Enthalpy and entropy changes in formation of gas phase

Transcription

1 JOURNAL OF GEOPHYSICAL RESEARCH, VOL. 105, NO. D15, PAGES 19,779-19,785, AUGUST 16, 2000 Enthalpy and entropy changes in formation of gas phase sulfuric acid monohydrates and dihydrates as a result of fitting to experimental pressure data Madis Noppel Institute of Environmental Physics, University of Tartu, Tartu, Estonia Abstract. The values of enthalpy change AH =-44.3 kj/mol and entropy change A& = J/mol in the gas phase addition of a water molecule to a sulfuric acid molecule and entropy change AS2 = -145 J/mol in hydration of a sulfuric acid monohydrate are obtained by a fitting procedure. The procedure uses experimental data from the literature on total pressure of sulfuric acid hydrates and free molecules. The enthalpy change in monohydrate hydration is taken, based on the results of ab initio calculations given in literature, to be AH2 = AH -10 kj/mol. The estimate of covariance matrix of uncertainties of these values is obtained. The standard errors of estimates of AH, AS, and AS2 are 43%, 63%, and 45% of the above values, respectively. Uncertainties are strongly correlated. The sulfuric acid hydration model is proposed for geophysical applications. The model converts total acid to free acid in accordance with experimental measurements and increases the predicted values of sulfuric acid-water nucleation rate by times compared with the classical liquid drop hydrate model. 1. Introduction liquid droplets having macroscopic properties such as liquid density and surface tension. This is a very rough approximation Atmospheric sulfate particles play important roles in for clusters containing only a few molecules. Calculations by the atmospherichemistry and in the Earth's radiative balance. They classical theory indicate the sensitivity of predicted nucleation influence the climate directly by scattering incoming solar rates to the variation of thermodynamical properties of radiation back to space and indirectly by acting as cloud monohydrates and dihydrates [Noppel, 1998]. condensation nuclei for cloud and fog droplets and thus affecting To date there are few molecular studies of sulfuric acid droplet concentrations, the optical properties, and the lifetimes of hydrates. Hale and Kathmann [1996] and Kusaka et al. [1996] clouds [Charlson et al., 1992]. It is believed that new sulfate studied clusters of varying sizes using classical Monte Carlo particles are formed in the gas phase via homogeneous or ion- simulation techniques. Kurdi and Kochansla'. [1989] performed induced nucleation of sulfuric acid and water vapors, possibly an ab initio study of the monohydrate assuming rigid molecules. aided by some other chemical species [Laaksonen et al., 1995]. Arstila et al. [1998] investigated the first three sulfuric acid The classical nucleation theory (including certain variants) is hydrates using ab initio density functional methods. All the atoms presently the only practical approach for predicting sulfuric were freely let to relax to the energetically optimal structure. The acid/water nucleation rates. Unfortunately, the predictions of the latter two papers provide binding energies of hydrates at the theory for this system have not been reliably tested so far, as the temperature of OK. Kusaka et al. [1998] developed a various laboratory measurements are not in quantitative computationally efficient Monte Carlo simulation method to agreement [Boulaud et al., 1977; Mirabel and Clavelin, 1978; study sulfuric acid/water nucleation. The enormous Wyslouzi! et al., 1991; Viisanen et al., 1997]. computational requirements and the lack of experimental and One of the difficulties for predicting the nucleation rate of quantum mechanical data necessitated the adoption of simple sulfuric acid and water vapors, compared with other binary model potentials for intermolecular interactions. The results systems, proceeds from the tendency of sulfuric acid to form indicate that the hydrates are highly nonspherical and the hydrates (small clusters of acid and water molecules) in the gas dissociation behavior of H2SO 4 in a cluster differs markedly from phase. The hydratestabilize the vapor. Their formation energy is that in bulk solution. Arstila et al. [ 1998] found that three to four negative, and therefore it is energetically more difficult to form a water molecules are needed to protonate the H2SO 4 molecule critical nucleus out of hydrates than out of free acid molecules. and, unlike Mirabel and Ponche [ 1991], that the classical values The ability of sulfuric acid to form hydrates was first recognized of the enthalpy of H20 addition for hydrates do not compare too by Doyle [ 1961 ]. The first thermodynamical