The sum of the second and last terms on the right-hand side of this equation is the integral of d{[A][B]}, so the above equation becomes:

Equality of the chemical potentials of water in the two aqueous phases is expressed by:

gwi[W]i = gwii[W]ii or lngwi + ln[W]i = lngwii + ln[W]n (11.58)

The molar concentration of water, [W], can be expressed in terms of mole fractions by:

[W] = Xw/Vw = (1 - Xa - Xb)/Vw so that ln[W] = -lnvw + ln(1 - xA - xB) @ -lnvw - xA - xB = -lnvw - ([A] + [B])vw Substituting this equation and Eq (11.57) into (11.58) yields:

[A], + [B], + 1 bAA [A]2 + 1 bBB [B]2 + b AB [A], [B], = [A]jj + [B]jj + 2 b AA [A]2i + 2 b BB [B]2 + bAB [A]ii [B]n

Equations (11.55a), (11.55b) and (11.59) provide three equations with four unknowns, [A]i, [B]i, [A]n and [B]ii. The fourth equation is a statement of species conservation.

Suppose that the system consists of a total volume V of water to which nA moles of A and nB moles of B are added. After splitting into two phases of volumes Vi and Vii, the species conservation equations are:

nA = [A]iVi + [A]nVn or [A]M = f [A]i + (1-f)[A]n nB = [B]iVi + [B]nVn or

[B]tot = f [B]i + (1-f)[B]n where f = VI/Vtot is the fraction of the total water volume converted to phase I and [A]tot = nA/Vtot and [B]tot = nB/Vtot. These are specified. Eliminating f between the two species conservation equations yields:

Solution of the phase-equilibrium equations Dimensionless forms of these equations result from the definitions:

X = bAA[A]i; Y = bAA[A]n; U = bee[B]i; V = bee^n ZA = bAA[A]tot;

Equations (11.57a) and (11.57b) become:

With some rearrangement, Eq (11.59) is converted to:

(1 + X) 2 J B + (1 + U) 2 JA + 2XU _ (1 + Y)2 J B + (1 + V) 2 J A + 2YV

Getting Started With Solar

Getting Started With Solar

Do we really want the one thing that gives us its resources unconditionally to suffer even more than it is suffering now? Nature, is a part of our being from the earliest human days. We respect Nature and it gives us its bounty, but in the recent past greedy money hungry corporations have made us all so destructive, so wasteful.

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