Ideal Mixtures - Dalton's, Raoult's Laws.pptx

Ideal Mixtures - Dalton's, Raoult's Laws

1. Ideal Mixtures - Dalton's, Raoult's Laws

2. Ideal mixtures, gas or liquid, consist of components that do not interact with each other chemically or physically. The concept of ideal mixtures has formed the basis for many quantitative relationships describing equilibrium. Of particular interest are Dalton's law of partial pressures and Raoult's law relating the pressure exerted by a component in the vapor phase to its concentration in the liquid phase. Dalton's law states that the total pressure of a mixture of gases is equal to the sum of the partial pressures of the mixed gases. Thus, PT = _PPi = PP1 + PP2 + PP3 + ... Dalton also postulated that the partial pressure of an ideal gas in a gas mixture is proportional to its mole fraction, that is, the relative number of molecules of that gas in the mixture. Thus, PPi = yi PT

3. Raoult's law, relating the partial pressure in the vapor phase to the liquid phase composition, is expressed as: PPi = xi VPi Combining Dalton's and Raoult's laws results in an expression describing mixtures of ideal vapors and liquids in equilibrium. PT = _ PPi = _ yi PT = _ xiVPi and for component i , yi = xi ( VPi /PT)

4. Equilibrium K-Values The definition of equilibrium K-value, also called K factor or distribution coefficient, of component i in a mixture is given in the following equation: Ki = yi xi The K-value is simply the ratio of the vapor to the liquid mole fraction of i . This ratio has no special thermodynamic significance, but has found extensive use in high-pressure VLE work. For ideal systems where Raoult's law applies, it can be expressed as: Ki = yi xi = VPi PT Equilibrium K values can be obtained from graphs or nomographs like the De Priester nomograph, Figure 4. K values are a function of temperature and pressure. For nonideal mixtures, K values are also a function of composition.

5. Two Component Example Let's assume that we have an ideal mixture of propane and n-butane at 100°F. The vapor pressures of the two components at 100°F are: • Propane 13 atm = 191 psia (Component 1). • n-Butane 3.5 atm = 52 psia (Component 2). The total pressure, PT, of the mixture can be calculated from, PT = PP1 + PP2 = x1VP1 + x2VP2 PT = 191x1 + 52(1-x1) = 52 + 139x1 This last equation indicates that the total pressure of an ideal binary mixture is a linear function of the composition. This relationship is illustrated in Figure 5, which shows that the total pressure is the sum of partial pressures and is a straight line between the vapor pressure of n-butane (x1 = 0) and propane (x1 = 1.0).

6. Mixtures Approximated as Ideal The mixtures that can be approximated as ideal must satisfy the following requirements: • Total pressure of the system must be below 200 psia. • The components must be chemically similar, for example, butane and pentane, both paraffins. A mixture of an aromatic component and a paraffin such as benzene and hexane cannot be approximated as ideal. • The components must be close boiling, that is, they must have similar boiling points. • The system pressure and temperature must not be near the critical pressure and temperature of the mixture. Using ideal mixture correlations in calculations results in approximate compositions or conditions (P, T). The error may be acceptable for a simple operation, such as a flash drum separation. The same correlations used in a superfractionator , where tray-to-tray calculations compound the error, may produce unacceptable results.

7. Equilibrium Diagram Figure 6 depicts a simple flash separation. The feed consists of two components, propane and n-butane. The feed temperature and composition vary. The table in Figure 6 lists vapor and liquid concentrations of propane and distribution coefficients (K1 and K2) for propane and nbutane , of the two components for five temperatures. Pressure is fixed at 100 psia. At 70°F, the mole fraction of propane in the liquid phase is 0.746. Its mole fraction in the vapor phase is higher, 0.907, since propane is the more volatile of the two components. The distribution coefficient K1 for propane is equal to the ratio y1/x1 = 0.907/0.746 = 1.22. As the temperature increases, the K values increase by a factor greater than two. From 70°F to 140°F, the value of the relative volatility, however, changes by only about 25%. The small effect of temperature on relative volatility is the reason for using relative volatility in shortcut distillation calculations. Relative volatility data for only two or three points in the column provide results of acceptable accuracy.

8. Figure 7 is an equilibrium diagram for the propane/n-butane system using the data from Figure 6 at 100 psia. The horizontal axis indicates the mole fraction of the more volatile component, propane, in the liquid phase. The vertical axis indicates its mole fraction in the vapor phase. The equilibrium line connects all the (x1, y1) points. Given the mole fraction in the liquid phase, the equilibrium line can be used to obtain the mole fraction in the vapor phase. Given the mole fraction in the vapor phase, the mole fraction in the liquid phase can be found.