CHAPTER 3: PHASE EQUILIBRIA 3.1 Introduction Multiphase and solution thermodynamics deal with the composition of two or more phases in equilibrium. Thus, the maximum concentration of a species in an aqueous stream in contact with an organic stream can be estimated by these calculations. This can establish the contaminant levels obtained in various wastewater streams. A second major application is in partitioning of a pollutant into various phases in the environment. These multiphase thermodynamic calculations are important in design of heterogeneous reactors. In this section, we provide some basic definitions and illustrate the applications of thermodynamic models to waste minimization. First, we provide various definitions for thermodynamic equilibrium and then illustrate them with applications. 3.2 Vapor-Liquid Equilibrium The ratio of the composition measure such as (mole fraction) in the vapor phase to that in the liquid phase at equilibrium is referred to as the K-value. Note that K y is dimensionless. y K i (1) yi x i eq where yi is the mole fraction of species i in the vapor phase and xi is the liquid. For ideal solutions, the Raoults law applies. This can be stated as follows. At equilibrium the partial pressure of a species in the gas phase, Pi , is equal to the mole fraction of the species in the liquid vap phase, , multiplied by its vapor pressure, Pi , at the given temperature. It is also equal to the product of the mole fraction in the gas phase, , and total pressure, P. vap Pi xi Pi yi P (2) Hence, Raoults law can also be stated as: P vap y x i (3) i i P Therefore, the K-factor for ideal mixtures is: Pvap K i (4) yi P For non-ideal solutions, the K-factors can be calculated using the activity coefficients. However, for many environmental applications, one can use experimentally reported K-factors. These calculations are, for example, useful to find the composition of the vapor phase in contact with the liquid in the reactor. If there is a fugitive emission, then we can estimate the amount of the toxic chemicals that has inadvertently escaped to the atmosphere. Vapor-liquid equilibrium is also useful in estimating the maximum concentration of VOC in a mixture. For dilute solutions, or for gaseous species, Henry’s law, given by Eq. (5) is more convenient. xi Hi yi P (5) -1 where Hi is the Henry’s law constant, (atm ) and P (atm) is the total pressure. With this definition, we find that Ky-factor is 1 1 K yi (6) H i P Vapor pressure data is often needed to estimate the levels of VOC emissions at various temperatures. The data for pure liquids are well represented by the Antoine equation. B log P* A (7a) 10 T C or by an empirical extended Antoine is equation: k vap 2 k7 ln P k1 k4T k5 ln T k6T (7b) k3 T 3.3 Gas-Liquid Systems Phase equilibrium between a dissolving gas, A, and its dissolved concentration, A H2 O or A , in water is often expressed (for dilute systems) through the equilibrium constant KA A H 2O 1 1 K A mol L atm (8) PA which is a function of temperature H d nKA Aab dT RT 2 (9) where HHHis the heat of absorption. Typically H 0 Aab Asolution A gas Aab so that as temperature increases the equilibrium constant decreases. Actually, this equilibrium constant is called Henry's constant in environmental chemistry and is independent of composition for dilute solutions. 1 A H2 O HAAA P H M atm (10) Note that the units of H A are now (mole/L atm) and M means mole/liter. Large HA implies a very soluble gas. Low HA is a slightly soluble gas. (Attention: some books define the reciprocal of HA as Henry's constant). The solubility of various gases is indicated in Table 3.1. ˆ A dimensionless Henry's constant, H A , is obtained if gas concentration is used in its definition instead of partial pressure, i.e [A]g = P A /R T AHO HHRTˆ 2 (11) AA A g Note that depending on the composition measures used to define equilibrium, Henry’s constants appear 1 1 ˆ with different units e.g. H A atm , H A M atm , H . It is unfortunate that the chemical engineering and some environmental engineering literature