# Saturation Vapour Pressure above a Solution Droplet

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1 Appendix C Saturation Vapour Pressure above a Solution Droplet In this appendix, the Köhler equation is derived. As in Appendix B, the theory summarised here can be found in Pruppacher & Klett (1980) and Seinfeld & Pandis (1997). C.1 Kelvin Effect In Appendix B, the saturation vapour pressure for a plane (flat) liquid surface has been derived. For a droplet, the electrostatic forces hindering the molecules from leaving the liquid are lower because there are fewer attracting neighbouring molecules within any chosen range (Figure C.1). The mechanical equilibrium given in (B.15) then changes to take account of the surface tension at the curved surface of the droplet. The new condition for mechanical equlibrium is called the Laplace formula: pw pv = 2 ff v a (C.1) where ffv is the surface tension between the liquid and vapour phase and a is the droplet radius. Combining (B.17) and (C.1), an extension to the Clausius-Clapeyron equation is obtained: dp dt = de a dt = de sat,w dt + 2 v w vv vw d ffv a dt (C.2) 67

2 68 Saturation Vapour Pressure above a Solution Droplet Figure C.1: Spatial distribution of molecules in a droplet with a tightly curved surface compared to the distribution in a water body with a plane surface (McIlveen, 1998). Here, e a refers to the equilibrium vapour pressure at the droplet surface. The equilibrium between a pure water droplet and humid air is given by (Pruppacher & Klett, 1980) d ffa e a + R d =0 (C.3) 2 v w T a where ff a is the surface tension at the interface between droplet and air, and R is the universal gas constant. (C.3) can be integrated between a and 1 to obtain e a e sat,w =exp p 2 vw ff a RT a (C.4) Substituting v w by Mw %w, where M w is the molar weight of water, yields the Kelvin equation in terms of molar units: e a 2 Mw ff a =exp e sat,w RT % w a (C.5) Equation (C.5) was named after Lord Kelvin ( ), then Sir William Thomson, who first studied the influence of curvature due to capillary effects on the equilibrium pressure (Thomson, 1870). The equation can either be used to determine the saturation vapour pressure e a over a droplet of specified radius a, or to calculate the radius a of a droplet in (unstable) equilibrium with air at the given vapour pressure e a. It is obvious from the equation that the ratio ea, which is the definition of the saturation ratio, increases for decreasing droplet radius. In Figure C.2, the relative humidity (RH) and the esat,w supersaturation (SS) with respect to a plane surface of water are plotted versus the droplet

3 C.1. Kelvin Effect 69 radius of a pure water droplet. Since RH = SS 100=100 ea e sat,w, Figure C.2 represents a graph of (C.5). Figure C.2: The Kelvin effect. Relative humidity and supersaturation with respect to a plane surface of water versus the radius of a pure water droplet (Wallace & Hobbs, 1977). The Kelvin effect is the reason why homogeneous or spontaneous nucleation, which occurs when a number of water molecules in the vapour phase randomly collide to form clusters or small embryonic water droplets, does not take place at the supersaturations observed in the atmosphere which seldom exceed 0:01 = 1 %. A critical saturation ratio of about 5 has been determined experimentally for the molecules to actually stick together after colliding so that droplets are formed (Fleagle & Businger, 1980). At saturation ratios larger than 5, the droplets continue to grow by condensation, below this value they evaporate again. Although such large saturation ratios are not realised in the atmosphere, the average earth cloud cover amounts to ß 50 %. This fact suggests that another process is at work in droplet formation.

