1. Introduction

The equation governing the growth of aerosols with relative humidity and activation of cloud droplets is generally known as the Kohler equation (Kohler 1936; Fletcher 1966; Rogers 1976; Pruppacher and Klett 1997). The erroneous use of this phase equilibrium condition has a long history. Howell (1949) in his work on cloud droplet growth did not include the van’t Hoff factor. McDonald (1953) pointed out the error. Low (1969) calculated values of the van’t Hoff factor from tables of mean ionic activities for different concentrations of solutions consisting of various salts. Low also implied that a variable van’t Hoff factor should be used in the Kohler equation for cloud and aerosol physics applications.

Reiss (1950) and Doyle (1961) derived a modified Kohler equation that contains a term proportional to the partial derivative of the droplet surface tension with respect to the droplet composition. Young and Warren (1992) and Young (1994) derived a modified Kohler equation that also contains the derivative of the van’t Hoff factor with respect to concentration.

We present analytical arguments suggesting that the modified Kohler equation derived by Young and Warren (1992) is not correct. We conclude that the original Kohler equation is the proper equation for applications in cloud and aerosol physics. We support our conclusions by comparing theoretical calculations with laboratory measurements.

2. Raoult’s law, van’t Hoff factor, and water activity

Raoult’s law is an empirical expression describing the observed changes in equilibrium vapor pressure over a flat surface of solution as a function of solution concentration. Raoult (Fletcher 1966; Rogers 1976; Adkins 1983; Pruppacher and Klett 1997) describes the results of his observations as

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where p^′_∞ and p_∞ are, respectively, the equilibrium vapor pressure over a flat surface of solution and over a flat surface of pure water, and n_L and n^′_L are the number of moles (or molecules) of solvent (water) and solute (dissolved salt) present in the solution.

For solutions in which dissolved solute molecules are partially dissociated, an empirical coefficient, the van’t Hoff factor i, is introduced to account for the dissociation and for the nonideal character of real solutions. To balance both sides of Eq. (1), the equation is modified to

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where the van’t Hoff factor is considered to be an empirical function of the concentration.

Except for the van’t Hoff factor, the right-hand side of Eq. (2) is known from the composition of the solution. The left-hand side can be measured experimentally. The experimental values, giving the fractional decrease of the equilibrium vapor pressure due to a definite amount of solute present in the solution, are generally known as water activities of the considered solution (Pruppacher and Klett 1997). The water activity a is thus given by

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where the dependence on solution concentration is expressed in terms of molality, M (moles of solute per kilogram of solvent).

It is the water activity that is usually measured experimentally. From the water activity other quantities of interest can be calculated. A useful expression for the van’t Hoff factor i as a function of solute concentration and water activity can be obtained from Eqs. (2)and (3) in the form

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where m_s and M_s are the mass and the molecular weight of the solute, m_w and M_w are the mass and the molecular weight of the solvent (water), and M is the molality of the solution. Equation (4) was used by Low (1969) and by Young and Warren (1992) to calculate the van’t Hoff factor from the water activity a.

3. Phase equilibrium condition

Substituting for chemical potentials into Eq. (5) and neglecting a very small term, exp[μ⁰_L(T, p) − μ⁰_L(T, p_∞)], for water molecules in liquid, we obtain the equilibrium condition in a form of the Kohler equation

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where R is a universal gas constant and ρ is the density of the solution.

A more formal derivation in line with the general structure of equilibrium thermodynamics is to look for an extreme in an appropriate thermodynamic potential. In the case of phase transition at given temperature and pressure, the corresponding thermodynamic potential is Gibbs free energy (Adkins 1983; Callen 1985). The partial derivative of the Gibbs free energy with respect to a suitable concentration variable determines the equilibrium condition and the sign of the second partial derivative determines the nature of the equilibrium (stable or unstable equilibrium). This procedure is more general than the one used above in the derivation of the equilibrium condition (8); however, it also provides more opportunities to obtain inaccurate results when various terms appearing in the equations are not treated in a consistent manner. In this way, several forms of modified Kohler equations have been derived (Reiss 1950; Doyle 1961; Young and Warren 1992) in the past.

The Gibbs free energy of a solution droplet surrounded by the vapor of its constituents may be written in the form

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where all quantities are evaluated at ambient temperature and pressure, n_L and n_V are the number of molecules of water (solvent) in the liquid and the vapor phase, n^′_L and n^′_V are the corresponding numbers of molecules of solute, μ_L and μ^′_L are chemical potentials of solvent and solute in the drop, and μ_V and μ^′_V are the corresponding chemical potentials in the vapor phase; A is the surface area of the solution droplet and σ is its surface tension.

We assume that the salt molecules are limited to the liquid phase and set n^′_υ = 0. Additional constrains are ∂n_L/∂n_υ = −1 (the total number of water molecules is a constant), ∂n^′_L/∂n_L = 0 (salt is limited to the liquid phase), and ∂μ_V/∂n_L = 0 (the chemical potential of water molecules in vapor is a function of T and p only). By taking the partial derivative of the Gibbs free energy with respect to the number of solute molecules in the droplet and using the above given constrains, we obtain

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When the partial derivative of the Gibbs free energy is set to zero and only the first three terms on the rhs of Eq. (10) are kept, we obtain the standard form of the Kohler equation (8).

