Introduction to Vapor/Liquid Equilibrium, part 2. Raoult s Law:

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1 CE304, Sprng 2004 Lecture 4 Introducton to Vapor/Lqud Equlbrum, part 2 Raoult s Law: The smplest model that allows us do VLE calculatons s obtaned when we assume that the vapor phase s an deal gas, and the lqud phase s an deal soluton. We have not talked about deal solutons yet, but we wll do so n the next few lectures. Bascally, an deal soluton s a mxture of lquds n whch the nteractons between molecules of dfferent speces are the same as the nteractons between molecules of the same speces. As a result, the energy and enthalpy of the mxture are just the sum of the mole fractons tmes the energy and enthalpy of the components, just as they are for a mxture of deal gases. Ths s only a good model for mxtures of thngs that are very chemcally smlar, lke dfferent somers of the same compound. However, t allows us set up vapor lqud equlbrum calculatons wth a smple, easy-tounderstand model that we can then extend to use more realstc models of the lqud phase behavor. Mathematcally, Raoult s law s expressed as x for all speces (, 2,..., ) where x s the lqud phase mole fracton, y s the vapor phase mole fracton, s the vapor pressure of pure component, and s the total pressure. In words, Raoult s law says that the partal pressure of each speces n the vapor phase s equal to ts mole fracton n the lqud phase tmes ts pure-component vapor pressure. Bubble ont and Dew ont Calculatons usng Raoult s Law: The most straghtforward, and perhaps the most commonly encountered, types of VLE calculatons are bubble pont and dew pont calculatons. There are 4 types of these, dependng on whch condtons are known. These are Bubble ont ressure calculaton (BUBL ): compute {y } and gven {x } and T. Dew ont ressure calculaton (DEW ): compute {x } and gven {y } and T. Bubble ont Temperature calculaton (BUBL T): compute {y } and T gven {x } and. Dew ont Temperature calculaton (DEW T): compute {x } and T gven {y } and. The phase rule tells us that we must fx F 2 - π + ndependent ntensve varables to specfy the state of the system. Specfyng the composton (mole fractons) n one of the phases sets ndependent ntensve varables (the th mole fracton doesn t count because t depends on the others snce the mole fractons have to sum to ). Thus, to specfy total ntensve varables, we can specfy the composton of ether the lqud or the vapor, plus ether the temperature or the pressure, leadng to the four combnatons lsted above. We could take a huge number of other combnatons (lke specfyng both T and and 2 mole fractons), but the four combnatons lsted are the most common specfcatons that are encountered n practce. Bubble ont ressure calculatons: p. of 6

2 CE304, Sprng 2004 Lecture 4 In a bubble pont pressure calculaton, we calculate the composton of the frst (nfntesmally small) bubble that would form as we decrease the pressure of a lqud mxture of specfed composton at constant temperature. Snce the amount vaporzed at that pont s very small, the lqud composton s known, as well as the temperature, and the unknowns are the pressure where the frst bt of vapor forms and ts composton. Ths calculaton s straghtforward and explct. Snce the uraton pressure s only a functon of temperature, we know all of the values of as well as all the x. We frst sum Raoult s law over all the speces to get x y x x where we have used the fact that the gas phase mole fractons have to sum to. Knowng all of the x and values, we can now compute each of the mole fractons drectly from Raoult s law wrtten separately for each speces: x y Dew ont ressure Calculatons: Here we compute the composton of the frst tny droplet of lqud that would form when we compress a gas mxture of specfed composton at fxed temperature. Snce the amount condensed at that pont s very small, the vapor composton s known, as well as the temperature, and the unknowns are the pressure where the frst bt of lqud forms and ts composton. Ths calculaton s also straghtforward and explct. Snce the uraton pressure s only a functon of temperature, we agan know all of the values of. Ths tme we know all of the y rather than all the x values. Ths tme, we wll dvde Raoult s law for each speces by the speces uraton pressure and then sum the results over all the speces to get the total pressure: for all speces (,2,..., x ) x y y Once we know, we can go back and compute the lqud phase mole fractons of each speces from Roult s law as x p. 2 of 6

