- \n
- All of the molecules of all types should have the same size.<\/li>\n\n\n\n
- All of the molecules should have the same intermolecular interactions.<\/li>\n<\/ul>\n\n\n\n
Most solutions are not ideal, but some real solutions are nearly ideal.<\/p>\n\n\n\n
Raoult\u2019s law<\/h4>\n\n\n\n
The best-known relation for ideal solutions ispi=xipi*(T)<\/p>\n\n\n\n
(7.8)<\/p>\n\n\n\n
wherepi=partial\u2009\u2009presure\u2009\u2009of\u2009\u2009component\u2009\u2009\u2009i\u2009\u2009in\u2009the\u2009\u2009vapor\u2009\u2009phasexi=mole\u2009fraction\u2009\u2009of\u2009\u2009component\u2009\u2009i\u2009\u2009in\u2009\u2009the\u2009\u2009liquid\u2009\u2009phasepi*(T)=vapor\u2009\u2009pressure\u2009\u2009of\u2009\u2009component\u2009\u2009i\u2009\u2009at\u2009\u2009T<\/p>\n\n\n\n
Figure 7.15\u00a0shows how the vapor pressure of the two components in an ideal binary solution sum to the total pressure at 80\u00b0C. Compare\u00a0Figure 7.15\u00a0with\u00a0Figure 7.16, which displays the pressures for a nonideal solution.<\/p>\n\n\n\n
![\"A](\"https:\/\/learning.oreilly.com\/api\/v2\/epubs\/urn:orm:book:9780137327164\/files\/graphics\/07fig15.jpg\")
Figure 7.15<\/strong> Application of Raoult\u2019s law to an ideal solution of benzene and toluene (like species) to get the total pressure as a function of composition. The respective vapor pressures are shown by points A and B at 0 and 1.0 mole fraction benzene.<\/figcaption><\/figure>\n\n\n\n ![\"A](\"https:\/\/learning.oreilly.com\/api\/v2\/epubs\/urn:orm:book:9780137327164\/files\/graphics\/07fig16.jpg\")
Figure 7.16<\/strong> Plot of the partial pressures and total pressure (solid lines) exerted by a solution of carbon disulfide (CS2<\/sub>) in methylal (CH2<\/sub>(OCH3<\/sub>)2<\/sub>) as a function of composition. The dashed lines represent the pressures that would exist if the solution were ideal.<\/figcaption><\/figure>\n\n\n\n Raoult\u2019s law is used primarily for a component whose mole fraction spans the full range from 0 to 1 for solutions of components quite similar in chemical nature, such as straight-chain hydrocarbons.<\/p>\n\n\n\n
Henry\u2019s law<\/h4>\n\n\n\n
Henry\u2019s law is used primarily for a component whose mole fraction approaches zero, such as a dilute gas dissolved in a liquid:pi=Hixi<\/p>\n\n\n\n
(7.9)<\/p>\n\n\n\n
where p<\/em>i<\/sub><\/em> is the partial pressure in the gas phase of the dilute component at equilibrium at some temperature, and H<\/em>i<\/sub><\/em> is the Henry\u2019s law constant<\/em>. Note that in the limit where xi<\/sub><\/em> \u2192 0, pi<\/sub><\/em> \u2192 0. Values of H<\/em>i<\/sub><\/em> can be found in several handbooks and on the Internet.<\/p>\n\n\n\n
Henry\u2019s law is quite simple to apply when you want to calculate the partial pressure of a gas that is in equilibrium with the gas dissolved the liquid phase. Take, for example, CO2<\/sub> dissolved in water at 40\u00b0C for which the value of H<\/em> is 69,600 atm\/mol fraction. (The large value of H<\/em> shows that CO2<\/sub>(g) is only sparing soluble in water.) If x<\/em>CO2<\/sub> = 4.2 \u00d7 10\u22126<\/sup>, the partial pressure of the CO2<\/sub> in the gas phase is<\/p>\n\n\n\n
