Equilibrium partial pressures of strong acids over concentrated solutions—III. The temperature variation of HNO3 solubility

Atmospheric Environment Vol. 24A, No. 7, pp. 1945 1955, 1990. Printed in Great Britain.

0004-6981/90 $3.00+0.00 © 1990 Pergamon Press plc

EQUILIBRIUM PARTIAL PRESSURES OF STRONG ACIDS OVER CONCENTRATED SOLUTIONS--III. THE TEMPERATURE VARIATION OF HNO3 SOLUBILITY PETER BRIMBLECOMBE* a n d SIMON L. CLEGGt *School of Environmental Sciences, University of East Anglia, Norwich NR4 7TJ, U.K. and ~fPlymouth Marine Laboratory, Citadel Hill, Plymouth PL1 2PB, U.K. (First received 11 November 1989 and received for publication ll January 1990) Abstract--The mole fraction Henry's Law constant (Knx/atm- 1) of nitric acid is described by the equation KHx= xH +.xNO~ .f,2/pHNO3, wheref * is the mean rational activity coefficient,prefix'x' indicates mole fraction (calculated on the basis of complete dissociation in solution), and 'p' denotes partial pressure. From 215 < T<400 K, ln(Kn~) is given by: 385.972199-3020.3522/T-71.001998In T+0.131442311 T-0.420928363 × 10-4 T2. The mean activity coefficientof pure aqueous HNO3 is described from 0 to 100% HNO3 by an empirical extension to the Pitzer and Simonson thermodynamic model, allowing activities of both HNO3 and H20 to be easily calculated. This approach, in addition to its utility at normal and high temperatures, offers a considerable improvement over the methods previously used for estimating the partial pressure of HNO3 over the low temperature, highly concentrated aerosols found in the stratosphere. Key word index. Nitric acid, Henry's Law, stratospheric aerosol, temperature effects, activity coefficients, Pitzer and Simonson model.

l. INTRODUCTION

Calculations of the solubility of volatile strong electrolytes are important in studies of elemental cycling (Moiler, 1984), the chemistry of seawater and brines, urban aerosol behaviour (Pilinis et al., 1987) and cloud chemical processes (e.g. Leaitch et al., 1986). Brines and atmospheric aerosols are frequently very concentrated with respect to dissolved constituents, thus complicating theoretical treatments. In earlier papers (Clegg and Brimblecombe, 1988a,b) we determined the solubility of nitric acid and hydrogen chloride in concentrated solutions of a range of nonvolatile electrolytes. These studies considered multicomponent solutions at ionic strengths less than about 10 mol kg- 1, which are chiefly relevant to the troposphere. However, some aerosols in very dry atmospheres may be more concentrated than this. For example, supercooled nitric acid aerosols, and nitric acid ices, are likely to be important features of stratospheric chemistry (McElroy et al., 1986; Hamill et al., 1988). Equilibrium partial pressures over aqueous HNO3 have been examined by Clavelin and Mirabel (1979) at 0°C and 50°C. However, their work is based upon a relatively restricted data set, and furthermore appears to be inconsistent with more recent data. It therefore seemed timely to re-examine the thermodynamics of aqueous nitric acid over the full concentration range, and at all the temperatures of interest to atmospheric chemists. This work was aided by new measurements of Tang et al. (1988) and Hanson and Mauersberger (1988) of the vapour pressures of HNO 3 and H 2 0 in

equilibrium with aqueous solutions. The latter study is of particular importance, as it extends our knowledge of temperatures as low as 220 K. Models of aqueous solution behaviour (e.g. Pitzer, 1979; Pitzer and Simonson, 1986) enable these newer data to be readily combined with a large number of earlier measurements over a wide range of temperatures. We have carried out a detailed study of the thermodynamics of pure aqueous nitric acid (Clegg and Brimblecombe, 1990a). Here we use this description to establish the Henry's Law constants and activity coefficients of nitric acid solutions from - 5 0 to 120°C over the entire concentration range, and derive equilibrium partial pressures of both HNO 3 and H20. 2. THEORY

The equilibrium of HNO3 between aqueous and gas phases may be represented by (Clegg and Brimblecombe, 1986): +

HNO 3Cg)= H (aq)+ NO 3(,q).

