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CHAPTER 13 Chemical-Reaction Equilibria
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present in each molecule of chemical species i The material balance on each element k may then be written:
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C n i a i k - Ak = O
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(k = 1 , 2 , , w)
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where w is the total number of elements comprising the system 2 Next, we introduce the Lagrange multipliers hk, one for each element, by multiplying each element balance by its hk:
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These equations are summed over k, giving:
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3 Then a new function F is formed by addition of this last sum to Gt Thus,
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This new function is identical with Gt, because the summation term is zero However, the partial derivatives of F and G' with respect to ni are different, because the function F incorporates the constraints of the material balances 4 The minimum value F (and G') occurs when all of the partial derivatives (aF/ani)T,p,nJ are zero We therefore differentiate the preceding equation, and set the resulting derivative equal to zero:
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Since the first term on the right is the definition of the chemical potential [see Eq (1 11)], this equation can be written:
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+ x h k a i k= O
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(i = 1,2, , N)
However, the chemical potential is given by Eq (139):
For gas-phase reactions and standard states as the pure ideal gases at 1 bar [or 1 atm]:
= GP
+ RT l n ( f i / ~ " )
If GP is arbitrarily set equal to zero for all elements in their standard states, then for compounds GP = AG>i, the standard Gibbs-energy change of formation for species i In
139 Multireaction Eauilibria
addition, the fugacity is eliminated in favor of the fugacity coefficient by Eq (1 148), = yi&i P With these substitutions, the equation for pi becomes:
Combination with Eq (1342) gives:
Note that P o is 1 bar, expressed in the units used for pressure If species i is an element, AGYi is zero Equation (1343) represents N equilibrium equations, one for each chemical species, and Eq (1341) represents w material-balance equations, one for each element- a total of N w equations The unknowns in these equations are the ni's (note that yi = n, / ni), of which there are N, and the hk's, of which there are w- a total of N w unknowns Thus the number of equations is sufficient for the determination of all unknowns
The foregoing discussion has presumed that each &i is known If the phase is an ideal gas, then for each species &i = 1 If the phase is an ideal solution, 6, = I#I~, values can at and least be estimated For real gases, $iis a function of the set {yi), which is being calculated Thus an iterative procedure is indicated The calculations are initiated with Ji = 1 for all i Solution of the equations then provides a preliminary set of {yi) For low pressures or high temperatures this result is usually adequate Where it is not satisfactory, an equation of state is used together with the calculated {yi) to give a new and more nearly correct set { J i ) for use in Eq (1343) Then a new set {yi) is determined The process is repeated until successive iterations produce no significant change in {yi) All calculations are well suited to computer solution, including the calculation of {Ji) equations such as Eq (1161) by In the procedure just described, the question of what chemical reactions are involved never enters directly into any of the equations However, the choice of a set of species is entirely equivalent to the choice of a set of independent reactions among the specieany event, a set of species or an equivalent set of independent reactions must always be assumed, and different assumptions produce different results
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