theory for predicting well with the ab initio values. Boehringer et al. [ 1984] measured the hydration energies and hydrate distributions was presented by the enthalpy and the entropy of H20 addition to HSO4-. To the Heist and Reiss [1974]. The theory was later refined by Jaecker- author's knowledge there have been no experimental studies of Voirol et al. [1987]. Their theory relies on the capillarity thermodynamic properties of uncharged sulfuric acid hydrates. approximation, assuming that the hydrates can be described as Data on vapor pressures above sulfuric acid solutions have been obtained by measurements related to the number concentration or pressure of free acid molecules, not to the total Copyright 2000 by the American Geophysical Union. concentration of sulfuric acid hydrates and free molecules Paper number 2000JD [Noppel, 1998]. Marti et al. [1997] measured total pressure of /00/2000JD sulfuric acid hydrates and free molecules for aqueous solutions 19,779

2 19,780 NOPPEL: HYDRATION ENTHALPIES AND ENTROPIES OF SULFURIC ACID between 55 and 77 wt % H2SO 4 (corresponding to about 5-25% relative humidity). The vapor pressure data agreed with the predictions of Ayers et al. [1980] at more concentrated solutions and had a positive deviation for more dilute solutions. This positive deviation can be attributed to the impact of hydrates. McGraw and Weber [1998] calculated the distribution of sulfuric acid-water hydrates using the liquid drop hydrate model [daecker-voirol et al., 1987; Kulmala et al., 1991] and compared the model-computed total acid concentrations (free acid plus hydrated acid) with the total concentrations of sulfuric acid over solutions of varying composition measured by Marti et al. [1997]. The results of this first direct comparison with experimental measurementsuggesthat the classical liquid drop model overestimates the extent of hydrate formation. McGraw and Weber [1998] explored the consequences of this overestimation on binary sulfuric acidwater nucleation rates and on higher-order multicomponent nucleation rates involving these and additional trace species in the atmosphere. They showed that the overestimation of hydrate formation leads to underestimation of free acid concentration and corresponding relative acidity. Relative acidity is needed for predicting nucleation rates, provided that the total acid vapor concentration is known. The underestimation of relative acidity results in a substantial under-prediction of nucleation rates. In particular, a half an order of magnitude underestimate of relative acidity by classicaliquid drop model at relative humidity RH=20% and 2. Fitting Procedure The pressure of free acid molecules in the vapor in equilibrium with the liquid aqueous acid solution is given by =?oa'a(xm) where P0a is the acid vapor pressure above pure liquid sulfuric acid and dta(xam) is the acid activity of the solution with the acid mass fraction Xam (100'x m is wt % of acid). The data of acid activity, commonly used in nucleation calculations, have been obtained by measurements related to the number concentration or pressure of free acid molecules, not to the total concentration of sulfuric acid hydrates and free acid molecules [Noppel, 1998]. Marti et al. [1997] measured total pressure P of free acid molecules and hydrates above aqueous sulfuric solutions with Xam = (corresponding to about 5-25% relative humidity) at temperatures , , and K. The ratio of total acid pressure to free acid pressure is given by [Noppel, 1998] = (2) where Pw = AwPow is the vapor pressure of water above acid solution with the water activity Aw, Pow is the pressure of pure water, and Ki are the equilibrium constants for the successive additions of water molecules to an acid molecule. The effect of hydrates with two and more acid molecules is considered negligible. Numerical estimation showed that in the composition range X m = the sum in (2) is basically determined by three terms 1 +K Pw+K K2Pw 2. The equilibrium constants (i= 1, 2) can be expressed as ln(k,) = (-a/-/, + (3) where AHi an A& are the enthalpy and entropy change per mole of addition of a water molecule to hydrate with i-1 water molecules, respectively, T is absolute temperature and R is the molar gas constant. The constants Ki (i = 3, 4, and 5) are calculated by the expression [daecker-voirol et al., 1987] ln(k ) = -ln(pw,sol ) - 2crv w/(k rr,), (4) where ri denotes the radius of the hydrate (assumed to be spherical), cr is the surface tension, Pw,sol is the saturated partial pressure of water in atmospheres above a solution having the composition of a hydrate, Vw is the partial molecular volume of water, and kb is Boltzmann's constant. The acid mole fraction of a hydrate is