(1)

A thermodynamic Henry's Law constant KH (mol 2 kg- 2 a t m - t) for HNO 3 can therefore be written: K n = ~ 2 mH + . m N O ~ / p H N O

a

(2)

where prefix 'm' represents molality, ~ ± is the mean activity coefficient of HNO3 in solution and pHNO3 the equilibrium partial pressure of HNO 3 (Clegg and Brimblecombe, 1986). At concentrations approaching pure nitric acid, values expressed on the molal scale tend to infinity. For the most general calculations it is necessary to use the mole fraction (rational) scale. The

1945

1946

PETER BRIMBLECOMBEand SIMON L. CLEGG

corresponding representation for Henry's Law may be written: KHx = X H + "x N 0 3

"j¢.2 ± /j p H N O 3

(3)

where f~: is the mean rational activity coefficient (infinite dilution standard state) and prefix 'x' indicates mole fraction. Complete dissociation in solution is assumed, and mole fractions calculated on the basis of the total number of components, thus: xH + = x N O ~ = n H + / ( 2 n H N 0 3 + n H 2 0 )

(4)

where prefix 'n' refers to the number of moles of each component present. Note that according to this definition both xH + and x N O 3 are equal to 0.5 for pure liquid H N O 3. The two Henry's Law constants are related by: g n x = K n ( M 1/ 1000) 2

(5)

and the mean activity coefficients by: f ~ = 7± (1 + (M1/1000)SZ m,)

(6)

where M1 is the molar mass of the solvent (18.0152 g for H 2 O), and the summation is over all solute species Strictly speaking, the H N O 3 partial pressure in Equations (2) and (3) should be expressed as fugacity, but it seems unlikely that the correction for gas-phase non-ideality will be significant at moderate temperatures and pressures up to a few atmospheres. Therefore, as with the other strong acid gases (Clegg and Brimblecombe, 1988a,b), we have used partial pressure (atm) throughout. (For conversion to S.I. units, atm = 101,325 Pa.) F r o m Equations (2) and (3) it is clear that, at a given temperature, calculation of pHNO3 over nitric acid solutions requires a knowledge of both mean activity coefficients and Kax (or KH) as functions of temperature. We have previously determined a value of 2.46 x 106 mol 2 kg -2 a t m - 1 (Kn) at 25°C, from available partial pressure data and the tabulated activity coefficients of Hamer and Wu (1972) (Brimblecombe and Clegg, 1988). More recently, Tang et al. (1988) have measured both p H N O 3 and p H 2 0 at 25°C, and evaluated ),+ to high concentration. Using these activ-

ity coefficients, Tang et al. obtained K H equal to 2.66 x 106, some 10% greater than our earlier estimate. However, this discrepancy is principally due to differences between the two sets of activity coefficients (see Tang et al. for a comparison), and partial pressures calculated using the two Henry's Law constants and related sets of activity coefficients are in good agreement. 2.1. The Henry's L a w constant o f H N O 3 as a function o f temperature The temperature variation of equilibrium constants can be obtained from the well known van't Hoff equation: dIn(K)/dT=AH°/RT

233.15 238.15 243.15 248.15 253.15 258.15 263.15 268.15 273.15

(7)

The enthalpy change AH ° cannot be treated as constant except over a very small temperature range. Here, it is necessary to account for the non-linear variation of AH ° with respect to temperature. An analysis of available heat capacities for aqueous HNO3, combined with tabulated values for gas phase HNO3, allowed us to determine the variation of Knx (Ciegg and Brimblecombe, 1990a), yielding the following equation: In (Knx) = 385.972199 - 3 0 2 0 . 3 5 2 2 / T - 71.001998 In T +0.131442311 T-0.420928363 x 10 -4 T 2

(8) the molal constant K n is also represented by the equation above but with the first term (385.972199) replaced by 394.0052779. Both Henry's Law constants are listed over a wide range of temperature in Table 1. 2.2. Activity coefficients over entire concentration range

It is frequently necessary to calculate the solubility of nitric acid in mixed electrolyte solutions of moderate concentration. Here the molal-based model of Pitzer provides a good description of activity and osmotic coefficients (necessary for calculating the equilibrium water vapour pressure) (Clegg and Brimblecombe, 1988a, b). Pitzer's equations for mixed solutions may be found in several reviews (e.g. Pitzer, 1986,

Table 1. The Henry's Law Constants (KHx and Ks) for nitric acid as a function of temperature T

2.