use the reciprocal of the above defined Henry’s constant under the same name! We have adopted here the definitions prevalent in the 2 environmental chemistry literature although the chemical engineering approach is the older and the better established one. A website posted as part of the course gives the various definitions and conversion factors (http://www.mpch-mainz.mpg.de/~sander/res/henry.html). The following relationship holds between the above constants ˆ K A H A H A H A (12) CL TOT CL TOT CL TOT RT Where CL TOT is the total liquid molar concentration. TABLE 3.1: Henry’s Law coefficients of atmospheric gases dissolving in liquid watera 3.3.1 Water Ionization The ionization or recombination reaction for water is for all practical purposes infinitely fast H2O H OH (13) so that equilibrium is always established ' 16 Kw H OH/ H2 O 1.82 x 10 M at 298 K (14) Since molar concentration of water is constant, we get 14 2 Kw H OH1.0 x 10 M at 298 K (15) 3 For pure water [H + ] = [O H - ] = 1.0 -7 M. Recall that p H = - log [H + ] (16) so that p H = 7 for pure water at 298K. 3.3.2 Carbon Dioxide/Water Equilibria Carbon dioxide is prevalent in the atmosphere and undergoes the following equilibrations in water (with equilibrium constant for each step indicated in parentheses) CO 2 (g) H 2O CO 2 H 2O (KCh) (17a) CO2 H 2O H HCO3 (KC1) (17b) 2 HCO3 H CO3 (KC2) (KC2) (17c) We can express now the concentration of dissolved carbon containing species (ions) in terms of these constants CO H O K p H p (18a) 22 ch CO2 CO 2 CO 2 K CO H O H K p HCO C1 2 2 CO22 C1 CO 3 HH (18b) K HCO H K K p 2 C 23CO22 C12 C CO CO 3 2 H H (18c) We also must always satisfy the electroneutrality relation 2 H HCO332 CO OH (19) Upon substitution of the above expressions into the electroneutrality equation we get: H K p2 H K K p CO2 C1 CO 2 CO 2 C 1 C 2 CO 2 Kw H 2 (19a) HHH Upon rearrangement a cubic equation for [H + ] results 3 H K H K p H20 H K K p (20) w CO2 C1 CO 2 CO 2 C 1 C 2 CO 2 Given the temperature, T, and partial pressure of CO, p , the hydrogen ion concentration [H+ ] can 2 CO2 be calculated from the above equation and pH obtained. (For temperature dependence see Table 3.2). At p = 330 ppm (= 3.3 x 10-4 atm) at 283oK the solution pH = 5.6 ([H +] = 2.51 x 10-6 CO2 (mol/L)). This is often called the pH of 'pure" rain water. 4 TABLE 3.2: Thermodynamic data for calculating temperature dependence of aqueous equilibrium constants 3.3.3 Sulfur Dioxide/Water Equilibrium The scenario is similar as in the case of CO2 absorption (21a) SO g H O SO H O H 2 2 2 2 SO2 (21b) SO2 H 2 O H HSO 3 KS 1 2 HSO3 H SO 3 KS 2 (21c) The concentrations of dissolved species are SO H O H p (22a) 22 SO22 SO H K p SO22 s1 SO HSO 3 H (22b) H K K p 2 SO22 S12 S SO SO 3 2 H (22c) Satisfying the electroneutrality relation requires 3 H K H K p H20 H K K p (23) w SO2 S1 SO 2 SO 2 S 1 S 2 SO 2 The total dissolved sulfur in oxidation state + 4 is S(IV) and its concentration is given by KKKSSS1 1 2 * S IV HSO p SO1 2 H S() IV p SO (24) 2 2H H 2 * where H S() IV is the modified Henry's constant which we can express as *2pH pH HHKKK1 10 10 (25) S( IV ) SO2 S 1 S 1 S 2 5 Clearly the modified constant H * is dependent on pH. The larger the pH, i.e the more alkaline the solution, the larger the equilibrium content of S (IV ). This is illustrated in Figure 3.2 for p =0.2 ppb and 200 ppb. SO2 FIGURE 3.2: Equilibrium dissolved S(IV) as a function of pH, gas-phase partial pressure of SO2 and temperature Let us express now mole fractions of various dissolved sulfur species as function of pH (really these are mole ratios, i.e moles of a particular sulfur species dividied by total moles of S (IV ) or mole fractions on water free basis since water is the dominant component. SO H O 1 (26a) XKKK22 1 10pH 10 2 pH 0SO22 H OS IV S 1 S 1 S 2 HSO (26b) 3 1 pH pH 1 1XKK1SS 1 10 2 10 HSO 3 S IV 2 (26c) SO 1 XKKK3 112 10pH1 10 pH 3SO 2 SSS 2 1 2 3 S IV Common notation for these are 0,, 1 2 in aquatic chemistry. 9 Figure 3.3 illustrates the above three mole fractions for p 10 atm (1 ppb).