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CE304, Spring 2004

Lecture 14

Introduction to Vapor/Liquid Equilibrium, part 2 Raoult's Law: The simplest model that allows us do VLE calculations is obtained when we assume that the vapor phase is an ideal gas, and the liquid phase is an ideal solution. We have not talked about ideal solutions yet, but we will do so in the next few lectures. Basically, an ideal solution is a mixture of liquids in which the interactions between molecules of different species are the same as the interactions between molecules of the same species. As a result, the energy and enthalpy of the mixture are just the sum of the mole fractions times the energy and enthalpy of the components, just as they are for a mixture of ideal gases. This is only a good model for mixtures of things that are very chemically similar, like different isomers of the same compound. However, it allows us set up vapor liquid equilibrium calculations with a simple, easy-tounderstand model that we can then extend to use more realistic models of the liquid phase behavior. Mathematically, Raoult's law is expressed as yi P = xi Pi sat for all species ( i = 1, 2,..., N ) where xi is the liquid phase mole fraction, yi is the vapor phase mole fraction, Pisat is the vapor pressure of pure component i, and P is the total pressure. In words, Raoult's law says that the partial pressure of each species in the vapor phase is equal to its mole fraction in the liquid phase times its pure-component vapor pressure.

Bubble Point and Dew Point Calculations using Raoult's Law: The most straightforward, and perhaps the most commonly encountered, types of VLE calculations are bubble point and dew point calculations. There are 4 types of these, depending on which conditions are known. These are Bubble Point Pressure calculation (BUBL P): compute {yi} and P given {xi} and T. Dew Point Pressure calculation (DEW P): compute {xi} and P given {yi} and T. Bubble Point Temperature calculation (BUBL T): compute {yi} and T given {xi} and P. Dew Point Temperature calculation (DEW T): compute {xi} and T given {yi} and P. The phase rule tells us that we must fix F=2-+N=N independent intensive variables to specify the state of the system. Specifying the composition (mole fractions) in one of the phases sets N 1 independent intensive variables (the Nth mole fraction doesn't count because it depends on the others since the mole fractions have to sum to 1). Thus, to specify N total intensive variables, we can specify the composition of either the liquid or the vapor, plus either the temperature or the pressure, leading to the four combinations listed above. We could take a huge number of other combinations (like specifying both T and P and N 2 mole fractions), but the four combinations listed are the most common specifications that are encountered in practice. Bubble Point Pressure calculations:

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Lecture 14

In a bubble point pressure calculation, we calculate the composition of the first (infinitesimally small) bubble that would form as we decrease the pressure of a liquid mixture of specified composition at constant temperature. Since the amount vaporized at that point is very small, the liquid composition is known, as well as the temperature, and the unknowns are the pressure where the first bit of vapor forms and its composition. This calculation is straightforward and explicit. Since the saturation pressure is only a function of temperature, we know all of the values of Pisat as well as all the xi. We first sum Raoult's law over all the species to get

y P =xP

i i i i =1 i =1 N N i =1 i =1 N

N

N

sat

P yi = xi Pi sat P = xi Pi sat

i =1

where we have used the fact that the gas phase mole fractions have to sum to 1. Knowing all of the xi and Pisat values, we can now compute each of the mole fractions directly from Raoult's law written separately for each species: x P sat yi = i i P Dew Point Pressure Calculations: Here we compute the composition of the first tiny droplet of liquid that would form when we compress a gas mixture of specified composition at fixed temperature. Since the amount condensed at that point is very small, the vapor composition is known, as well as the temperature, and the unknowns are the pressure where the first bit of liquid forms and its composition. This calculation is also straightforward and explicit. Since the saturation pressure is only a function of temperature, we again know all of the values of Pisat. This time we know all of the yi rather than all the xi values. This time, we will divide Raoult's law for each species by the species saturation pressure and then sum the results over all the species to get the total pressure: yi P = xi for all species ( i = 1, 2,..., N ) Pi sat

yi P N P sat = xi = 1 i =1 i i =1 P

i =1 N N

yi =1 Pi sat 1

N

P=

P

i =1 i

yi

sat

Once we know P, we can go back and compute the liquid phase mole fractions of each species from Roult's law as yP xi = isat Pi