Kax

Kn

T

Kin,

KH

1.615 × 10 6 8.211 x 105 4.250 x 105 2.239 x 105 1.199 x 10s 6.528 x 104 3.610 x 104 2.028 x 104 1.156 x 104

4.977 X 10 9 2.530 x 109 1.310 x 109 6.897 x 10s 3.694 x 10s 2.011 x 10a 1.112 x 10s 6.247 x 107 3.560 x 107

278.15 283.15 288.15 293.15 298.15 303.15 308.15 313.15 318.15 323.15

6680 3916 2326 1400 853.1 526.3 328.5 207.4 132.4 85.44

2.058 x 107 1.207 x 107 7.167 x 106 4.313 X 10 6 2.629 x 10 6 1.622 X 10 6 1.012 x 106 6.390 x 105 4.080 x 105 2.633 x 105

Equilibrium partial pressures of strong acids--III 1987). However, this model is only applicable to concentrations < 6 mol k g - ~ for pure aqueous solutions and < 10-15 mol k g - 1 ionic strength for multicomponent ones. Pitzer and Simonson (1986), and Simonson and Pitzer (1986) have shown that it is possible to describe activity and osmotic coefficients of aqueous single salt solutions and three component mixtures, on a mole fraction basis, over the entire concentration range. Despite the fact that nitric and other acids undergo measurable association in aqueous solution (e.g. Davis and DeBruin, 1963), Clegg and Brimblecombe (1990a) have successfully applied the same approach to nitric acid, although with an extension to the model in order to obtain agreement for f * at concentrations up to the pure acid. 2.2.1. Pitzer and Simonson model applied to nitric acid. Satisfactory descriptions of solution behaviour at moderate concentrations can be achieved by using the unmodified Pitzer and Simonson (1986) model, which has the advantage of a relatively simple representation of the temperature variation of activity coefficients. In the model, the water activity (al) of a pure aqueous solution of H N O 3 is given by: In (a 1) = In (x 1) + 2A x Iax/2/(1 + pl~/2) -~-X2 (WI,HNO3 "1-(2x x- 1) Ul.nNo~ ) (9)

where p, WI,HNO3 and ULnNO ~ are adjustable parameters. The D e b y e - H u c k e l constant on a mole fraction basis (A~) is equal to (lO00/Ml) 1/2 A ~, and the mole fraction ionic strength (Ix) is given by: ~x = (1/2) Zz~x~

(10)

1947

Table 2. Pitzer and Simonson model parameters for 298.15K Parameter p WLn~o 3 UI.nNo3

Value

Standard error

14.9420 -3.5424 0.23256

0.315 0.0137 0.0479

p above yields a value of 'a' (Equation (14)) equal to 2287.66. Parameter U1. nr~o~was retained, despite its relatively high standard error, as it was found to be significant at other temperatures.

mole fraction (x~) of about 0.6, leading to a satisfactory representation of H N O 3 and H 2 0 partial pressures. Calculations of activity at other temperatures require that the variation of the parameters p, Wl.nNo3 and Ul.m~o, be determined as a function of temperature. Pitzer and Simonson (1986), and earlier Pitzer and Li (1983) have defined the functionality of p from theory, while determining the absolute value at some temperature T(K) from empirical data:

p = a(dx/DT) t/2,

(14)

where d~ and D are the density ( g c m - 3) and dielectric constant of the solvent, respectively, and 'a' is a constant (see Table 2). The parameters W1.HNOa and U1,HNOa are given as functions of temperature by (Clegg and Brimblecombe, 1990a): UI,HNO~ , WI,HNO3 = r 1 + r 2 ( T 3 In( T)-- T~ in(T~))

+ r3(T 5 - T~)+ r4(T 3 - T 3) where z~ is the magnitude of the charge on ionic component i, and the sum is over all ionic components. We also define the composition variable xt, the total mole fraction of ions, here equivalent to ( 1 - x 1) or 2I~. The mean mole fraction activity coefficient of H N O 3 (referenced to ideal behaviour at infinite dilution) may be written: In ( f * ) = - A~ { (2/p)In (1 + pl~/2)

+rs(T2-

T~)+r6(T-T0

(15)

where T r is equal to 298.15 K. The coefficients r 1 _ 6 are given in Table 3, together with equations for Ax and p from 273.15 to 393.15 K. For convenience, values of all the parameters over a range of temperatures of atmospheric interest are listed in Table 4. It is important to note that a restricted fit to apparent molal enthalpy

+ (IxL'2 - 213/2)/(1 + pI~/2 ) } + X21(WI.rlNO3+2xIU1.HN03)

- - WI.I.INO3.