p<\/em>CO2<\/sub> = 69,600(4.2 \u00d7 10\u22126<\/sup>) 0.29 atm<\/p>\n\n\n\n
That is, the gas phase contains almost 30% CO2<\/sub> for 1 atm pressure, but the liquid phase would contain only 0.00042% CO2<\/sub>.<\/p>\n\n\n\n
7.5.2 Vapor-Liquid Equilibria Phase Diagrams<\/h4>\n\n\n\n
The phase diagrams discussed in\u00a0Section 7.2\u00a0for a pure component can be extended to cover binary mixtures. Experimental data usually are presented as pressure as a function of composition at a constant temperature, or temperature as a function of composition at a constant pressure. For a pure component, vapor-liquid equilibrium occurs with only one degree of freedom:<\/p>\n\n\n\n
F<\/em> = 2 \u2212 P<\/em> + C<\/em> = 2 \u2212 2 + 1 = 1<\/p>\n\n\n\n
At 1 atm pressure, vapor-liquid equilibrium will occur at only one temperature: the normal boiling point. However, if you have a binary solution, you have two degrees of freedom:<\/p>\n\n\n\n
F<\/em> = 2 \u2212 2 + 2 = 2<\/p>\n\n\n\n
For a system at a fixed pressure, both the phase compositions and the temperature can be varied over a finite range.\u00a0Figures 7.17\u00a0and\u00a07.18\u00a0show the vapor-liquid envelope for a binary mixture of benzene and toluene, which is essentially ideal.<\/p>\n\n\n\n
![\"A](\"https:\/\/learning.oreilly.com\/api\/v2\/epubs\/urn:orm:book:9780137327164\/files\/graphics\/07fig17.jpg\")
Figure 7.17<\/strong> Phase diagram for a mixture of benzene and toluene at 80\u00b0C. At 0 mol fraction of benzene (point A), the pressure is the vapor pressure of toluene at 80\u00b0C. At a mole fraction of benzene of 1 (point B), the pressure is the vapor pressure of benzene at 80\u00b0C. The tie line shows the liquid and vapor compositions that are in equilibrium at a pressure of 0.62 atm (and 80\u00b0C).<\/figcaption><\/figure>\n\n\n\n ![\"A](\"https:\/\/learning.oreilly.com\/api\/v2\/epubs\/urn:orm:book:9780137327164\/files\/graphics\/07fig18.jpg\")
Figure 7.18<\/strong> Phase diagram for a mixture of benzene and toluene at 0.50 atm. At 0 mole fraction of benzene (point A), the temperature is that when the vapor pressure of toluene is 0.50 atm. At a mole fraction of benzene of 1 (point B), the temperature is that when the vapor pressure of benzene is 0.50 atm. The tie line shows the liquid and vapor compositions that are in equilibrium at a temperature of 80\u00b0C (and 0.50 atm).<\/figcaption><\/figure>\n\n\n\n You can interpret the information on the phase diagrams as follows: Suppose you start in\u00a0Figure 7.17\u00a0at a 50-50 mixture of benzene-toluene at 80\u00b0C and 0.30 atm in the vapor phase. Then you increase the pressure on the system until you reach the dew point at about 0.47 atm, at which point the vapor starts to condense. At 0.62 atm the mole fraction in the vapor phase will be about 0.75, and the mole fraction in the liquid phase will be about 0.38 as indicated by the tie line. As you increase the pressure from 0.70 atm, all of the vapor will have condensed to liquid. What will the composition of the liquid be? 0.50 benzene, of course! Can you carry out an analogous conversion of vapor to liquid on\u00a0Figure 7.18, the temperature-composition diagram?<\/p>\n\n\n\n
