excess gibbs free energy models
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EXCESS GIBBS FREE ENERGY MODELS
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CONTENT• EAXCESS GIBBS FREE ENERGY MODELS• MARGULES EQUATION• REDLICH-KISTER EQUATION • VAN LAAR EQUATION• WILSON AND “NRTL” EQUATION• UNIversal QUAsi Chemical equation
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Excess Gibbs Energy ModelsPracticing engineers find most of the liquid-phase information needed for equilibrium calculations in the form of excess Gibbs Energy models. These models:
reduce vast quantities of experimental data into a few empirical parameters,
provide information an equation format that can be used in thermodynamic simulation packages (Provision)
“Simple” empirical models Symmetric, Margule’s, vanLaar No fundamental basis but easy to use Parameters apply to a given temperature, and the models
usually cannot be extended beyond binary systems.
Local composition models Wilsons, NRTL, Uniquac Some fundamental basis Parameters are temperature dependent, and multi-component
behaviour can be predicted from binary data.
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Margule’s Equations
21212121
ExAxA
xRTxG
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While the simplest Redlich/Kister-type expansion is the Symmetric Equation, a more accurate model is the Margule’s expression:
(11.7a)
Note that as x1 goes to zero,
and from L’hopital’s rule we know:
therefore,
and similarly
1
210xln
xRTxG
limE
1
120x21
E
AxRTx
G
1
112 lnA 221 lnA
Margule’s Equations
21212121
ExAxA
xRTxG
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If you have Margule’s parameters, the activity coefficients are easily derived from the excess Gibbs energy expression:
(11.7a)
to yield:
(11.8ab)
These empirical equations are widely used to describe binary solutions. A knowledge of A12 and A21 at the given T is all we require to calculate activity coefficients for a given solution composition.
]x)AA(2A[xln 1122112221
]x)AA(2A[xln 2211221212
Redlich-Kister equationThe Margules equation adequately represents the behavior of liquid solutions, the components of which are of similar size, shape and chemical nature. For several solutions, the free energy is not symmetric about x1= 0.5 the activity coefficients do not appear as mirror images of To behavior solutions, and Kister each other. represent the of general Redlich expansion an equation where the parameter A in the equation is replaced with a is given by in (x1-x2) and introduced additional parameters. The Redlich-Kister equation
The activity coefficients ɣ1, ɣ2 can be obtained form above eq and it can be shown that
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(X)
Redlich-Kister equation
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Substracting eq (2), form eq (1) , we get
(1)
(2)
Redlich-Kister equation
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Redlich-Kister equation
van Laar Equations
2/121
/21
/21
/12
21
E
xAxAAA
xRTxG
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Another two-parameter excess Gibbs energy model is developed from an expansion of (RTx1x2)/GE instead of GE/RTx1x2. The end results are:
(1)for the excess Gibbs energy and:
(2)
(3)
for the activity coefficients.
Note that: as x10, ln1 A’12
and as x2 0, ln2 A’21
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2/21
1/12/
121 xAxA1Aln
2
1/12
2/21/
212 xAxA1Aln
van Laar Equations
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Wilson’s Equations for Binary Solution Activity
)xxln(x)xxln(xRTG
2112212211
E
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A versatile and reasonably accurate model of excess Gibbs Energy was developed by Wilson in 1964. For a binary system, GE is provided by:
(1)
where(2)
Vi is the molar volume at T of the pure component i.aij is determined from experimental data.
RTaexp
VV
RTaexp
VV 21
2
121
12
1
212
Wilson’s Equations for Binary Solution Activity
2112
21
1221
12212211 xxxx
x)xxln(ln
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Activity coefficients are derived from the excess Gibbs energy using the definition of a partial molar property:
When applied to equation 11.16, we obtain:
(3)
(4)
2112
21
1221
12121122 xxxx
x)xxln(ln
jn,P,Ti
EEii n
nGGlnRT
Wilson’s Equations for Binary Solution Activity
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UNIversal QUAsi Chemical (UNIQUACK) equation
The to QUAsi Chemical (UNIQUAC) model was developed by Abrams and for express the excess Gibbs free energy of a binary mixture. The UNIQUAC equation The (gE/RT) contains two parts, a part and a residual The combinatorial part takes into account the composition, size and shape of the constituent molecules and contains pure component properties only. The residual part takes into account the intermolecular forces and contains two adjustable parameters. The UNIQUAC equation is given by
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UNIversal QUAsi Chemical (UNIQUACK) equation
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(1)
(2)
(3)(4)
(5)
(6)
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from eq. (1) – (3) One can obtain the activity coefficients as
UNIversal QUAsi Chemical (UNIQUACK) equation
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where vk is the number of groups of type in a molecule of component i. The UNI QUAC contains only two adjustable parameters Ʈ12 and Ʈ 21. The UNIQUAC equation is applicable to a wide variety of liquid solutions commonly encountered by chemical engineers. Although UNIQUAC is mathematically more complex.