(11) The activity of the solvent is related to the rational osmotic coefficient 0 by:

Table 3. Coefficients for the temperature variation of the Pitzer and Simonson model parameters for 273.15 ~
ln(al)=ln(fl xl)=oln(xl)

where fx is the activity coefficient of the solvent. The following equation gives the relationship between q~ (the molal osmotic coefficient) and 0 for finite molal concentrations (m) (Robinson and Stokes, 1965):

g = -(Zm~)(M~/lOOO)~b/ln(x~). The model parameters commended for nitric Brimblecombe (1990a) temperature the model

W1, HNO3

(12)

(13)

p, WLnNO ~ and ULnNo ~ reacid at 25°C by Clegg and are given in Table 2. At this performs well up to a solute

r1 r2 r3 r4 rs r6

0.2325640 - 1.662045 x 10 -5 4.403503 × 10-12 1.233135 x 10 -a --0.01481856 2.576840

- 3.542378 - 1.957717 x 10 -6 0.0 1.535269x 10- 5 --2.493699 x 10 -3 0.59837350

Over the same temperature range parameters p and A~ are given by the following equations: p = 8.75365 --0.0084483 T+ 1.222965 In(T) + 1.95626 x 10- s T 2 and: A~ = 4.1725332- 0.1481291T 1/2 + 1.5188505 x 10- 5 T 2 - 1.8016317 x 10 -s T 3 +9.3816144 × 10- lo T3.5.

1948

PETER BRIMBLECOMBE and SIMON L. CLEGG

Table 4. Pitzer and Simonson model parameters for 273.15 ~
A~

p

273.15 278.15 283.15 288.15 293.15 298.15 303.15 308.15 313.15 318.15 323.15

14.766 14.801 14.836 14.871 14.907 14.942 14.979 15.016 15.054 15.093 15.133

2.806 2.826 2.847 2.869 2.892 2.917 2.942 2.969 2.997 3.026 3.057

~V1,HN03

U 1,HNOa

-5.1007 -4.7663 --4.4438 -4.1326 --3.8323 -3.5424 -3.2623 --2.9917 --2.7301 -- 2.4771 --2.2324

-1.7955 - 1.3351 -0.90394 -0.50017 -0.12191 0.23256 0.56485 0.87644 1.1687 1.4430 1.7004

data (Clegg and Brimblecombe, 1990a) means that Equations (9) and (11) are limited to x t < 0 . 2 4 for T#298.15 K. 2.2.2. Extending the model to higher concentrations. The need to treat extremely concentrated solutions over a wide temperature range led us to extend the Pitzer and Simonson description with a single additional term: ln(f~ )= - A~{(2/o~)ln(1 + ~ I 1/2) +(i~/2 _ 2I~/2)/(1 + o9it/2)}

+x2(W+2x, U+x 2 IO- W

(16)

where co, W and U are again adjustable parameters. The symbol V represents an additional parameter to the ones used by Pitzer and Simonson. We employ a different symbolism to avoid confusion with parameters of the original model. Values of co, W, U and V are listed in Table 5. The corresponding equation for the activity of the solvent is given by: In(at ) = ln(xt ) + 2Axial2~(1 + col~ 12)

+ x][{W+(2x,-

l)U}

- Vxi(xt - I/3)].

(17)

While the current treatment of HNO a solubility employed is unrealistic in assuming complete dissociation at all concentrations, it provides a satisfactory empirical description. Activities at temperatures other than 298.15 K are calculated by the following method, valid for all

Table 5. Extended Pitzer and Simonson model parameters at 298.15 K Parameter co W U V

Value 14.0645 - 3.82721 0.202749 -- 2.57620

Standard error 0.368 0.0193 0.0498 0.231

Equations only valid for T equal to 298.15 K. Values at other temperatures to be obtained by application of partial molal functions (Equations (18) and (19)).

concentrations. The mean activity coefficient ( f ~ , r ) at some temperature T, is related to that at a reference temperature Tr (here 298.15 K) by the equation (Harned and Owen, 1958; Klotz and Rosenberg, 1972): log (f$. r) = log(f~:. Tr) + (Y/v)L2

-- (z/v)J 2 + (f~/v)r 2

(18a)

T)/(2.303 R T~T)

(18b)

where:

y=(T,-

z = T~y - log ( Tr/T)/R

(18c)

fl = T~(z + ( 1 / 2 ) ( T - T~)y)

(18d)

A corresponding equation applies to the solvent activity: l°g(aa, r) = log(al, r,) + y/L1 - z / J l +f~F l , (19) where L t and L 2 are the partial molal enthalpies of the solvent and solute, respectively, J1 and J2 the corresponding partial molal heat capacities, v the number of ions produced by the complete dissociation of one molecule of solute and R(J mol - t K - t ) is the gas constant. The terms F 2 and Ft are equal to aJ2/t3T and aJ 1/aT, respectively, at the reference temperature and concentration of interest. Here they are assumed to be invariant with temperature. Note that Equation (18) is valid for activity coefficients on both molai and rational scales. The functions F 2 and F t (Equations (18a) and (19)) are given below (Clegg and Brimblecombe, 1990a): F2 = 1.7722xi(xl - 2)-0.35496x25(5xt - 7) (20) F t = x2(0.8861-0.8874x]'5).

