. Czech. Chem. Commun. 1967, 32, 2397-2404. (62) Sproesser, W. C.; Taylor, G. B. J . Am. Chem. SOC. 1921, 43, 1782-1787. (63) Pabalan, R. T.; Pitzer, K. S. J . Chem. Eng. Data 1988,33, 354-362.

The Journal of Physical Chemistry, Vol. 94, No. 13, 1990 5377

Thermodynamics of 0-100% Nitric Acid I1

I

3-

I

I

I

I

20

I

I

I

I

I

I

I

16

12 m

8 ', I

8

I 0

@"'

4

c

L 0

0.2

0.6 0.8

0.L

u

0

1

1.0

I

I

0.2

0.L

0.6 0.8

1

XI

XI

Figure 8. Rational osmotic coefficients g, calculated from p(H20)data at all temperatures (except T = 298.15 K) and corrected to 298.15 K using eq 19. Symbols: cross (X), T < 273.15 K; plus (+), 273.15 IT < 298.15 K; diamond, 298.15 < T I323.15 K;large square, 323.15 < T 5 373.15 K;small square, T > 373.15 K. Line (with dashed extrapolation for x , > 0.6) was calculated using eq 9 with parameters listed in Table 11.

Figure 9. Mean activity coefficientsf,', calculated from p(HN0,) data at all temperatures (except T = 298.15 K) and corrected to 298.15 K using eq 18. Symbols: cross (X), T < 273.15 K; plus (+), 273.15 IT < 298.15 K; diamond, 298.15 < T 5 323.15 K; large square, 323.15 < T I373.15 K; small square, T > 373.15 K. Line (with dashed extrapolation for x I > 0.6) was calculated using eq 13 with parameters listed in Table 11.

the value of AHo at 298.15 K from the tabulation of Wagman et al.,39 and integrating eq 37 AHo = 25112.418 - 590.339T 1.0928639p - (6.99953 X 1 0 - 4 ) p (39) Recalling that 298KHxis equal to 853.1 atm-' and integrating eq 38 yields an expression for TKHx: In ( T K H x )= 385.972199 - 3020.3522/T - 71.001998 In (7') 0.13144231 1 T - (0.420928363 X 1 0 - 4 ) p (40)

parameters listed in Table 11. Figure 8 compares data for the solvent in terms of g, the rational osmotic coefficient, as a function of concentration. For xI less than about 0.6 there are no apparent systematic deviations between experimentally derived and predicted values, although there is quite a large degree of scatter. Neither do there appear to be systematic errors with respect to temperature. For higher concentrations the combined data have very poor precision, as the vapor phase consists almost entirely of HNO, (small amounts of nitrogen oxides are also present in some cases), and it is not possible to come to any firm conclusions. Similar comparisons are now made for p ( H N 0 3 ) in terms of f**. In this case the results are a test of the prediction of KHx in addition to the activity coefficient model and partial molal functions. Figure 9 showsf,* at 298.15 K derived from p ( H N 0 3 ) at all temperatures and values calculated by use of the Pitzer and Simonson model. At low concentrations agreement is good, although there appear to be some small deviations (positive below xI = 0.4). However, for xI > 0.6 the model extrapolation is clearly in error, yielding values considerably lower than the data indicate. In the light of this, it was decided to use the estimates off,* for which T # 298.15 K to obtain an accurate representation of activity and osmotic coefficients at high concentration. While a purely empirical polynomial could be used to represent f,*,we have adopted a single term extension to the Pitzer and Simonson model, which yields satisfactory results:

+

+

Equation 40, together with other equations given above for f,*, make it possible to estimate equilibrium p ( H N 0 3 ) for aqueous solutions over a wide range of temperatures and concentrations. This can also be done for the solvent, using tabulated equilibrium water vapor pressures23 and the relationship al = p ( H 2 0 ) / p (H20)0, where p ( H 2 0 ) is the experimental water vapor pressure and p(H20)0 the value over the pure solvent at the same temperature. Polynomial equations for p(H20)0, used in the analysis below, are given in Appendix 3.

V. Data Comparison and Extension of Activity Coefficients * to 100% HNOj There are a number of assumptions inherent in the thermodynamic treatment above, notably those concerning heat capacity and its variation with temperature. Activity coefficients,estimated for temperatures other than 298.15 K, and the Henry's law constant KHxare affected by this. In addition, activity and osmotic coefficients at 298.1 5 K are uncertain at high concentrations due to paucity of data at that temperature and an apparently poor fit by the model. Consequently it is essential that the results be tested against available p ( H 2 0 ) and p ( H N 0 3 ) data at all temperatures. The concentration and temperature ranges of each individual data set are listed in Table 1. Taking the data as a whole, p ( H 2 0 ) varies over greater than 5 orders of magnitude, and p ( H N 0 3 ) varies by more than 7 orders of magnitude. For both components the maximum partial pressure encountered is about 1.13 atm. Rather than compare actual and predicted partial pressures at the temperature of measurement ( T ) , experimental partial pressures were first used to derive f,* and al at temperature T using eqs 3 and 40 (f**)and pure water vapor pressures given by the equations in Appendix 3 ( a , ) . These were then corrected to 298.15 K, using eqs 18 and 19, for direct comparison with values calculated by the Pitzer and Simonson model equations using the

In

In (1 + w I , ' / + ~ )(Ix1l2 + w ~ x ~ / 2 ) )+ x 1 2 ( w + 2 x , u + X

Vi*)= -A .((2/w) 2zx3/2)/(1

- w (41)

r 2 ~

where Vis the additional unknown parameter. Here we use the symbol w in place of p and unsigned constants Wand U to avoid confusion with the original model parameters, whose temperature variation has been determined. The corresponding equation for the activity of the solvent is now given by In ( a l ) = In (x,)

