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Chemical Engineering Thermodynamics II Content Chemical Engineering Thermodynamics II Lec 2: Phase Equilibria: Thermodynamics of Mixtures Dr.-Eng. Zayed Al-Hamamre 1 Chemical Engineering Department | University of Jordan | Amman 11942, Jordan Tel. +962 6 535 5000 | 22888 Content Learning Objectives Introduction Partial Molar Properties Determination of Partial Molar Properties Relations Among Partial Molar Quantities 2 Chemical Engineering Department | University of Jordan | Amman 11942, Jordan Tel. +962 6 535 5000 | 22888 Learning Objectives Apply thermodynamics to mixtures. Write the differential for any extensive property, dK, in terms of m + 2 independent variables, where m is the number of species in the mixture. Define and find values for pure species properties, total solution properties, partial molar properties, and property changes of mixing. Define a partial molar property and describe its role in determining the properties of mixtures. Calculate the value of a partial molar property for a species in a mixture from analytical and graphical methods. Apply the Gibbs–Duhem equation to relate the partial molar properties of different species. 3 Chemical Engineering Department | University of Jordan | Amman 11942, Jordan Tel. +962 6 535 5000 | 22888 Introduction For a pure species i, all the intermolecular interactions are identical. The resulting thermodynamic properties—such as —are a manifestation of those interactions. Mixture contains more than one species and more complex than pure species Mixture properties are determined only in part by an average of each of the pure species (i-i) interactions. We must now also take into account how each of the species interacts with the other species in the mixture, that is, the unlike (i-j) interactions Hence, the properties of a mixture depend on the nature and amount of each of the species in the mixture. The values of the mixture’s properties will be affected not only by how those species behave by themselves but also by how they interact with each other. 4 Chemical Engineering Department | University of Jordan | Amman 11942, Jordan Tel. +962 6 535 5000 | 22888 Introduction Upon mixing, o The mass remains constant since it is a conserved quantity o However, the mixture volume is different from the sum of the pure species volume. o The ethanol and water can pack together more tightly than can each species by itself. o This is due to the nature of the hydrogen bonding involved in the structure of the liquid. 5 Chemical Engineering Department | University of Jordan | Amman 11942, Jordan Tel. +962 6 535 5000 | 22888 Introduction When a species becomes part of a mixture, it loses its identity; Yet it still contributes to the properties of the mixture, since the total solution properties of the mixture depend on the amount present of each species and its resultant interactions 6 Chemical Engineering Department | University of Jordan | Amman 11942, Jordan Tel. +962 6 535 5000 | 22888 Partial Molar Properties Any intensive thermodynamic property can be mathematically described in terms of partial derivatives of two independent, intensive properties. Since we are now concerned with thermal and mechanical equilibrium, it makes sense to choose T and P as the independent, intensive properties. For extensive properties, the total number of moles much be also specified. For mixture, in addition to specifying two independent properties, the number of moles of each species in the mixture should be considered. Mathematically, we can write the extensive total solution property K in terms of T, P, and the number of moles of m different species: 7 Chemical Engineering Department | University of Jordan | Amman 11942, Jordan Tel. +962 6 535 5000 | 22888 Partial Molar Properties The differential of K can then be written as the sum of partial derivatives of each of these independent variables, as follows: 8 Chemical Engineering Department | University of Jordan | Amman 11942, Jordan Tel. +962 6 535 5000 | 22888 Partial Molar Properties Define a new thermodynamic function, the partial molar property, as partial derivatives with respect to moles, K denotes for any extensive properties partial molar property The partial molar properties are governed by how a species behaves in the mixture. 9 Chemical Engineering Department | University of Jordan | Amman 11942, Jordan Tel. +962 6 535 5000 | 22888 Partial Molar Properties A partial molar property is always defined at constant temperature and pressure, two of the criteria for phase equilibrium. Partial molar properties are also defined with respect to number of moles The number of moles of all other j species in the mixture are held constant; it is only the number of moles of species i that is changed. It is a response function, i.e., a measure of the response of total property K to the addition at constant T and P of a differential amount of species I to a finite amount of solution. 