(21)

We have evaluated the partial molal functions from heat of dilution and specific heat data over the full concentration range, and for practical calculations represent them in terms of Chebychev polynomials. The polynomial equation is given in the Appendix, as are the coefficients for calculating each of the partial molal functions, see Tables A l-A4. Plots of the partial molar quantities L and J for both HNO 3 and H20 as functions of concentration are also given in the Appendix (Figs Al and A2). Calculated values o f f * and a I over a range of temperatures are shown in Fig. 1. At concentrations below about 0.10 mole fraction the temperature dependence for both solute and solvent is small. However, it is clear that at higher aqueous phase concentrations the variation of solute and solvent activity becomes marked, and should be taken into account in all calculations. So far we have not applied the model of Pitzer and Simonson to multicomponent electrolyte solutions. As noted earlier, at moderate ionic strengths ( < 10-15 mol k g - t ) activity and osmotic coefficients can be calculated using the molal based Pitzer model. Extensions to temperatures other than 298.15 K can be made using the derivatives (with respect to temperature) of the pure electrolyte model parameters (see Pitzer, 1979; Clegg and Brimblecombe, 1990b).

Equilibrium partial pressures of strong acids--III I

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3. VALIDITYOF THE MODEL A number of assumptions are inherent in the present thermodynamic treatment. The most important are those concerning heat capacity and its variation with temperature (Clegg and Brimblecombe, 1990a). In addition, activity and osmotic coefficients at 298.15 K are relatively uncertain at high concentrations because of the paucity of data. We now compare the temperature variation of the Henry's Law constant described by Equation (8), and estimates obtained with Equation (3) using observed values of pHNO 3 and activity coefficients derived as recommended above (see Fig. 2). It is clear that there is good agreement between theory and the experimental data over almost seven orders of magnitude of Kax, although the data show some scatter, chiefly at high temperatures. In practical calculations of the properties of aqueous HNO3 the equilibrium partial pressures of both solvent and solute are often required. (Recall that the partial pressure of the solvent is given by al PH2 O°, where pH 2 0 ° is the equilibrium vapour pressure over pure water.) In Figs 3 and 4 we compare calculated partial pressures of HNO3 and H20, respectively, with available data. Comparisons are made in terms of the percentage deviation of the measured partial pressure from the calculated value, and the results are plotted independently against both temperature and concentration. The ranges in measured partial pressures are 7.5 orders of magnitude (pHNO3) and 5.5 orders of magnitude (pH2 O). It can be seen that, while there is a large amount of scatter (as was the case in Fig. 2), there do not appear to be any systematic

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1950

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lation of equilibrium partial pressures. The equations are valid over the full concentration range, and from approximately 215 K to 400 K. Contour plots of (calculated) p H N O 3 and p H z O are shown in Fig. 5 for the range in T and Xz given

above. Calculations were extended to supercooled solutions (below the freezing point curve shown in the figure), as some partial pressure measurements have been made in this region for concentrated solutions, see below.

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TO THE STRATOSPHERE

In calculations for stratospheric systems the vapour pressure over nitric acid droplets and ices at extremely low temperatures has been predicted by extrapolating the results of Clavelin and Mirabel (1979) at 0°C and 50°C to temperatures as low as 190 K (Toon et al., 1986). These extrapolations, as loglo P against 1/T, include a slope correction at the freezing point of the solution, to account for the effect of a transition to a solid solution (Toon et al., 1986; Hamill et al., 1988). However, as Hansen and Mauersberger (1988) point out, there is considerable disagreement between the results of this extrapolation and their own measurements at low temperatures over aqueous droplets. In Fig. 6 the data of H a n s o n and Mauersberger (for both p H N O 3 and p H 2 0 ) are compared with calculations made using the equations given in this paper. It is clear that, except in the case of pHNO3 over 0.1818 and 0.276 mole fraction solutions, there is good agreement for both solute and solvent partial pressures. In Fig. 7 a further comparison is made with partial pressures extrapolated from those given by Clavelin and Mirabel (1979), as described above. Not surprisingly, there are considerable differences between the two sets of calculations. It is clear from the earlier figures that the approach described in this paper, while more complex than that of other workers such as Clavelin and

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Fig. 5. Calculated values of loglo(pHNO3) and loglo(pH20 ) as functions of temperature and concentration. Full lines pHNO3; fine lines-pH20. The upper dotted line marks the boiling point of aqueous HNO3 at atmospheric pressure, and the lower dotted line marks the freezing point. Dashed extensions of the contours below the freezing point give values for hypothetical supercooled solutions.

