Thermodynamics and Chemistry

CHAPTER 11 REACTIONS AND OTHER CHEMICAL PROCESSES

11.1 MIXINGPROCESSES 311

plot, the tangent to the curve at any given composition has intercepts equal to.BB/at
xAD 0 and.AA/atxAD 1.
In order for two binary liquid phases to be in transfer equilibrium,Amust be the
same in both phases andBmust also be the same in both phases. The dashed line in the
figure is a common tangent to the curve at the points labeledíandì. These two points
are the only ones having a common tangent, and what makes the common tangent possible
is the downward concavity (negative curvature) of a portion of the curve between these
points. Because the tangents at these points have the same intercepts, phasesíandìof
compositionsxAíandxAìcan be in equilibrium with one another: the necessary conditions

íADìAandíBDìBare satisfied.
Now consider point 1 on the curve. A phase of this composition is unstable. It will
spontaneously separate into the two phases of compositionsxíAandxìA, because the Gibbs
energy per total amount then decreases to the extent indicated by the vertical arrow from
point 1 to point 2. We know that a process in whichGdecreases at constantT andpin a
closed system, with expansion work only, is a spontaneous process (Sec.5.8).

To show that the arrow in Fig.11.4represents the change inG=nfor phase separa-

tion, we letyrepresent the vertical ordinate and write the equation of the dashed line

through pointsíandì(yas a function ofxA):

yDyíC

yìyí

xAìxíA

!

.xAxAí/ (11.1.38)

In the system both before and after phase separation occurs,xAis the mole fraction of

component A in the system as a whole. When phasesíandìare present, containing

amountsníandnì,xAis given by the expression

xAD

xíAníCxìAnì

níCnì

(11.1.39)

By substituting this expression forxAin Eq.11.1.38, after some rearrangement and

usingníCnìDn, we obtain

yD

1

n



níyíCnìyì



(11.1.40)

which equatesyfor a point on the dashed line to the Gibbs energy change for mixing

pure components to form an amountníof phaseíand an amountnìof phaseì,

divided by the total amountn. Thus, the difference between the values ofyat points

1 and 2 is the decrease inG=nwhen a single phase separates into two equilibrated

phases.

Any mixture with a value ofxAbetweenxíAandxAìis unstable with respect to separation

into two phases of compositionsxAíandxAì. Phase separation occurs only if the curve of
the plot ofÅGm(mix) versusxAis concave downward, which requires the curve to have at
least two inflection points. The compositions of the two phases are not the compositions at
the inflection points, nor in the case of the curve shown in Fig.11.4are these compositions
the same as those of the two local minima.
By varying the values of parameters in an expression for the excess molar Gibbs energy,
we can model the onset of phase separation caused by a temperature change. Figure11.5
shows the results of using the two-parameter Redlich–Kister series (Eq.11.1.36).