taken to be 1/(1+i). The total pressure data of Marti et al. [1997] (47 data at temperature T= K corresponds to a factor of 105 points Pat, Xami; the data point X m = 0.541, T= K was underestimate of nucleation rate. The comparison by McGraw excluded due to remarkable deviation of this point from the and Weber of experimental and classicaliquid drop hydrate trend of other points) and (1), (2), and (3) give an model results suggests that improved models of hydration are overdetermined system of equations to determine A/-/, AS, required. Such models would include the conversion from AH2, and AS2. The pressures of pure water [Preining et al., total acid to free acid required in simulating geophysical 1981] and pure acid [Ayers et al., 1980] and the activities of processesuch as the formation of atmospheric aerosols from acid solutions [Zeleznik, 1991] are considered to be known, the oxidation of SO2. and the temperature dependence of A/-/, A&, AH2, and AS2 is In this paper the thermodynamic proprieties of hydrates with ignored in the temperature range of K. one and two water molecules are estimated by fitting the Marti et al. [1997] estimated the composition uncertainty values of these proprieties to the experimental vapor pressure of solutions in their experiments to be 15% and the data of sulfuric acid solutions by Marti et al. [1997]. These uncertainty of obtained total vapor pressures of acid to be proprieties can be used in calculation of sulfuric acid-water 36%. Due to a steep dependence of activity Aa on composition hydrate distributions and in the conversion from total acid to X m the uncertainty of 15% gives rise to the uncertainty of acid free acid in agreement with the measurement data of Marti et pressure of %. It is therefore appropriate to solve al. [1997]. the above mentioned system by minimizing the sum ; (X m -Xamp(Pai)) 2, where X m is the mass fraction of i data point and Xamp(Pai ) is the mass fraction that satisfies (2) for P =Pa,. The contribution of larger hydrates in the total vapor pressure of acid grows with vapor pressure of water. The quantities AH2 and AS2, determined by the least squares method, depend essentially on the values of a few experimental data points close to Xam=0.55. The positive value of enthalpy AH2 obtained by fitting is physically unrealistic. Arstila et al. [1998] obtained by ab initio calculations that at T = 0 K the enthalpy AH2 is about-10 kj/mol lower than A/-/ due to interaction between water molecules of dihydrate. The temperature dependence of zk/-/, was expected to be of the order of 4%/300 K as it was found for Ca 2+ hydrates [Kochanski, 1989]. The change in enthalpy with temperature within the same order is also obtained for monohydrates of ions C1-and NO3- [Castleman et al., 1982]. The total bond energy of-38 kj/mol, calculated by Arstila et al. for sulfuric acid monohydrate, is 70% smaller in magnitude than the value by Kurdi and Kochanski [1989]. It is believed that this difference is due to the different treatment of the electron

3 NOPPEL: HYDRATION ENTHALPIES AND ENTROPIES OF SULFURIC ACID 19,781 exchange correlation. The test calculations of Arstila et al. by different programs gave a difference of about 20% in the binding energy of a monohydrate. The binding energy of a water dimer by Arstila et al. is about 20% smaller in magnitude than experimental binding energy. The value of- 43 kj/mol obtained by Arstila et al. for the total bond energy (enthalpy) of protonized monohydrate HSO4-'H20 at 0 K is 16% smaller in magnitude than the experimental value of kj/mol ( K, uncertainty about 11%) [Boehringer et al., 1984]. Taking into account the above, the enthalpy change Z r-/2 was chosen equal to AH -10 kj/mol and the values of AHb AS, z 2 were subjected to fitting. 3. Results The results of the fitting procedure are presented in the first two rows of Table 1. The results of other authors and the results of the classicaliquid drop model by different activities are also presented in the Table 1. The thermodynamical data needed in the classicaliquid drop _model are taken from the collection of Kulmala at al. [1998]. The obtained enthalpy value AH = kj/mol is 17% larger in magnitude thml the ab initio value of Arstila et al. [1998]. Taking into account that the enthalpy value for HSO4 'tt20 by Arstila et al. is 16% smaller than the experimental one [Boehringer et al., 1984] and the variation of the above mentioned test calculation values by Arstila et al. was 20%, the obtained value = kj/mol is in rather good accordance with the restilt of Arstila et al. [1998]. When the activities by Taleb et al. [ 1996] 'are applied instead of activities by Zeleznil [ 1991 ], the agreement disappears (see the second row of