Phase diagrams for nonideal solutions abound.\u00a0Figure 7.19\u00a0shows the temperature-composition diagram for isopropanol in water at 1 atm. Note the minimum boiling point at a mole fraction of isopropanol of about 0.68, a point called an\u00a0azeotrope<\/strong>\u00a0(a point at which on a\u00a0y<\/em>i<\/sub>-versus-x<\/em>i<\/sub>\u00a0plot the function of (y<\/em>i<\/sub>\/x<\/em>i<\/sub>) crosses the function\u00a0yi<\/sub><\/em>\u00a0=\u00a0xi<\/sub><\/em>, a straight line). An azeotrope makes separation by simple distillation difficult because it creates a pinch point because when the dew point and the bubble point coincide so that separation between the more volatile and the less volatile components does not occur.<\/p>\n\n\n\n
![\"A](\"https:\/\/learning.oreilly.com\/api\/v2\/epubs\/urn:orm:book:9780137327164\/files\/graphics\/07fig19.jpg\")
Figure 7.19<\/strong> Phase diagram for a nonideal mixture of isopropanol and water at 1 atm<\/figcaption><\/figure>\n\n\n\n 7.5.3 K<\/em>-value (Vapor-Liquid Equilibrium Ratio)<\/h4>\n\n\n\n
For nonideal as well as ideal mixtures that comprise two (or more) phases, it proves to be convenient to express the ratio of the mole fraction in one phase to the mole fraction of the same component in another phase in terms of a\u00a0distribution coefficient<\/strong>\u00a0or\u00a0equilibrium ratio<\/strong>\u00a0K<\/strong>,<\/em>\u00a0usually called a\u00a0K<\/strong><\/em>-value<\/strong>. For example:Vapor-liquid\u00a0ratio of\u00a0component\u2009i=yixi=Ki<\/p>\n\n\n\n
(7.10)<\/p>\n\n\n\n
and so on. If the ideal gas law p<\/em>i<\/sub><\/em> = y<\/em>i<\/sub><\/em> p<\/em>total<\/sub> applies to the gas phase and the ideal Raoult\u2019s law pi=xipi*(T) applies to the liquid phase, then for an ideal<\/em> systemKi=yixi=pi*(T)ptotal<\/p>\n\n\n\n
(7.10a)<\/p>\n\n\n\n
Equation (7.10a) gives reasonable estimates of\u00a0Ki<\/sub><\/em>\u00a0values at low pressures for components well below their critical temperatures but yields values too large for components above their critical temperatures, at high pressures, and\/or for polar compounds. For nonideal mixtures, Equation (7.10) can be employed if\u00a0Ki<\/sub><\/em>\u00a0is made a function of temperature, pressure, and composition so that relations for\u00a0Ki<\/sub><\/em>\u00a0can be fit by equations to experimental data and used directly, or in the form of charts, for design calculations, as explained in some of the references at the end of this chapter.\u00a0Figure 7.20\u00a0shows how\u00a0K<\/em>\u00a0varies for the nonideal mixture of acetone and water at 1 atm.\u00a0K<\/em>\u00a0can be greater or less than 1 but never negative.<\/p>\n\n\n\n
![\"A](\"https:\/\/learning.oreilly.com\/api\/v2\/epubs\/urn:orm:book:9780137327164\/files\/graphics\/07fig20.jpg\")
Figure 7.20<\/strong> Change of K<\/em> of water with composition at p<\/em> = 1 atm<\/figcaption><\/figure>\n\n\n\n For ideal solutions, you can calculate values of K<\/em> using Equation (7.10a<\/a>). For nonideal solutions you can get approximate K<\/em>-values from<\/p>\n\n\n\n
- \n