4 70 Saturation Vapour Pressure above a Solution Droplet C.2 Raoult Effect This other process involves aerosol particles that act as condensation nuclei. Since substances other than water are present, the term heterogeneous nucleation is used. The saturation vapour pressure above a solution, e sat,s, at the surface of a solution droplet is reduced by the following two mechanisms. On the one hand, the inherent size of the particles ensures that the supersaturations needed for condensation on the particle surface are not as high as for embryonic droplets because the particles are larger than the droplet embryos. On the other hand, water molecules at the surface are replaced by solute molecules. Hence, less water molecules are able to break free from the liquid and the equilibrium vapour pressure decreases. The quantitative explanation of the second mechanism is complex and requires a foray into physical chemistry. A solution is regarded as ideal if the chemical potential of every component, μ k;l,isa linear function of its aqueous mole fraction, ψ k;l : μ k;l = μ ffi k;l (T;p)+RT ln ψ k;l (C.6) where μ ffi k;l (T;p) is the standard chemical potential of component k. In general, the more dilute a solution is, the more it approaches ideal behaviour. Similarly, a gas mixture is ideal if its c components satisfy the relation μ k = μ k;0 (T )+RT ln p k (C.7) where μ k;0 (T ) is the standard chemical potential of component k and p k is its partial pressure. Equation (C.7) follows from (B.13), = v k, upon integration T;n j6=k v k = RT p k. Note that the standard chemical potential of a vapour is a function of temperature alone, whereas for a liquid it is a function of both temperature and pressure. For an ideal solution of component k in equilibrium with an ideal gas mixture follows from (B.16) that μ k = μ k;l (C.8)

5 C.2. Raoult Effect 71 and therefore μ k;0(t )+RT ln p k = μ ffi (T;p)+RT ln ψ k;l k;l or μ ffi μ k;l k;0 p k =exp ψ k;l = K k (T;p) ψ k;l (C.9) RT For ψ k;l =1, K k equals the partial pressure of component k in equilibrium with the pure liquid phase of k at the same temperature, p ffi, so that (C.9) can be rewritten as k p k = p ffi k ψ k;l (C.10) Equation (C.10) is known as Raoult s law. Figure C.3: Variation of the equilibrium partial pressures of components A and B of an ideal binary mixture (left) and a non-ideal binary mixture (right) with respect to the mole fraction of A, x A. The dashed lines in the right picture reflect ideal behaviour (Seinfeld & Pandis, 1997). Consider a binary solution of A and B. In the case of ideality, the partial pressures p A and p B will vary linearly according to (C.10) which can be seen in the left panel of Figure C.3. In the case of non-ideality, though, this linearity is no longer maintained, see the right panel in Figure C.3. Hence, (C.10) is no longer applicable. The deviation from ideality is commonly taken into account by introducing the rational activity coefficient,

6 72 Saturation Vapour Pressure above a Solution Droplet f k. Then, the chemical potential of component k in a non-ideal solution is given by μ k;l = μ ffi k;l (T;p)+RT ln(f kψ k;l ) (C.11) The standard chemical potential μ ffi (T;p) is defined by taking the limits f k;l k! 1 and ψ k;l! 1. In general, f k = f k (T;p;ψ k;l ). For ideal solutions f k = 1. The product of the activity coefficient and the corresponding mole fraction, f k ψ k;l, is the definition of the activity a k : Therefore, (C.11) can be written as a k f k ψ k;l (C.12) μ k;l = μ ffi k;l (T;p)+RT ln a k (C.13) Consider a binary solution of water and salt, denoted by subscripts w and s, respectively. The chemical potentials for this aqueous salt solution are characterised by μ w (T;p;a w )=μ ffi w (T;p)+RT ln a w; a w = f w ψ w (C.14) μ s (T;p;a s )=μ ffi s (T;p)+RT ln a s; a s = f s ψ s (C.15) From (B.16) follows for a system of water vapour in equilibrium with such an aqueous salt solution that μ v (T;e sat,s ;a w )=μ w (T; p; a w ) (C.16) Hence the substitution of this equilibrium condition in (C.14) yields for p = e sat,s μ v (T;e sat,s ;a w )=μ ffi w (T;e sat,s;a w )+RT ln a w (C.17) For fixed temperature, the equilibrium can be maintained for variable e sat,s ffi ln sat,s T de sat,s sat,s T + sat,s T de sat,s (C.18) Since the variation of chemical potential with vapour pressure can be substituted by