Table 1). The lnodel expressions for activities by Taleb et al. [1996] have attractively simple fbnn and they have been used in recent hydration and nucleation calculations [Arstila et al., 1998' Kulmala et al., 1998]. The relative chemical potentials by 7½tleb el al. [1996] and by Zeleznik [1991] differ up to 11% over the range of composition and temperature of fitting. The relative enthalpies differ up to 75%. A simulation was carried out to estimate the influence of measurement errors of Xami, Pal. The standard error of mass fractions Xam was estimated, using the sum of squares Ei(Xami - Xamp(Pai)) 2, to be Normal random numbers with a zero mean and a standard deviation of were added to Xamp(Pai) calculated by (2) at the values of AHb AS, AH2, and 33'2 given in the first row of Table 1. Normal random values with the standard deviation (0.32'P i)/2 and a zero mean were added to the values Pal. The total H2SO 4 pressure in the gas flow stream of experimental instrument was measured with an uncertainty of about 32% [Marti et al., 1997]. Altogether 220 sets of data points (X mi, t i) were simulated and the corresponding values of AH, AS, and AS2 were calculated. On about 10% of occasions it was not possible to obtain the values for AS2. The fitting method led to large growing negative values of AS2 or to a zero value of K2. The temperature range of the data points within K is small. The effect of the alteration of AH on the sum of squares to be minimized can largely be compensated by the change in A& and AS2 (see(2) and (3)). Therefore the obtained values of AH, A&, and AS2 were rather scattered and strongly correlated. The standard deviations of AH, AS, and AS2 were 38%, 59%, and 36% of the corresponding values, respectively. The uncertainty of the added 10 kj/mol (A/J2 =AHl-10 kj/mol) was taken to be 20%. The changing of the added 10 kj/mol by 20% led to the change of fitted values of AH, AS, and '2 by 0.8%, 1.3%, and 4.0%, correspondingly. The product of vectors (AAH, AA&, z Am 2)T(z Z -/i, Z i, Z Z 2), as a covariance matrix due to the uncertainty of the added 10 kj/xnol, was surmned to the covariance matrix evoked by the simulated data points (x ',m,,/,,). Here the superscript T denotes transposing, and the extra character A denotes the changes in the fitted values of AH, A&, and AS2. Table 1. Gas Phase Hydration Enthalpies AH, AH2, Entropies.&%', &5'2, Free Energies AG, AG2, and Equilibrium Constants Kl, K2 at K for the Water Molecule Addition to an Acid Molecule and to a Monohydrate, Respectively AH, A&, A (71, K, MrI2, AS2, A (72, K2, kj/mol J/mol kj/mol atm - kj/mol J/tool kj/mol atto - Reference Fitting results (this work), activities by Zeleznik [1991] Fitting results (this work), activities by Taleb et al. [1996] -38* Arstila et al. [1998] -64.8* Kurdi and Kochanski [1989] # Classical liqui drop model, activities by Zeleznik [1991] Classical liqui drop model, -23.4& a -22.3& 7930 & *Evaluated by ab initio at T = 0 K. Obtained from (3),(4), and the Gibbs-Helholtz equation (8). activities by Taleb et al. [1996] Kusaka et al. [1998], Monte &The values cited here are calculated by the concentration values of hydrates estimated from Figures 17, 19, and (38) of Kusaka et al. [1998]. Carlo

4 19,782 NOPPEL: HYDRATION ENTHALPIES AND ENTROPIES OF SULFURIC ACID To estimate the effect of constants K3, K4, and K5 on the fitted values of M-/1, ZXS1, and zxs2, the values of chemical Table 2. Covariance Matrix of Uncertainties for the Fitted Values of Enthalpy and Entropy Changes With the Water constants were doubled and set to zero. The evoked changes Molecule Addition to an Acid Molecule and in zkh1, zxs1, zxs2 were 0.15%, 0.5%, and 1%, respectively. a Monohydrate M-/1/(R.K), ZXS1/R, ZXS2/R, The covariance matrix, analogous to the one due to the added Respectively 10 kj/mol in the above paragraph, was added to the sum of M-/1/(R.K) A&/R AS2/R covariance matrixes. Based on the result ofayers et al. [1980](Poa in atm), ln(p0a ) = , (5) T M--/ /(R. K) A&/R AS2/R 6O.9O the uncertainty of the value of vapor pressure of pure sulfuric R is molar gas constant acid was described with a standard deviation of 175*R'K/1.96 for the enthalpy change upon vaporization, and with a standardeviation of 0.437'R/1.96 for the entropy change. The number 1.96 is the 97.5th percentile of the chemical potentials at temperatures and K standard normal distribution. The uncertainties of enthalpy were obtained from the above random functions by the sum and entropy changes upon pure acid vaporizfition were f(xam, K) +fr(xam, )(+5), where +5 stands for considered independent. The change in vaporization enthalpy T= and -5 for T= K. The fitting procedure was by the value of the above standard deviation