- Empirical equations such as4<\/a><\/sup>4<\/a><\/sup>S. I. Sandler, in Foundations of Computer Aided Design<\/em>, Vol. 2, edited by R. H. S. Mah and W. D. Seider, American Institute of Chemical Engineers, New York (1981), p. 83.If\u2009Tc,i\/T\u2009>1.2:\u2009\u2009Ki=(pc,i)exp[7.224\u22127.534\/Tr.i\u22122.598\u2009ln\u2009Tr,i]ptotal<\/li>\n\n\n\n
- Databases refer to the supplementary references (at the end of the chapter).<\/li>\n\n\n\n
- Charts such as\u00a0Figure 7.21
![\"A](\"https:\/\/learning.oreilly.com\/api\/v2\/epubs\/urn:orm:book:9780137327164\/files\/graphics\/07fig21.jpg\")
Figure 7.21<\/strong>\u00a0K<\/em>-values for isobutane as a function of temperature and pressure. From\u00a0Natural Gasoline Association of America Technical Manual<\/em>, 4th ed. (1941) (based on data provided by George Granger Brown).<\/li>\n\n\n\n- Thermodynamic relations refer to the references (at the end of the chapter).<\/li>\n<\/ol>\n\n\n\n
7.5.4 Bubble Point and Dew Point Calculations<\/h4>\n\n\n\n
Here are some typical problems you should be able to solve that involve the use of the equilibrium coefficient K<\/em>i<\/sub><\/em> and material balances:<\/p>\n\n\n\n
1. Calculate the bubble point temperature of a liquid mixture given the total pressure and liquid composition.<\/strong><\/p>\n\n\n\n
To calculate the\u00a0bubble point temperature<\/strong>\u00a0(given the total pressure and liquid composition), you can write Equation (7.10) as\u00a0yi<\/sub><\/em>\u00a0=\u00a0Ki<\/sub>xi<\/sub><\/em>. Also, you know that \u03a3yi<\/sub><\/em>\u00a0= 1 in the vapor phase. Thus, for a binary,1=K1x1+K2x2<\/p>\n\n\n\n
(7.11)<\/p>\n\n\n\n
in which the\u00a0Ki<\/sub><\/em>\u00a0are functions of solely the temperature. Because each of the\u00a0K<\/em>i<\/sub>\u00a0increases with temperature, Equation (7.11) has only one positive root. A trial-and-error procedure is required to determine the bubble point temperature, which can be performed on the computer or by hand. If you choose to solve for the bubble point by hand, you have to assume varying temperatures so that you can look up or calculate\u00a0Ki<\/sub><\/em>, and then calculate each term in Equation (7.11). After the sum (K<\/em>1<\/sub>x<\/em>1<\/sub>\u00a0+\u00a0K<\/em>2<\/sub>x<\/em>2<\/sub>) brackets 1, you can interpolate to get a\u00a0T<\/em>\u00a0that satisfies Equation (7.11).<\/p>\n\n\n\n
For an ideal solution, Equation (7.11) becomesptotal=p1*x1+p2*x2<\/p>\n\n\n\n
(7.12)<\/p>\n\n\n\n
and you might use Antoine\u2019s equation for p<\/em>*i<\/sub><\/em>. Once the bubble point temperature is determined, the vapor composition can be calculated fromyi=pi*xiptotal<\/p>\n\n\n\n
A degree-of-freedom analysis for the bubble point temperature for a binary mixture shows that the degrees of freedom are zero:<\/p>\n\n\n\n
Total variables=2\u00d72+2=6variables:x1,\u2009\u2009x2;\u2009y1,\u2009\u2009y2;\u2009ptotal;\u2009\u2009T<\/p>\n\n\n\n
Prespecified values of = 2 + 1 = 3 variables: x<\/em>1<\/sub>, x<\/em>2<\/sub>; p<\/em>Total<\/sub>Independent equations=2+1=3\u2009\u2009equations: \u2009y1=K1x1,\u2009\u2009\u2009y2=K2x2;\u2009\u2009y1+y2=1<\/p>\n\n\n\n
Therefore, there are three unknowns and three equations with which to determine their values.<\/p>\n\n\n\n
