Chemical Engineering Thermodynamics II
Lec 2: Phase Equilibria: Thermodynamics of Mixtures
Dr.-Eng. Zayed Al-Hamamre
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Content
Learning Objectives Introduction Partial Molar Properties Determination of Partial Molar Properties Relations Among Partial Molar Quantities
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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.
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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.
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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
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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:
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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:
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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.
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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
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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
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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
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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.
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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
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Example Cont.
But and
Since
and
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xi dMi 0 i dM1 dM2 For binary mixture x1 dM1 x2 dM2 0 x1 x2 0 dx1 dx1
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Example Cont.
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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
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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. +962 6 535 5000 | 22888
Graphical Determination of Partial Molar Properties
Eq. B becomes dk K1 K 2 dx1 dk From Eq A and D dk K1 k x2 K 2 k x1 dx1 dx1 These equations can be used to obtain partial molar properties from solution property. dk dk Or k K1 x2 k K 2 x1 dx1 dx1
intercept slope Similarly,
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K 1
k1 k k K 2
k2
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Graphical Determination of Partial Molar Properties
In the figure, the slope of the line at point k is
dk k I2 dk I1 I2 dx1 x1 dx1
dk dk I2 k x1 I1 k (1 x1) dx1 dx1
Comparing these equation with dk dk K1 k x2 K 2 k x1 dx1 dx1
K 2 I K1 I1 2
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The limiting cases of the partial molar volume of species 1 in terms of composition of 1.
In the limit as x1 goes to 1, we have all 1 and no 2 in the mixture. In this case, the partial molar volume just equals the pure species molar volume:
lim k lim K i Ki xi 1 xi 1
In the limit of infinite dilution
lim K i K i Ki xi 0
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Example The need arise in a laboratory for 2000 cm3 of an antifreeze solution consisting of 30 mol % methanol in water. What volumes of pure methanol and of pure water at 25 C must be mixed to form the of antifreeze, also at 25 C? Partial molar volumes for methanol and water in a 30 mol % methanol solution and their pure-species molar volume, both at 25 C , are: Methanol (1) and water (2):
3 1 3 1 V1 38.632 cm mol V1 40.727 cm mol 3 1 3 1 V2 17.765 cm mol V2 18.068 cm mol
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V x1V1 x2V2 (0.3)(38.632) (0.7)(17.765) V 24.025 cm3mol1 V t 2000 cm3 n 83.246 cm3 V 24.025 cm3mol1
n1 x1n (0.3)(83.246 ) 24.974 mol
n2 x2n (0.7)(83.246 ) 58.272 mol
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Example Cont. Solution The line drawn tangent to the V-x1 curve at x1=0.30, illustrates the values of
V1=40.272 cm3 mol-1 and V2=18.068 cm3 mol-1.
t 3 1 V1 (24.497 mol)(40.727 cm mol ) t 3 V1 1017 cm t 3 1 V2 (58.272 mol)(18.068 cm mol ) t 3 V2 1053 cm
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Find the partial molar volume of species b in a binary solution when we know the partial molar volume of species a, , as a function of composition
Applying Gibbs-Duhem equation at constant T and P
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Summary of the Different Types of Thermodynamic Properties
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Vi ,Ui , Hi ,Si , Ai ,Gi these properties must be intensive
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Relations Among Partial Molar Quantities
Partial molar properties can be related to each other, considering the equation
Since the pressure is constant,
Applying the definition of a partial molar quantity
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Similarly,
What is the most important property ? ……G………. For pure component; G = G (T, P)
Also, for closed system: no mass transfer across boundary or in a single-phase fluid in a closed system wherein no chemical reactions occur
When there is no change in composition, dG VdP SdT 35 Chemical Engineering Department | University of Jordan | Amman 11942, Jordan Tel. +962 6 535 5000 | 22888
Relations Among Partial Molar Quantities
For a homogeneous mixture e.g. containing i components mixture;
G = G (T, P, n1, n2, …, ni)
ni is the number of moles of species i
(G) (G) (G) d(G) dP dT dni P T ,n T P,n i ni T ,P,n j
all mole numbers held constant all mole numbers except ni held constant
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Also,
G G G V S Gi P T ,n T P,n ni P,T ,n j
Application of the criterion of exactness
V S Gi (S) T ni T P ,n P T ,n P ,n P,T ,n j
G i Si T P ,x
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Relations Among Partial Molar Quantities
Gi (V ) G i Vi P ni T ,n P,T ,n j P T ,x
38 Chemical Engineering Department | University of Jordan | Amman 11942, Jordan Tel. +962 6 535 5000 | 22888 Relations Among Partial Molar Quantities Thus, every equation that provides a linear relation among thermodynamic properties of a constant-composition solution has as its counterpart an equation connecting the corresponding partial properties of each species in the solution. For example,
dG VdP SdT dGi VidP SidT
H U PV Hi Ui PVi
G H TS Gi Hi TSi
G Gi V Vi P T ,x P T ,x 39 Chemical Engineering Department | University of Jordan | Amman 11942, Jordan Tel. +962 6 535 5000 | 22888
Relations Among Partial Molar Quantities
Also, for closed system: no mass transfer across boundary or in a single-phase fluid in a closed system wherein no chemical reactions occur
For a mixture,
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dG VdP SdT Gi dNi i
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Relations Among Partial Molar Quantities
42 Chemical Engineering Department | University of Jordan | Amman 11942, Jordan Tel. +962 6 535 5000 | 22888 Partial Properties in Binary Solutions
For a binary (For two components) solution
dH TdS VdP(μ1 )S,Pdn1 (μ2 )S,Pdn2
dU TdS PdV (μ1 )V,Sdn1 (μ2 )V,Sdn2
dGVdP -SdT(μ1 )T,Pdn1 (μ2 )T,Pdn2
dA PdV -SdT(μ1 )T,V dn1 (μ2 )T,V dn2 H U G A G1 1 n1 n1 n1 n1 S,P,n2 V ,S,n2 T ,P,n2 T ,V ,n2
Keep in mind that,
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Excursion
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