resulted in repeated many times and standardeviations of change in the changes in zxh1, ZXS1, and zxs2 by 16%, 17%, and 22%, fitted values of zkh1, zxs1, and zxs2 were determined to be respectively. In the case of vaporization entropy, the changes 1.7%, 1.8%, and 4%, respectively. A corresponding covariance matrix was added to the sum of covariance were 10%, 11%, and 15%, respectively. The corresponding matrixes described before. estimates of covariance matrixes in the form of (AM-/1, A S1, Az 2)T(A i, Az i, Az 2) were added to the sum of The sum of covariance matrixes is presented in Table 2. covariance matrixes described above. The effect due to the The standard errors (the standardeviations) of zx/-/1, zxs1, and uncertainties of saturated vapor pressure of pure water was zxs2 are 43%, 63%, and 45% of the corresponding values, considered negligible. respectively. It follows, for example, from the obtained values In the implementation of the above fitting procedure the of zx/-/1, zxs, and Z 2 and Table 2 that the free energies AG,=z,-T. ZXS, of the water molecule addition to an acid activities by Zeleznik [1991] were used. Zeleznik [1991] correlated experimental data of dilution enthalpy, heat molecule and to a monohydrate K are kj/mol capacities, electromotive force and solution freezing points as and-11.0 kj/mol, respectively. The estimated standard errors a function of temperature and composition. The correlation of these free energies are 17% and 42% of the above values, respectively. The correlation coefficient between uncertainties yielded mutually consistent expressions that generally reproduce the experimental data to +0.75%. To estimate of these free energies is Due to a strong correlation approximately the effect of the uncertainty of water and acid between zkh1, ZXS1, and zxs2 the relative standard errors of AG activities on the fitted values, random parabolic polynomials and AG2 are smaller than the corresponding values for were added to the expression of relative chemical potential of zxs1, and zxs2 but still large. sulfuric acid RTInAa [Zeleznik, 1991] in the composition The obtained values of ZXH1, ZX/-/2, ZX&, and / S 2 and Eqs. range X m = at K. Integration of the Gibbs (1), (2), (3), and (4) represent a model for sulfuric acid - Duhem relation hydrate distributions. The number of calculations can be (1- xa)dlnaw + xadlnaa = 0, (6) reduced by estimating the hydration enthalpies zx/-//. and entropies fisi of larger hydrates before the application of the model on the base of (3), (4), and (8) and the liquid drop where Xa is the acid mole fraction, with a random model, and then using these values and (3) in calculations. integration constant was used to obtain random functions for The values zk/--/3/(r.k) = -5124, zxs3/r = ,/ /4/(R.K) the relative chemical potential of water RTInAw from the = -4902, / S 4/g = , ZX/-/5/(R-K) = -4742, and zxs5/r above parabolic polynomials. These added random functions = , the results of the liquid drop model at T = K, J(X, m, K) were chosen to cause the variation of the were used in the fitting procedure. The difference between the values of the relative chemical potentials with a standard model predictions of the rati of total acid to free acid deviation of 0.75/2% to reflect the exactness of reproduction with fixed and unfixed values of MY/ and zxsi (i= 3-5) of experimental data of the electromotive force. Random parabolic polynomials were generated to represent the reaches up to 8 % at T= 200 K and RH=100%, which is the maximum value in the ranges of T= K and RH=0 - uncertainty of the derivative of the relative chemical potential 100%. Ignorance of the temperature dependence of enthalpies of acid with respecto temperature. Corresponding random zkh/and entropies zxsi for all values of i leads, according to the functions to representhe uncertainty of the derivative of liquid drop model (see also (2),(3),(4)), to an error of 15% in relative chemical potential of water were obtained by the the ratio of P,,/,, at RH=100% and T=200 K. The classical integration of the Gibbs - Duhem relation with a random value of P,,/ at T=200 K was obtained by the extrapolated integration constant. The parameters of these last random values of the surface tension fitting function. The surface functions fr(xam, K) were chosen so that the standard tensions of sulfuric acid solutions have been measured at deviation of the changes in enthalpy of aqueousulfuric acid temperatures between 283 and 323 K. Therefore the relative to pure species, caused by these functions, was 0.5% - 0.6% in the range Xam = The random functions for extrapolated values of the surface tension, but also the classical value of P,,/, may be in error at T=200 K.