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1000/T

Fig. 6. Comparison of calculated pHNO 3 (a) and p a 2 0 (b) with measurements of Hanson and Mauersberger (1988). Symbols--measured partial pressures at seven concentrations, each given a different symbol; lines--calculated values. Solution concentrations (x~) are marked on the graphs. Dotted lines give freezing points of solutions at the given temperature, and in equilibrium with the given vapour pressure. Values to the right of these lines are for supercooled solutions.

1952

PETER BRIMBLECOMRE and SIMON L. CLEGG

10 -2

i0-~

10-3 e

1ff s

E

C

1G~

Z

•-r-

6

%

"-r-

~-iffs

10-6 .~-8

'

/U

0~s,8

270 I

3,6

250 I

I

L,,0

T

0.276

230

I

I

t~,k

o33,

\

210 I

I

/.,8

270 I

1000/T

I

3,6

250 I

I

t~,0

T

o.

230

I

I

4,/.

210 I

I

Z~,8

1000/T

Fig. 7. Comparison of pHNOa (a) and pH20 (b) calculated in the present work, with extrapolations of partial pressures given by Clavelin and Mirabel (1979). Solution concentrations are the same as those in Fig. 6. Full lines--partial pressures calculated using equations given in the present work; dotted lines--values extrapolated (as loglo P against 1/T) from the results of Clavelin and Mirabel. Solution concentrations (x~) are marked on the graphs, as are freezing points. Mirabel, gives an improved description of the H N O 3 - H 2 0 system over a very wide range of temperature.

5. CONCLUSIONS The variation of the Henry's Law constant for nitric acid is described by an equation that is valid over the range 215-400K. Nitric acid mean activity and osmotic coefficients can be represented using thermodynamic models for electrolyte solution behaviour developed by Pitzcr and Simonson. Combining estimates of the Henry's Law constant and activity c o d ficients enables the equilibrium partial pressures of H 2 0 and H N O a over aqueous H N O a to be estimated over a wide temperature and at all concentrations, with reasonable accuracy. REFERENCES

Brimblecomb¢ P. and Clegg S. L. (1988) The solubility and behaviour of acid gases in the marine aerosol. J. atmos. Chem. 7, 1-18. Ciavelin J.-L. and Mirab¢! P. (1979) Determination des pressions partielles du melange eau-acide nitrique. J. Chim. Phys. 76, 533-537. Clegg S. L. and Brimblecomb¢ P. (1986) The dissociation constant and Henry's law constant of HC1 in aqueous solution. Atmospheric Environment 20, 2483-2485.

Ciegg S. L. and Brimblecomb¢ P. (1988a) Equilibrium partial pressures of strong acids over concentrated saline solutions. Part I. HNO 3. Atmospheric Environment 22, 91-100. Clegg S. L. and Brimblecomb¢ P. (1988b) Equilibrium partial pressures of strong acids over concentrated saline solutions. II. HC1. Atmospheric Environment 22, 117-129. Clegg S. L. and Brimblecomb¢ P. (1990a) Equilibrium partial pressures, and mean activity and osmotic coefficients of 0-100% nitric acid as a function of temperature. J. phys. Chem. (in press). Clegg S. L. and Brimblecomb¢ P. (1990b) The solubility of volatile electrolytes in multicomponent solutions, with atmospheric applications. In Chemical Modellino in Aqueous Systems II (edited by Bassett R. L. and Melchior D.). American Chemical Society, Washington. Davis W. and DeBruin H. J. (1963) New activity coefficients of 0-100% aqueous nitric acid. J. lnorg. Nucl. Chem. 26, 1069-1083. Hamer W. J. and Wu Y-C. (1972) Osmotic coefficients and mean activity coefficients of uni-univalent electrolytes in water at 25°C. J. Phys. Chem. Ref Data 1, 1047-1099. Hamill P., Turco R. P. and Toon O. B. (1988) On the growth of nitric acid and sulphuric acid aerosol particles under stratospheric conditions. J. atmos. Chem. 7, 287-315. Hanson D. and Mauersbcrger K. (1988) Vapour pressures of HNO 3/H 2 0 solutions at low temperatures. J. phys. Chem. 92, 6167-6170. Harned H. S. and Owen B. B. (1958) The Physical Chemistry of Electrolytic Solutions. Reinhold, New York. Klotz I. M. and Roscnberg R. M. (1972) Chemical Thermodynamics, Basic Theory and Methods. Benjamin/Cummings, Menlo Park. Lcaltch W. R., Strapp J. W., Wieb¢ H. A., Anlauf K. G. and Isaac G. A. (1986) Chemical and microphysical studies of