3 CE304, Sprng 2004 Lecture 4 Bubble ont Temperature and Dew ont Temperature calculatons: In a bubble pont temperature calculaton, we compute the temperature at whch the frst tny bt of vapor forms when a lqud mxture of specfed composton s heated at constant pressure, as well as the composton of that frst bt of vapor. Snce the vapor pressures of the components depend on temperature n some way that we have not yet specfed, we can t necessarly solve explctly for the temperature. One approach to ths s to start from an ntal guess for the temperature and then do a bubble pont pressure calculaton, compare the computed total pressure to the specfed total pressure, and then change the temperature (terate) untl the computed pressure matches the specfed pressure. Once we know the temperature and pressure, we can compute the vapor mole fractons from x y Smlarly, n a dew pont temperature calculaton, we compute the temperature at whch the frst tny bt of lqud forms when a vapor mxture of specfed composton s cooled at constant pressure, as well as the composton of the lqud. Agan, we have to do ths teratvely, because the uraton pressures depend on the temperature. We guess a temperature, do a dew pont pressure calculaton, compare the pressure to the specfed pressure, and terate untl they match. Once we know the temperature and pressure, we can compute the lqud phase mole fractons from x Example0. n SVA. Henry s Law: To apply Raoult s law, or extensons of t based on the same dea, we must have the uraton vapor pressure of each speces. Thus, t can t be appled to speces above ther crtcal temperature (where there s no uraton pressure). Thus, for example f we have a contaner contanng water and ntrogen at room temperature, Raoult s law can apply to the water (whch s below ts crtcal temperature) but not to the ntrogen (whch has a crtcal temperature of 26.2 K). To compute the mole fracton of water vapor n the vapor phase, we could assume that the lqud phase mole fracton of water s almost and wrte xho 2 HO 2 HO 2 y HO 2 However, we mght also want to know how much ntrogen can dssolve n the water. We can t wrte Raoult s law for t, because ts vapor pressure s not defned above ts crtcal temperature. Henry s law s devsed for just such a stuaton. It smply says that for a speces present as a very dlute solute n a lqud phase, the mole fracton n the lqud phase s drectly proportonal to ts partal pressure n the vapor phase (just as t s n Raoult s law, but wth a dfferent proportonalty constant). That s xh p. 3 of 6

4 CE304, Sprng 2004 Lecture 4 where H s the Henry s law constant for speces n that partcular soluton. Henry s law constants must generally be determned expermentally. Some are gven n Table 0. on page 348 of SVA. Example 0.2 n SVA. Modfed Raoult s Law formulatons of VLE: There are a wde range of stuatons where the pressure s low enough that the vapor phase s nearly deal (the assumpton of an deal gas mxture n the vapor phase s good), but the lqud phase s not an deal soluton. Thus, much more realstc VLE calculatons can often be done usng a modfed verson of Raoult s Law that can be stated as xγ for all speces (, 2,..., ) where γ s called the actvty coeffcent of speces n the soluton, and generally depends on both temperature and the soluton composton. The actvty coeffcents must be determned from experment, usually va an actvty coeffcent model ftted to expermental data. Ths wll be dscussed n great detal n upcomng lectures on soluton thermodynamcs. For the moment, we wll assume that we know the actvty coeffcents. Then, bubble pont pressure and dew pont pressure calculatons can be done just as we dd wth Raoult s law, summng over all the speces to get xγ y xγ xγ for the bubble pont pressure calculaton and x for all speces, 2,..., γ x γ y γ y γ for dew pont pressure calculatons. ( ) As was the case for Raoult s law wthout the actvty coeffcents, for bubble pont temperature calculatons and dew pont temperature calculatons we wll usually want to use an teratve p. 4 of 6

5 CE304, Sprng 2004 Lecture 4 soluton strategy n whch we perform a seres of bubble pont pressure or dew pont pressure calculatons untl our computed pressure matches the specfed pressure. Example 0.3 n SVA K-values as a descrpton of VLE: The K-value of a substance n a vapor/lqud system s defned as y K x Ths number provdes a convenent relatve measure of the lghtness of a component. Thngs wth K-values greater than one favor the vapor phase, whle those wth K-values greater than favor the lqud phase. For a system that obeys Raoult s law, we have y K x and for a system that obeys the modfed form of Raoult s law dscussed above, we have y γ K x Charts of K-values for mxtures of lght hydrocarbons (where use of these s most common) are gven n SVA on pages 356 and 357. To use these, you use a straght-edge to connect the pressure and temperature of nterest and then read off the K-values where the straght-edge crosses the curve for the speces of nterest. Generally only one or the other of T and s known, so ths requres teratve graph readng/straght-edge use. Flash Calculatons: Another mportant type of calculaton s the flash calculaton, n whch we specfy the temperature and pressure and total amounts of each speces and want to compute the composton and total amounts of each phase. We know from Duhem s theorem that specfyng 2 ntensve varables plus the total amount of each speces n the system determnes the state of the system. We wll call the set of (known) overall mole fractons {z }, and call the lqud phase fracton L and the vapor phase fracton V. Then we have the followng equatons: L + V z x L + y V (for,2,,) as well as Raoult s law (or modfed Raoult s law) for each speces and the requrement that the mole fractons n each phase sum to. If we substtute x y /K nto the above and then solve for y, we get y L+ VK z L+ yv y K K y L + VK Then substtutng L V p. 5 of 6

6 CE304, Sprng 2004 Lecture 4 y V + VK + V ( K ) Summng ths over all the speces gves y + V ( K ) Ths s a sngle equaton n whch the only unknown s V. After solvng t for V, we could use the precedng equatons to fnd L and all of the mole fractons. Example 0.5 n SVA p. 6 of 6