5 NOPPEL' HYDRATION ENTHALPIES AND ENTROPIES OF SULFURIC ACID 19,783 The proposed model converts total acid to free acid in accordance with experimental measurements by Marti et al. [1997]. The predictions of the ratio of total acid to free acid P / are times lower than the predictions by the explored possible systematic errors and found no significant bias in their experiments. Despite these potential disadvantages, the author recommends that the model herein be used in geophysical applications such as simulation of the formation of atmospheric aerosols from the oxidation of SO2. The considerations for this recommendation will now be summarized. The hydrate formation has a significant effect on the nucleation of sulfuric acid. More sophisticated calculations of hydration do not yet give results directly suitable for geophysical applications. The liquid drop model is presently the only practical approach for predicting the hydration of sulfuric acid, but the liquid drop model is a rough approximation for hydrates containing only a few molecules. Their predictions are therefore less reliable than the results of the proposed model. The proposed model is based on the experimental data by Marti e! a/. [1997] and agrees rather well with the ab initio results by Arstila e! a/. [1998]. It is better to use partially verified models based on experiments and ab initio calculations than to use the liquid drop model and to introduce corrections later on. Only the introduction of better estimates of enthalpies and entropies of hydration when they become available is needed to make the model more exact. pressure of acid solutions. Figure 1 presents the relative change of equilibrium constant K due to the vapor pressure change APa = /[aa (CS'xXam )- aa (Xam )]2 + [CS. pa (Xam )]2, (7) classical liquid drop model. As a result the binary sulfuric acid-water nucleation rates of classical theory with the of aqueou sulfuric acid at various values of mass fraction hydration correction by the proposed model are Xam. Here the coefficient 5'x describes the change in pressure times higher than the classical rates with the hydration due to the uncertainty of composition and the coefficient 5e correction by the liquid drop model in the ranges of T=200 - characterizes the measurement uncertainty of total vapor 350 K and RH = 0-100%. The largest differences ( 8.5x 106 pressure of acid. The coefficient 5'x is taken to be 1.03 and times) in nucleation rates are occurring around the 1.01 and the coefficient 5'p and The acid vapor temperature T- 255 K and relative humidity RH = 8%. At pressure Pa(Xam) is given by (1) and (2) and the derivative temperature T- 300 K the difference of 5x 10 s times at RH = (1/K1)(dK /dpa) = (1/KO/(dPa/dK ). The relative change 10% changes to 2x 103 times at RH = 100%. The classical (1/KO(dK /dp,,)ap,,, given in Figure 1 at Xam=0.55, 4=1.03, theory used here has been presented by Noppel [1998]. Based and 52=0.32 is Considering only the experimental on the analyses by McGraw and Weber [1998] the effect of uncertainties of acid vapor pressure and mass fraction, the the same order of magnitude is expected also for higher-order relative uncertainty (two standard deviations divided by the multicomponent nucleation rates. value of K0 for the fitted vane of K is As one can see The proposed model converts total acid to free acid in from Figure 1, the relative change in equilibrium constant K accordance with experimental measurements by Marti et al. decreases toward the smaller values of mass fraction. At first, [ 1997] but, due to the scatter of the experimental data points, down to about Xam--0.6, the decrease is rapid and then it the uncertainty of the conversion is large. For instance, at becomes remarkably slower. At the values of X,,m greater than T= K the ratio of total acid to free acid P / is about 0.6 only the more exact determination of solution predicted with uncertainty factors of about 5, 6, and 8 for the composition will improve the accuracy of equilibrium relative humidities of 25%, 50%, and 100%, respectively. constant K1, whereas at the small values of Xam < 0.2 the more These factors correspond to the uncertainty of two standard exact determination of acid vapor is essential. In the range Xam deviations of In 5, In 6 and In 8, respectively, in ln(p / ) = , both composition and vapor pressure should be estimated by the covariance matrix of Table 2. In addition to determined more exactly. The vapor pressure measurements the scatter of the data points, the data of Marti et al. [1997] of solutions with mass fraction values smaller than 0.5 are may contain significant systematic errors, since these were more important but also more difficult owing to the small very difficult experiments. However, Marti et al. [1997] magnitude of acid partial vapor pressure of these solutions. loo Q,. 10 <:I 1 0.1,, I i,, Acid mass fraction x am Analyses of the Effects of Uncertainties in Vapor Pressure and Composition The estimated uncertainties for equilibrium constants of hydration are large. Here the essential part is brought about by the experimental uncertainties in composition and vapor Figure 1. Relative change in the values of equilibrium constant K of the gas phase water molecule addition to an acid molecule due to the vapor pressure change APa of aqueous sulfuric acid given by (7) at various solution compositions and at T= K. Curve 1, 8x=1.03, 8e=0.32; curve 2, 8x=1.03, 82=0.16; curve 3, 8x=l.01, 82=0.32; curve 4, 8x=l.01, 8/,=0.16.