Equilibrium partial pressures of strong acids--III nonprecipitating s u m m e r cloud in Ontario, Canada. J. geophys. Res. 91, 11,821-11,831. McElroy M. B., Salawitch R. J., Wofsy S. C. and Logan J. A. (1986) Antarctic ozone: chemical mechanisms for the spring decrease. Geophys. Res. Lett. 13, 1296-1299. Moiler D. (1984) O n the global natural sulphur emission. Atmospheric Environment 18, 19-27. Pilinis C., Seinfeld J. H. and Seigneur C. (1987) Mathematical modelling of the dynamics of multicomponent aerosols. Atmospheric Environment 21, 943-955. Pitzer K. S. (1979) Theory: ion interaction approach. In Activity Coefficients in Electrolyte Solutions (edited by Pytkowicz R. M.), Vol. 1, pp. 209-265. C R C Press, Boca Raton. Pitzer K. S. (1986) Theoretical considerations of solubility with emphasis on mixed aqueous electrolytes. Pure appl. Chem. 58, 1599-1610. Pitzer K. S. (1987) T h e r m o d y n a m i c model for aqueous solutions of liquid-like density. Rev. Mineral. 17, 97-142. Pitzer K. S. and Li Y-G. (1983) Thermodynamics of aqueous sodium chloride to 823 K and 1 kilobar (100 MPa). Proc. Natl. Acad. Sci. 80, 7689-7693. Pitzer K. S. and Simonson J. M. (1986) Thermodynamics of multicomponent, miscible, ionic systems: theory and equations. J. phys. Chem. 90, 3005-3009. Robinson R. A. and Stokes R. H. (1965) Electrolyte Solutions. Butterworth, London. Simonson J. M. and Pitzer K. S. (1986) Thermodynamics of multicomponent, miscible, ionic systems: the system L i N O 3 - K N O 3 - H 2 0 . J. phys. Chem. 90, 3009-3013. T a n g I. N., Munkelwitz H. R. and Lee J. H. (1988) Vapour-liquid equilibrium measurements for dilute nitric acid solutions. Atmospheric Environment 22, 2579-2585. T o o n O. B., Hamill P., Turco R. P. and Pinto J. (1986) Condensation of H N O 3 and HCI in the winter polar

1953

stratosphere. Geophys. Res. Lett. 13, 1284-1287. Weast R. C. (Ed.) (1983) CRC Handbook of Chemistry and Physics. C R C Press, Boca Raton.

APPENDIX

Tables A1-A4 list the Chebychev polynomial coefficients for the calculation of the partial molal functions L 2, L 1, J2 and J1 for 0~
-

where X,~.x and Xm~. are the upper and lower limits of the fit, respectively (given in each table). The polynomial representation, with N + 1 coefficients, of each partial molal function f(x) is given by:

f(x)=O.5ao To(x)+al Tl(x)+a2 T2(x) + a3 Ts(x) • • • + aN TN(x)

Xmi. Xmax ao a1 a2 a3 a4 a5 a6

0.0 0.1146991 753.0043515 177.7416892 --154.9534807 36.7922356 --4.8568604 1.238092 --0.9198174

7",(x) = 2x T, _ 1(x) - T. _ 2(x).

0.1 ~
0.25 ~
0.5 ~
0.2366268 0.5096280 23013.1055374 11257.5112646 751.9024962 --370.2494804 70.6666189 -- 34.7281052 0.0

0.4867724 0.7071067 56318.7624896 5820.4448643 --1236.8416632 --107.8051159 5.6112619 0.2606108 0.0

Table A2. Chebychev polynomial coefficients for calculation of partial molal enthalpy function L1 0~
0.1 ~
0.25 ~
0.5 ~
0.0 0.1146991 0.0632680 0.2108328 0.3093970 0.1341192 --0.0093489 --0.0146750 -- 0.0024496 - 0.0010445

0.09406486 0.2509652 --40.0719174 --35.1358810 -- 21.4663058 -- 7.7409019 -- 1.5137565 --0.1444610 0.0 0.0

0.2366268 0.509628 --3951.4949599 --2589.8022600 -- 729.7017104 -- 63.0933078 0.0 0.0 0.0 0.0

0.4867724 0.7071067 --22416.4074106 --7673.2532247 -- 873.4022440 -- 15.5786947 3.7112558 0.0 0.0 0.0

Independent variable X is equal to I~/2.

