## HoHHorinp

because the ratio pw/p°w is by definition equal to the relative humidity expressed in the 0 to 1 scale. Thus the water activity in an atmospheric aerosol solution is equal to the RH (in the 0.0-1.0 scale). This result simplifies significantly equilibrium calculations for atmospheric aerosol, because for each RH the water activity for any liquid aerosol solution is fixed.

Water equilibrium between the gas and aerosol phases at the point of deliquescence requires that the deliquescence relative humidity of a salt will then satisfy

 Salt n (at 298 K) A B c (NH4)2S04 0.104 0.1149 -4.489 x 10 -4 1.385 x 10" -6 Na2S04 0.065 0.3754 -1.763 x 10 -3 2.424 x 10- -6 NaNOa 0.194 0.1868 -1.677 x 10" -3 5.714 x 10- -6 NH4N03 0.475 4.298 -3.623 x 10" -2 7.853 x 10" -5 KCl 0.086 - 0.2368 1.453 x 10" -3 -1.238 x 10" -6 NaCl 0.111 0.1805 -5.310 x 10- -4 9.965 x 10" -7

where a.ws is the water activity of the saturated solution of the salt at that temperature. The water activity values can be calculated from thermodynamic arguments using aqueous salt solubility data (Cohen et al. 1987; Pilinis and Seinfeld 1987; Pilinis et al. 1989).

10.2.2 Temperature Dependence of the DRH

The DRH for a single salt varies with temperature. The vapor-liquid equilibrium of a salt S can be expressed by the following reactions

where n is the solubility of S in water in moles of solute per mole of water (Table 10.2). The energy that is released in reaction (a) is the heat of condensation of water vapor, which is equal to the negative value of its heat of vaporization, — AHv. The heat that is absorbed in reaction (b) is the enthalpy of solution of the salt AHs. This enthalpy can be readily calculated from the heats of formation tabulated in standard thermodynamic tables. Values of AHs are shown in Table 10.3 (Wagman et al. 1966). The overall enthalpy change AH for the two reactions is

The change of the vapor pressure of water over a solution with temperature is given by the Clausius-Clapeyron equation (Denbigh 1981)

d ln Pw AH

 Salt AH,,kJ mol-1 (NH4)2S04 6.32 Na2S04 -9.76 NaN03 13.24 nh4no3 16.27 KCl 15.34 NaCl 1.88

which for this case becomes dlnpK A Hv AHS ~dT = ~RT2 'RT2 (,°-67)

Applying the Clausius-Clapeyron equation to pure water, we also obtain d In pi = AH„ dT RT2

where p°u, is the saturation vapor pressure of water at temperature T. Combining (10.67) and (10.68)

dT RT2

and substituting (10.63) into (10.69) and applying it to the DRH, we obtain

rf1n(DRH/100)

The solubility n can be written as a polynomial in T (see Table 10.2), and the equation can be integrated from To = 298 K to T to give The only assumption used in (10.72) is that the heat of solution is almost constant from 298 K to T. Wexler and Seinfeld (1991) proposed a similar expression assuming constant solubility, namely, B = C ~ 0 in (10.72), while expression (10.72) was derived by Tang and Munkelwitz (1993).

Dependence of the (NH4)2S04, NH4N03, and NaN03 DRH on Temperature

Calculate the DRH of (NH4)3S04, NH4N03, and NaN03 at 0°C. 15°C, and 30°C.

The DRH at 25*C for the three salts is given in Table 10.1. Using (10.72) and the corresponding parameter values from Tables 10,3 and 10.2, we find that

Deliquescence Relative Humidity

NH4N03 76.6 68.1 58.5

NaN03 80.9 76.9 73.0

(NH4)2S04

o International Critical Tables V Wexlerand Hasegawa (1954) • Tang and Munkelwitz (1993)

Theory

Temperature, °C

FIGURE 10.6 Deliquescence RH as a function of temperature for (NH^SO* (Reprinted from Atmos. Environ. 27A, Tang, I. N., and Munkelwitz, H. R„ Composition and temperature dependence of the deliquescence properties of hygroscopic aerosols, 467-473. Copyright 1993, with kind permission from Elsevier Science Ltd., The Boulevard, Langford Lane, Kidlington 0X5 1GB, UK.)

These results indicate that the DRH for (NH4)2S04 is practically constant within the temperature range of atmospheric interest while for the other two salts it varies significantly.

The predictions of (10.72) for ammonium sulfate are compared with measurements from a series of investigators in Figure 10.6.

### 10.2.3 Deliquescence of Multicomponent Aerosols

Multicomponent aerosol particles exhibit behavior similar to that of single-component salts. As the ambient RH increases the salt mixture is solid, until the ambient RH reaches the deliquescence point of the mixture, at which the aerosol absorbs atmospheric moisture and produces a saturated solution. A typical set of data of multicomponent particle deliquescence, growth, evaporation, and then crystallization is shown in Figure 10.7 for a KCl-NaCl particle. Note that the DRH for the mixed-salt particle occurs at 72.7% RH, which is lower than the DRH of either NaCl (75,3%) or KC1 (84.2%).

Following Wexler and Seinfeld (1991), let us consider two electrolytes in a solution exposed to the atmosphere. The change of the DRH of a single-solute aqueous solution when another electrolyte is added can be calculated using theGibbs-Duhem equation, (10.21). For constant T and p and for a solution containing two electrolytes (1 and 2) and water (w): FIGURE 10.7 Hygroscopic growth and evaporation of a mixed-salt particle composed initially of 66% mass KC1 and 34% mass NaCl. (Reprinted from Atmos. Environ., 27A, Tang I. N., and Munkelwitz, H. R., Composition and temperature dependence of the deliquescence properties of hygroscopic aerosols, 467^-73. Copyright 1993, with kind permission from Elsevier Science Ltd., The Boulevard, Langford Lane, Kidlington OX5 1GB, UK.)

Relative Humidity, %

FIGURE 10.7 Hygroscopic growth and evaporation of a mixed-salt particle composed initially of 66% mass KC1 and 34% mass NaCl. (Reprinted from Atmos. Environ., 27A, Tang I. N., and Munkelwitz, H. R., Composition and temperature dependence of the deliquescence properties of hygroscopic aerosols, 467^-73. Copyright 1993, with kind permission from Elsevier Science Ltd., The Boulevard, Langford Lane, Kidlington OX5 1GB, UK.)

where nun2, and nw are the numbers of moles of electrolytes 1, 2, and water, respectively, while P!, p2> and are the corresponding chemical potentials. Let us assume that initially electrolyte 1 is in equilibrium with solid salt 1 and the solution does not yet contain electrolyte 2. As electrolyte 2 is added to the solution, the chemical potential of electrolyte

1 does not change, because it remains in equilibrium with its solid phase. Thus d\ix = 0 in (10.73). The chemical potentials of electrolyte 2 and water can be expressed using (10.51) to get

Accounting for the fact that «2/»™ = M„,m2/] 000, where m2 is the molality of electrolyte

2 and Mw is the molecular weight of water, we obtain m2 d In oc2 -I--d In aw = 0

Integration of the last equation from m'2 = 0 to m'2 = m2 gives aw(m2) Mw fm2 m2 da2(m'2) , lnW=_T000/0 a2(m>2) dm'2 2 (1°'76)

Wexler and Seinfeld (1991) have argued that da2/dm2 > 0 and therefore the integral is positive, and then 