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Bubble Point Temperature and Dew Point Temperature calculations: In a bubble point temperature calculation, we compute the temperature at which the first tiny bit of vapor forms when a liquid mixture of specified composition is heated at constant pressure, as well as the composition of that first bit of vapor. Since the vapor pressures of the components depend on temperature in some way that we have not yet specified, we can't necessarily solve explicitly for the temperature. One approach to this is to start from an initial guess for the temperature and then do a bubble point pressure calculation, compare the computed total pressure to the specified total pressure, and then change the temperature (iterate) until the computed pressure matches the specified pressure. Once we know the temperature and pressure, we can compute the vapor mole fractions from x P sat yi = i i P Similarly, in a dew point temperature calculation, we compute the temperature at which the first tiny bit of liquid forms when a vapor mixture of specified composition is cooled at constant pressure, as well as the composition of the liquid. Again, we have to do this iteratively, because the saturation pressures depend on the temperature. We guess a temperature, do a dew point pressure calculation, compare the pressure to the specified pressure, and iterate until they match. Once we know the temperature and pressure, we can compute the liquid phase mole fractions from yP xi = isat Pi Example10.1 in SVA. Henry's Law: To apply Raoult's law, or extensions of it based on the same idea, we must have the saturation vapor pressure of each species. Thus, it can't be applied to species above their critical temperature (where there is no saturation pressure). Thus, for example if we have a container containing water and nitrogen at room temperature, Raoult's law can apply to the water (which is below its critical temperature) but not to the nitrogen (which has a critical temperature of 126.2 K). To compute the mole fraction of water vapor in the vapor phase, we could assume that the liquid phase mole fraction of water is almost 1 and write sat sat xH O PH O PH O y H 2O = 2 2 2 P P However, we might also want to know how much nitrogen can dissolve in the water. We can't write Raoult's law for it, because its vapor pressure is not defined above its critical temperature. Henry's law is devised for just such a situation. It simply says that for a species present as a very dilute solute in a liquid phase, the mole fraction in the liquid phase is directly proportional to its partial pressure in the vapor phase (just as it is in Raoult's law, but with a different proportionality constant). That is yi P = xi H i

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Lecture 14

where Hi is the Henry's law constant for species i in that particular solution. Henry's law constants must generally be determined experimentally. Some are given in Table 10.1 on page 348 of SVA. Example 10.2 in SVA. Modified Raoult's Law formulations of VLE: There are a wide range of situations where the pressure is low enough that the vapor phase is nearly ideal (the assumption of an ideal gas mixture in the vapor phase is good), but the liquid phase is not an ideal solution. Thus, much more realistic VLE calculations can often be done using a modified version of Raoult's Law that can be stated as yi P = xi i Pi sat for all species ( i = 1, 2,..., N ) where i is called the activity coefficient of species i in the solution, and generally depends on both temperature and the solution composition. The activity coefficients must be determined from experiment, usually via an activity coefficient model fitted to experimental data. This will be discussed in great detail in upcoming lectures on solution thermodynamics. For the moment, we will assume that we know the activity coefficients. Then, bubble point pressure and dew point pressure calculations can be done just as we did with Raoult's law, summing over all the species to get

y P = x P

i i i i i =1 i =1 N N i =1 i =1 N

N

N

sat

P yi = xi i Pi sat P = xi i Pi sat

i =1

for the bubble point pressure calculation and yi P = xi for all species ( i = 1, 2,..., N ) i Pi sat

N yi P = xi = 1 P sat i=1 i =1 i i N

P P=

yi =1 sat i =1 i P i

N

1

P

i =1 i i

N

yi

sat

for dew point pressure calculations. As was the case for Raoult's law without the activity coefficients, for bubble point temperature calculations and dew point temperature calculations we will usually want to use an iterative

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Lecture 14

solution strategy in which we perform a series of bubble point pressure or dew point pressure calculations until our computed pressure matches the specified pressure. Example 10.3 in SVA K-values as a description of VLE: The K-value of a substance in a vapor/liquid system is defined as y Ki i xi This number provides a convenient relative measure of the `lightness' of a component. Things with K-values greater than one favor the vapor phase, while those with K-values greater than 1 favor the liquid phase. For a system that obeys Raoult's law, we have y P sat Ki i = i xi P and for a system that obeys the modified form of Raoult's law discussed above, we have yi i Pi sat Ki = xi P Charts of K-values for mixtures of light hydrocarbons (where use of these is most common) are given in SVA on pages 356 and 357. To use these, you use a straight-edge to connect the pressure and temperature of interest and then read off the K-values where the straight-edge crosses the curve for the species of interest. Generally only one or the other of T and P is known, so this requires iterative graph reading/straight-edge use. Flash Calculations: Another important type of calculation is the flash calculation, in which we specify the temperature and pressure and total amounts of each species and want to compute the composition and total amounts of each phase. We know from Duhem's theorem that specifying 2 intensive variables plus the total amount of each species in the system determines the state of the system. We will call the set of (known) overall mole fractions {zi}, and call the liquid phase fraction L and the vapor phase fraction V. Then we have the following equations: L+V=1 zi = xiL + yiV (for i = 1,2,...,N) as well as Raoult's law (or modified Raoult's law) for each species and the requirement that the mole fractions in each phase sum to 1. If we substitute xi = yi/Ki into the above and then solve for yi, we get L + VK i y zi = i L + yiV = yi Ki Ki

zi K i L + VK i Then substituting L = 1 V yi =

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zi K i zi K i = 1 - V + VK i 1 + V ( K i - 1) Summing this over all the species gives N N zK yi = 1 = 1 + V (i Ki - 1) i =1 i =1 i This is a single equation in which the only unknown is V. After solving it for V, we could use the preceding equations to find L and all of the mole fractions. yi = Example 10.5 in SVA

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