6 19,784 NOPPEL: HYDRATION ENTHALPIES AND ENTROPIES OF SULFURIC ACID The relative changes in equilibrium constant K2 are larger but water molecule to a sulfuric acid molecule, and the hydration the dependencies are similar such that the same conclusions entropy 2 = -145 J/mol of the addition of a water molecule can be drawn. The enthalpy of acid molecule hydration 3J-/1 can be obtained from the Gibbs - Helmholtz equation to a sulfuric acid monohydrate are obtained by a fitting procedure. These values were fitted to the experimental data of total pressure of sulfuric acid hydrates and free acid molecules [Marti et al., 1997]. The estimate of covariance matrix of the uncertainties of these values is obtained. The AH = O(G /T) = R O(-lnK ) -R [nk. (8) 0(I/T) 0(l/T) A(I/T) standard errors are 43%, 63%, and 45% of the estimate values Taking the change Aln K] equal to (d In K]/dPa)APa(Jx, J?), of AH,, and 2, respectively. The uncertainties are we get from the last approximatequality the temperature strongly correlated. The precision of estimates can be change AT that is needed to get the estimate of AH with the uncertainty of 100% at the pressure uncertainty of APa. The improved by increasing the temperature range of experimental pressure data, by measuring the total acid vapor pressure of estimate of AH is obtained by the same approximation. The acid solutions with acid mass fraction of about 0.5 and temperature change AT depending on a mass fraction Xam is presented in Figure 2. In Figure 2 the temperature change AT at xw =0.55, Jx=l.03, and 8 =0.32 is 10 K. The relative uncertainty (two standar deviations divided by the value of AH ) for the fitted estimate of AH is 0.86 (0.76, if only the smaller and, most importantly, by reducing the uncertainties in acid solution composition. A revised model of sulfuric acid hydration, using the derived values of enthalpy and entropy and (1), (2), and (3), is proposed to be suitable for geophysical application such as uncertainties of experimental values of acid vapor pressure simulating the formation of atmospheric aerosols from the and mass fraction are taken into account). The temperature oxidation of SO2. This model converts from total acid to free range of the experimental data points is 10 K. If this temperature range were increased from 10 to 100 K, it would acid in agreement with experimental results by Marti et al. [1997] and ab initio results of Arstila et al. [1998] and be possible to diminish the uncertainty of 3Jf by a factor of increases the predicted values of sulfuric acid-water 10. It follows from Figure 2 that diminishing of the nucleation rate by times. The model is applicable experimental uncertainty of vapor pressure values by a factor also for higher-order multicomponent nucleation. of 2 and the experimental uncertainty of mass fraction by a Acknowledgments. This research has in part been supported by factor of 3 at xw =0.55 would diminish the uncertainty of the the Estonian Science Fotmdation grant estimate of AH by a factor of 2.7. Conclusions drawn from Figure I are also confmned by the behavior of curves in Figure 2. References 5. Conclusions The values of gas phase hydration enthalpy 3J-/ =-44.3 kj/mol and entropy AS1 = J/mol of the addition of a.l IO i... I i 1 I _ Acid mass fraction Xam Figure 2. The temperature difference AT that is needed to get the estimate of hydration enthalpy AH] with the uncertainty of 100% at the pressure uncertainty of APa given by (7) at various values of mass fraction Xam and at T= K. Curve 1, 6x=1.03, 6e=0.32; curve 2, 6x=1.03, $e=0.16; curve 3, 6x=l.01, =0.32; curve 4, 6x=l.01, $p=0.16. 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