2. Calculate the dew point temperature of a vapor mixture given the total pressure and vapor composition.<\/strong><\/p>\n\n\n\n
To calculate the\u00a0dew point temperature<\/strong>\u00a0(given the total pressure and vapor composition), you can write Equation (7.10) as\u00a0xi<\/sub><\/em>\u00a0=\u00a0yi<\/sub><\/em>\/Ki<\/sub><\/em>, and you know \u03a3xi<\/sub><\/em>\u00a0= 1 in the liquid phase. Consequently, you want to solve the equation1=y1K1+y2K2<\/p>\n\n\n\n
(7.13)<\/p>\n\n\n\n
in which the K<\/em>s are functions of temperature as explained for the bubble point temperature calculation. For an ideal solution,1=ptotal[y1p1*+y2p2*]<\/p>\n\n\n\n
(7.13a)<\/p>\n\n\n\n
The degree-of-freedom analysis is similar to that for the bubble point temperature calculation.<\/p>\n\n\n\n
In selecting a particular form of the equation to be used for your equilibrium calculations, you must select a method of solving the equation that has desirable convergence characteristics. Convergency to the solution should<\/p>\n\n\n\n
- \n
- Lead to the desired root if the equation has multiple roots<\/li>\n\n\n\n
- Be stable, that is, approach the desired root asymptotically rather than by oscillating<\/li>\n\n\n\n
- Be rapid, and not become slower as the solution is approached<\/li>\n<\/ul>\n\n\n\n
You can use MATLAB\u2019s\u00a0fzero<\/strong>\u00a0function or Python\u2019s\u00a0scipy.optimize.newton<\/strong>, which were introduced in\u00a0Chapter 6,\u00a0Section 6.2.2, to solve bubble point and dew point equations because in both cases the equation to be solved is a single nonlinear equation in which the temperature is unknown. These equations can be more efficiently and reliably solved by first transforming them into a more linear form.5<\/sup><\/p>\n\n\n\n
5<\/a><\/sup> J. B. Riggs, Computational Methods for Chemical Engineers<\/em> (Austin, TX: Ferret Publishing, 2020), 95\u201397.<\/p>\n\n\n\n
Table 7.1\u00a0summarizes the usual phase equilibrium calculations.<\/p>\n\n\n\n
Table 7.1 Summary of the Information Associated with Typical Phase Equilibrium Calculations<\/strong><\/p>\n\n\n\n
Type<\/th> Known*<\/a><\/sup> Information<\/th> Variables to Be Calculated<\/th> Equation(s) to Use<\/th> Convergence Characteristics<\/th><\/tr><\/thead> Bubble point temperature<\/td> p<\/em>total<\/sub>, x<\/em>i<\/em><\/sub><\/td> T<\/em>, y<\/em>i<\/em><\/sub><\/td> 7.10<\/td> Good<\/td><\/tr> Dew point temperature<\/td> p<\/em>total<\/sub>, y<\/em>i<\/em><\/sub><\/td> T<\/em>, x<\/em>i<\/em><\/sub><\/td> 7.12<\/td> Good<\/td><\/tr> Bubble point pressure<\/td> T<\/em>, x<\/em>i<\/em><\/sub><\/td> p<\/em>total<\/sub>, y<\/em>i<\/em><\/sub><\/td> 7.10<\/td> Fair<\/td><\/tr> Dew point pressure<\/td> T<\/em>, y<\/em>i<\/em><\/sub><\/td> p<\/em>total<\/sub>, x<\/em>i<\/em><\/sub><\/td> 7.12<\/td> Fair<\/td><\/tr><\/tbody><\/table><\/figure>\n\n\n\n
- Empirical equations such as4<\/a><\/sup>4<\/a><\/sup>S. I. Sandler, in Foundations of Computer Aided Design<\/em>, Vol. 2, edited by R. H. S. Mah and W. D. Seider, American Institute of Chemical Engineers, New York (1981), p. 83.If\u2009Tc,i\/T\u2009>1.2:\u2009\u2009Ki=(pc,i)exp[7.224\u22127.534\/Tr.i\u22122.598\u2009ln\u2009Tr,i]ptotal<\/li>\n\n\n\n