The last term in Eq. (10) contains the partial derivative of surface tension with respect to the concentration variable. When this term is retained, the modified Kohler equation containing the term ∂σ/∂n_L is obtained (Reiss 1950; Doyle 1961). Theoretical results obtained using such a modified Kohler equation are in disagreement with experimental measurements (Flageollet et al. 1980; Wilemski 1988).

The water molecule chemical potential μ_L is a function of water activity, a, and thus it can be considered to be a function of the van’t Hoff factor i. Consequently, the term ∂μ_L/∂n_L in Eq. (10) contains the derivative of the van’t Hoff factor with respect to the droplet composition variable. The terms containing a derivative of the surface tension and of the van’t Hoff factor were retained by Young and Warren (1992) in the derivation of their version of the modified Kohler equation.

In thermodynamically consistent treatment, however, the last three terms on the rhs of Eq. (10) will cancel each other. Using the Gibbs–Duhem identity (Chue 1977; Adkins 1983; Callen 1985) and the Gibbs adsorption equation leads to the following relation derived by Wilemski (1984) and Reiss and Koper (1995):

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Thus, setting ∂G/∂n_L = 0 and using Eqs. (10) and (11), the original Kohler equation (8) is recovered. The correction terms obtained by Young and Warren (1992)cancel each other out. The same conclusion was reached recently by Konopka (1996). Consequently, it is the Kohler equation in form (8) that provides the correct equilibrium condition for the solution droplet for both supersaturated and subsaturated water vapor pressure.

4. Numerical results and experimental measurements

To compare results predicted by the Kohler equation (8) and by the modified Kohler equation derived by Young and Warren (1992) a solution of ammonium sulfate in water is considered. Figure 1 shows the equilibrium water vapor pressure as a function of solution droplet radius for relative humidity slightly above 100%. The solid curve is for the Kohler equation (8) with the van’t Hoff factor as a function of molality as given by Young and Warren (1992). The dotted line is for the Kohler equation (8) with the constant van’t Hoff factor (i = 3), and the dashed curve is for the case of the modified Kohler equation containing the additional di/dM term. Results displayed in Fig. 1 are similar to those obtained by Young and Warren.

Young and Warren (1992) assumed that the modified Kohler equilibrium curve (dashed line in Fig. 1) is the correct solution. They observed that the Kohler equation (8) with constant van’t Hoff factor i = 3 (dotted line in Fig. 1) is very close to the modified Kohler equation results, while the Kohler equation (8) with a concentration-dependent van’t Hoff factor i (solid line in Fig. 1) deviates considerably from the modified Kohler equation results. They concluded that it is better for cloud physics applications to consider i to be a constant equal to its assumed value at infinite dilution (i = 3 for ammonium sulfate) rather than take i to be a function of molality, as suggested by McDonald (1953) and Low (1969). Unfortunately, there are no sufficiently accurate measurements at supersaturated vapor pressure that could be used to show which of the theoretical results is correct.

With the development of laser and electromagnetic levitation of individual droplets (Ashkin and Dziedzic 1977; Chylek et al. 1978; Chylek et al. 1983; Fung et al. 1987), it became possible to accurately measure the mass of individual levitated droplets as a function of relative humidity. Tang and Munkelwitz (1994) measured the mass of liquid water condensed on an individual ammonium sulfate particle as a function of relative humidity. Their results are shown as solid circles in Fig. 2. The mass of the (NH₄)₂SO₄ aerosol remains unchanged as it is exposed to an increasing relative humidity, until the deliquescence point at about 80% RH is reached. The condensed water dissolves ammonium sulfate and the solution droplet continues to grow with the increasing relative humidity. When relative humidity is decreased, the solution droplet evaporates until the recrystalization point around RH = 37% is reached. This set of measurements provides a suitable set of results against which the Kohler equation (8) and modified Kohler equation (Young and Warren 1992; Young 1994) can be tested.

Figure 2 shows the experimental measurements together with numerical results obtained using the modified Kohler equation of Young and Warren (1992)(dashed line), the Kohler equation (8) with a variable van’t Hoff factor (solid line), and the Kohler equation with a constant i = 3 (dotted line). The measurements of water activity by Tang and Munkelwitz (1994) are used to calculate the van’t Hoff factor for an ammonium sulfate solution using Eq. (4). It is the Kohler equation (8), with the variable van’t Hoff factor i, that provides the best agreement with experimental data. The Kohler equation with a constant van’t Hoff factor and Young and Warren’s modified Kohler equation lead to considerable errors at higher relative humidities.

5. Conclusions

The application of equilibrium thermodynamics leads to the Kohler equation (8) as an equilibrium condition between the vapor and solution droplet. The modified Kohler equation derived recently by Young and Warren (1992) is theoretically inconsistent and is in disagreement with experimental data. Also, the suggestion (Young and Warren 1992; Young 1994) that it is more accurate to use the Kohler equation with a constant van’t Hoff factor rather than with the van’t Hoff factor dependent on the solution concentration is not supported by theory or experiment.

We conclude that the original Kohler equation with the van’t Hoff factor dependent on concentration, as suggested by McDonald (1953) and Low (1969), is the proper equilibrium condition and that the numerical results following from this equation are in agreement with experimental data at subsaturated conditions.