AE(A) 24:7-W

(a4)

Values of partial molal enthalpies L~ and L 2 over the entire concentration range are shown in Fig. AI, and partial molal heat capacities Jl and J2 in Fig. A2. To assist in program evaluation, values of the polynomial approximations of the partial molal functions are given for different X in Table A5.

Independent variable X is equal to llx/2.

Xr, i, Xmax ao a1 a2 a3 a4 a5 a6 a7

(a3)

where T0(x)= 1, Tl(x)=x and, for n>~2:

Table A1. Chebychev polynomial coefficients for calculation of partial molal enthalpy function L2 0~
(al)

The polynomial is represented in Chebychev series form by the normalized variable x: x = ( 2 X - X m , x Xmin)/(Xm.x -- Xml.) (a2)

1954

PETER BRIMBLECOMBEand SIMON L. CLEGG

Table A3. Chebychev polynomial coefficients for calculation of partial molal heat capacity function J2 0~
0.0 0.4275968 223.9622636456 115.6006435535 --7.2798475853 -13.1878737137 -1.3364758484 0.6920809739 -0.1175789636 0.0719146717 -0.0353268030 0.0171526066 -0.0092635246 0.0041271283 -0.0045938767

Independent variable X is equal to

0.4~
0.5~
0.3884932 0.5291503 392.1697985322 --14.8062775012 --0.3832410917 3.2428027637 -0.1312662761 -0.8440563447 0.0 0.0 0.0 0.0 0.0 0.0 0.0

0.4898979 0.7071067 365.5752776353 --1.4841020659 0.8923170148 -0.3932543734 0.0653659365 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0

11/2

Table A4. Chebychev polynomial coefficients for calculation of partial molal heat capacity function J~ 0<~X<0.4 Xmi. Xmax ao aI a2 a3 a4 a5 a6 av as a9 alo

0.4 ~
0.0 0.4094472 -- 8.7953754943 - 6.3276558665 -7.2798475853 0.4431250357 0.5198346132 0.1445081118 0.01211134284 -- 0.0003030311 0.0004235444 0.0006508587 0.0002910300

0.475 ~
0.3884932 0.479484 -- 18.4332904575 3.2973006370 -0.3832410917 - 0.0588858992 - 0.1759351492 - 0.1159043522 --0.0699845756 0.0 0.0 0.0 0.0

Independent variable X is equal to

0.4701978 0.7071067 -- 5.5836609353 2.0078082282 0.8923170148 0.2717589033 - 0.1063168995 0.0660843697 --0.0469639109 0.0 0.0 0.0 0.0

11/2.

Table AS. Evaluated Chebychev polynomials for partial molal functions L~, L2, J l and J2 X 0.05 0.20 0.45 0.60

L1 -0.242593056 -9.695495670 -0.310587675 x 104 -0.105443281 × l05

L2 0.513111689 0.685255050 0.178754866 0.295705939

Jl x x x x

103 l03 l05 l05

-0.0151525305 -- 1.85944617 -9.09209270 -2.07060976

J2 0.174630437 0.107795519 0.199889498 0.181956731

x x x x

102 10 a 10 a 10 a

Equilibrium partial pressures of strong acids--III

1955

I i

i

i

i

i

(n)

i

I

!

!

(b) 0

3 -&

0" 2

...T

'

'

'

-E

-0,2 -12 I

/ 0 . 1 0 ~

I

,

, 0.04

I

I

I

0,2

0,4

0,6

,

-16

, 0.08

I

O,B

I

-20

b-I

1,0

I

I

0,2

0,4

, I

I

0,6

0,8

1,0

xz

xz

Fig. A1. Partial molal enthalpies at 298.15 K of HNO 3 (L2) and H 2 0 (LI) in pure aqueous HNO 3 over the entire concentration range. Note different scale factor in each case. Insets give function values at low concentrations. (a)--L2. (b) - - L 1.

I

i

i

i

I

i

240

I

I

I

!

(a)

200

160

120

-6

80

-8

40

-10

I

I

I

I

I

0

0,2

0,4,

0,6

0.8

xz

-12 1,0

I

I

i

i

I

0,2

0,~

0,6

0,8

1,0

xz

Fig. A2. Partial molal heat capacities at 298.15 K of HNO3 (J2) and H 2 0 (J~) in pure aqueous HNO3 over the entire concentration range. (a) --J2, (b) --J1-