Thermodynamics

Chapter 13 | 701gory of ideal solutions. Another major category of ideal solutions is the diluteliquid solutions, such as the saline water. It can be shown that the enthalpy ofmixing and the volume change due to mixing are zero for ideal solutions (seeWark, 1995). That is,¢V mixing,ideal aiN i 1v i v i2 0and¢H mixing,ideal aiN i 1h i h i 2 0(13–46)Then it follows that v i v– i and h i h– i . That is, the partial molar volumeand the partial molar enthalpy of a component in a solution equal the specificvolume and enthalpy of that component when it existed alone as a pure substanceat the mixture temperature and pressure. Therefore, the specific volumeand enthalpy of individual components do not change during mixing if theyform an ideal solution. Then the specific volume and enthalpy of an idealsolution can be expressed as (Fig. 13–20)V mixing,ideal 0v i,mixture v i,purev mixture y i v i,pureiH mixing,ideal 0 h i,mixing h i,pureh mixture y i h i,pureiv mixing,ideal aiy i v i aiy i v i,pureandh mixture,ideal aiy i h i aiy i h i,pure(13–47)Note that this is not the case for entropy and the properties that involveentropy such as the Gibbs function, even for ideal solutions. To obtain a relationfor the entropy of a mixture, we differentiate Eq. 13–45 with respect totemperature at constant pressure and mole fraction,a 0 m i,mixing 1T, P i 20Tb a 0 m i,pure 1T, P2b RP,y 0Tu ln y iP,y(13–48)We note from Eq. 13–38 that the two partial derivatives above are simply thenegative of the partial molar entropies. Substituting,si,mixture,ideal 1T, P i 2 s i,pure 1T, P2 R u ln y 1 1ideal solution2(13–49)Note that ln y i is a negative quantity since y i 1, and thus R u ln y i is alwayspositive. Therefore, the entropy of a component in a mixture is always greaterthan the entropy of that component when it exists alone at the mixture temperatureand pressure. Then the entropy of mixing of an ideal solution is determinedby substituting Eq. 13–49 into Eq. 13–34 to beFIGURE 13–20The specific volume and enthalpy ofindividual components do not changeduring mixing if they form an idealsolution (this is not the case forentropy).¢S mixing,ideal aiN i 1 s i s i2 R u aiN i ln y i 1ideal solution2(13–50a)or, dividing by the total number of moles of the mixture N m ,¢ s mixing,ideal aiy i 1 s i s i2 R u aiy i ln y i 1per unit mole of mixture2(13–50b)Minimum Work of Separation of MixturesThe entropy balance for a steady-flow system simplifies to S in S out S gen 0. Noting that entropy can be transferred by heat and mass only, the