+ 2Ax12/2/(1 + wIx1/2) + x,'[{W

+ (2x1 - 1 ) q - V X ~ (-X1/3)] ~

(42)

Equations consistent with those above could also be derived for apparent molal enthalpy and heat capacity. The additional high-concentration p ( H N 0 3 ) data used to fit the model are shown in Figure 10. (Of the available data, those obtained from experiments on boiling mixtures at reduced pressure were of relatively low precision and were excluded from the fit.) Equation 41 was fitted to the 298.15 K data set, with the addition of that shown in Figure IO. Weighting of the data was

5378

The Journal of Physical Chemistry, Vol. 94, No. 13, 1990

Clegg and Brimblecombe

7 3c

j . -30

0

02

06

OL

08

-

+”

1

i 10

XX

T , ?

30

06

08

07

05

i.‘---

10

XI

Figure 10. Values off,* (298.15 K) at high concentration, estimated from p(HN03) data for which T # 298.15 K and used to fit the extended Pitzer and Simonson model (eq 41). Symbols: small square, Potier? large open square, Vandoni and L a ~ d iplus ; ~ ~(+), Yakimov and Mishin;26diamond, Wilson and Miles;s6large solid square, pure HNOl at 298 IS K. Haase et al ,24 CRC Handbook.23

--’1

3 . 5 L

-.

1

ic! ,

.

0

02

06

OL

08

10

XI

Figure 12. Percentage deviation in osmotic coefficient g and activity coefficientf** at 298.15 K (experimental - predicted), calculated from partial pressure data for all T and all xi and corrected to 298.15 K using eqs 19 and 18, respectively. (a) Osmotic coefficient g; (b) osmotic coefficient g; (c) activity coefficient f**;(d) activity coefficient f+*.

1.6

--

D+I L

C

0.6 I

i

q I

-3.4

I

0

I

I

0.6

0.2 0.L

I

vi*)at 298.15 K, 0 5 x, 5 1.0. Squares, data from all

sources used in the fit; line, fitted values (eq 41). Inset: residuals (experimental- predicted) for model fit. Symbols: small square. freezing point data; asterisk. p(H20);large square, p(HN03). TABLE IX: Extended Pitzer and Simonson Model, Parameters for 298.15 KQ

param __I____.__ (d

W L

v

value 14.064 5 -3.827 21 0.202 749 -2.57620

HN03 IO4 a2P(o)/dT2 -0.1390 IO4 a2P(’)/dT2 0.0 105 a 2 c * / a T * 0.01632

Multicomponent molality based model. See Pitzer6 for the osmotic and activity coefficient equations and Pabalan and PitzeP for corresponding expressions for apparent molar properties.

0.8 1 0

XI

Figure 11. In

TABLE X: Pitzer Model’ Parameters for /3(”) 0.12556 IO4 aP(O)/aT 2.4401 /3‘l) 0.28778 IO4 afi(’)/dT 13.396 ca -0.00559 105 a c + / a T -0.9274

standard error____

-

0.368 0.0193 0 0498 0.231

“Equations only valid for T = 298.15 K. Values at other temperatures to be obtained by application of partial molal functions (eqs 18 and 19) the same as previously. Fitted model parameters are listed in Table IX. Results are shown in Figure 11 as In(fi*) and (inset) as residuals from the fitted model. It can be seen that these are very similar to those obtained earlier when fitting eq 13 to data at lower concentrations (Figure 3b). I t is now possible to compare all partial pressure data with predictions using the activity and osmotic coefficients, and Henry’s law constants, derived here. This was carried by first determining g and fi* from the experimental data and correcting to 298.1 5

TABLE XI: Estimates of Pitzer and Simonson Debye-Huckel Parameters below 273.15 K 1OSp” I0 3 ~ 1 P Ax ALr AJ x TI K 273. I5 2.807 9408 183 14.763 -1.019 6.955 268.15 2.788 8525 170 14.729 -2.954 7.121 263.15 2.771 158 14.686 7703 -5.367 7.396 258.15 2.755 6942 I46 14.652 -8.377 7.805 253.15 2.740 6243 134 14.610 8.383 -1 2.1 5 1 248.15 2.126 5605 121 14.567 9.173 -16.924 243.15 2.712 109 14.516 10.236 -23.051 5028 238.15 2.700 4512 97 14.465 I 1.649 -31.075 233.15 2.689 4058 85 14.397 13.529 -41.898

K as described previously. Percentage deviations from predicted values (eqs 41, 42, and 14) were then calculated. The results for g are shown in Figure 12a,b, plotted first against xI(irrespective of temperature) and then against temperature (irrespective of concentration). It is clear from Figure 12a that, while the scatter in the data increases greatly as xI 1 .O, g is satisfactorily predicted by the model. Figure 12b shows that there are no systematic changes in the percentage deviation with temperature, except perhaps at the very lowest temperatures where a positive deviation is apparent. This may be due to errors in the calculation of p(H,O)O, which was obtained by extrapolation (see Appendix 3). Percentage deviations inf,* are shown in Figure 12c,d. It can be seen in part c of the figure that the data show quite large positive deviations for xI = 0.2, and Figure 12d shows that these

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The Journal of Physical Chemistry, Vol. 94, No. 13, 1990 5319

Thermodynamics of 0-100% Nitric Acid

TABLE XII: Chebvchev Polvnomial Coefficients for Calculation of Partial Molal Enthalpy Function L ," OIX 373.15 K p(H2O)" = -324.706 - (1.06781 X 10-5)T,3.5 + (4.133 X 10-7)Tx4,5 45.1066Tx1/2(A3.1)

+

for 298.15