10 Chemical Engineering Department | University of Jordan | Amman 11942, Jordan Tel. +962 6 535 5000 | 22888 Partial Molar Properties Note that Because in changing the number of moles of species i, we change the mole fractions of all the other species in the mixture as well, since the sum of the mole fractions must equal 1. Using the definition of partial molar properties, the total differential of the variable K becomes: differentiation at constant composition 11 Chemical Engineering Department | University of Jordan | Amman 11942, Jordan Tel. +962 6 535 5000 | 22888 Partial Molar Properties (k) (k) Also dK d(nk) n dP n dT Ki dni (11.9) P T,x T P,x At constant temperature and pressure, this equation reduces to Since ni = xin dni xidnndxi and d(nk) ndk kdn k k k k Ki Rearrange, k k k Ki k Ki 12 Chemical Engineering Department |=0.0 University of Jordan | Amman 11942, Jordan =0.0 Tel. +962 6 535 5000 | 22888 Partial Molar Properties k k dk- dP- dT-Kidxi 0 P T,x T P,x i k k (I) dk dP dT Kidxi P T,x T P,x i k xi Ki 0 i (II) per mole of mixture k xi Ki i These equations show that the calculation of nk K ni Ki (III) i 13 Chemical Engineering Department | University of Jordan | Amman 11942, Jordan Tel. +962 6 535 5000 | 22888 Partial Molar Properties The extensive total solution property K is equal to the sum of the partial molar properties of its constituent species, each adjusted in proportion to the quantity of that species present. Similarly, the intensive solution property k is simply the weighted average of the partial molar properties of each of the species present. For example, Where k is the corresponding intensive property to K and xi is the mole fraction of species i. 14 Chemical Engineering Department | University of Jordan | Amman 11942, Jordan Tel. +962 6 535 5000 | 22888 Partial Molar Properties Differentiating Eq (II) dk xi d K i K i dxi i i Comparison of this equation with Eq. (I) (Subtraction gives) yields k k dP dT xi dKi 0 P T ,x T P,x i This equation must be satisfied for all changes in P, T, and the ki caused by changes of state in a homogeneous phase Gibbs-Duhem equation As a special case at constant T and P: x d K i 0 i at constant T and P i Provides a very useful relationship between the partial molar properties of different species in a mixture 15 Chemical Engineering Department | University of Jordan | Amman 11942, Jordan Tel. +962 6 535 5000 | 22888 Partial Molar Properties Gibbs-Duhem equation at constant T and P for other thermodynamics properties As a solution becomes pure in species i, both properties approach pure species property lim k lim K i Ki xi 1 xi 1 In the limit of infinite dilution lim K i K i Ki xi 0 16 Chemical Engineering Department | University of Jordan | Amman 11942, Jordan Tel. +962 6 535 5000 | 22888 Analytical Determination of Partial Molar Properties Often an analytical expression for the total solution property, k, is known as a function of composition. In that case, the partial molar property, , can be found by differentiation of the extensive K i expression for K with respect to ni, holding T, P, and the number of moles of the other j species constant, Example 17 Chemical Engineering Department | University of Jordan | Amman 11942, Jordan Tel. +962 6 535 5000 | 22888 Example Cont. But and Since and 18 Chemical Engineering Department | University of Jordan | Amman 11942, Jordan Tel. +962 6 535 5000 | 22888 Example Cont. xi dMi 0 i dM1 dM2 For binary mixture x1 dM1 x2 dM2 0 x1 x2 0 dx1 dx1 19 Chemical Engineering Department | University of Jordan | Amman 11942, Jordan Tel. +962 6 535 5000 | 22888 Example Cont. 20 Chemical Engineering Department | University of Jordan | Amman 11942, Jordan Tel. +962 6 535 5000 | 22888 Example Cont. 21 Chemical Engineering Department | University of Jordan | Amman 11942, Jordan Tel. +962 6 535 5000 | 22888 Graphical Determination of Partial Molar Properties Used to calculate the partial molar volume (or any other partial molar property) for a binary mixture when we have a graph of the molar volume (or whatever molar property) vs. mole fraction of one component 22 Chemical Engineering Department | University of Jordan | Amman 11942, Jordan Tel. +962 6 535 5000 | 22888 Graphical Determination of Partial Molar Properties For a binary solution A k x1 K1 x2 K 2 B dk x1 dK1 K1dx1 x2 dK 2 K 2dx2 Gibbs-Duhem equation is x1 dK1 x2 dK 2 0 Dividing by dx1, we have the Gibbs-Duhem equation in derivative forms dK1 dK 2 C x1 x2 0 dx1 dx1 Because x1 x2 1 dx1 dx2 23 Chemical Engineering Department | University of Jordan | Amman 11942, Jordan Tel.
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