7 C.2. Raoult Effect = v k and the molar volume of water is small compared to the molar volume of water vapour, (C.18) simplifies to T;n ln sat,s T = v v RT = 1 e sat,s (C.19) The integration of (C.19) results in ln a w =lne sat,s + g(t ) (C.20) The integration constant g(t ) is an unknown function of temperature. It can be determined by taking the limit a w! 1. Since this corresponds to the case of pure water, g(t )= ln e sat;w so that e sat,s = a w (C.21) e sat,w Equation (C.21) can be considered as an extension of Raoult s law for non-ideal solutions. It should be noted that the ratio esat,s e sat,w is independent of chemical potential. To determine a w, the molality (m mol ), another concentration scale, is introduced. It is defined as the number of moles of salt dissolved in a kilogram of water. When the molar weights M w and M s are given in kg mol 1, the molality of the solution is m mol = n s n w M w = m s m w M s (C.22) When the molality is used, the rational activity coefficient f k has to be replaced by the mean ionic activity coefficient fl±. Then, after e.g. Robinson & Stokes (1970) and Lewis & Randall (1961): ln a w = ν n s n w 1 m mol Z m mol 0 m mol d ln fl± 1 A (C.23) where ν is the total number of ions the salt molecule dissociates into. The expression in parentheses in (C.23) defines the molal or practical osmotic coefficient of the salt in

8 74 Saturation Vapour Pressure above a Solution Droplet R m solution, ' s =1+ 1 mol m mmol 0 mol d ln fl±. Hence, a w =exp ν ' s n s n w =exp m s M w ν ' s m w M s (C.24) For an aquerous salt solution droplet with radius a and density % s, ms Mw Ms( 4 3 ßa3 %s ms ). Therefore a w =exp ν' s m s M w M s ( 4 3 ßa 3 % s m s ) ns nw is given by (C.25) Another expression is needed for a w for a solution containing insoluble substances. With the assumption that the insoluble portion does not adsorb salt ions or take up any water and that it is completely submerged, the volume of the liquid part of the solution is given by V (`) d = n w v w + n s v s = m w M w v w + m s M s v s = 4 3 ß (a3 r 3 u ) (C.26) where r u is the radius of the insoluble portion assumed to possess spherical shape. Equation (C.26) can be used to obtain an expression for m w : m w = 4 3 ß (a3 r 3 u ) M w v w m s M w v s M s v w which on substitution into (C.24) leads to a w = exp ψ 4 ν' s ms Mw Ms %w 3 ß (a3 r 3 u ) ms vs Ms! (C.27) The size of the insoluble particle is difficult to determine, though. For a mixed aerosol particle of radius r N, the mass fraction " m ms and the volume fraction " %N V = " m are mn %s therefore introduced. The total volume of the drop is approximated by mw + m N which %w %N

9 C.3. Köhler Curves 75 leads to an expression for m N m w : m N m w ß % N r 3 N % w (a 3 r 3 N ) and from there to or a w =exp ν' s " m M w % N rn 3 M s % w (a 3 rn 3 ) ν' s " V M w % s rn 3 a w = exp M s % w (a 3 r 3 N ) (C.28) (C.29) With the substitutions m s = % s rn 3 and m w = % w (a 3 rn 3 ), (C.29) yields a w =exp ν' s " V ψ s ψ w (C.30) C.3 Köhler Curves Above, the change in vapour pressure at the flat surface of an aqueous solution and at the curved surface of a droplet have been determined separately. Now, both effects are combined to derive the vapour pressure above a droplet formed by water vapour condensing on a salt condensation nucleus. For fixed temperature T, total air pressure p and number of salt moles n s, the dependence of the saturation vapour pressure e a on radius is given by d ln e a = 2 v w,0 RT d ffsol + d ln a w (C.31) a where ff soo is the surface tension between an aqueous salt solution and air. Equation (C.31) is integrated from the conditions for the droplet (vapour pressure e a, radius a, and activity a w ) to the conditions for a plane surface of pure water (e = e sat,w ;a!1;a w =1): e a 2 Mw ff sol = a w exp e sat,w RT % w a (C.32)

10 76 Saturation Vapour Pressure above a Solution Droplet and on substituting (C.30) e a 2 Mw ff sol = S p =exp ν' s " V ψ s e sat,w RT % w a ψ w (C.33) where S p denotes the saturation ratio at the droplet surface. Equation (C.33) is the Köhler equation given in Chapter 2, Equation (2.11).

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