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136 5 SIMPLEMIXTURES
�e mole fractions are rewritten in terms of nA and nB to give
G = (nA + nB)gRT ( nA
nA + nB
)( nB
nA + nB
)
+ (nA + nB)RT [( nA
nA + nB
) ln( nA
nA + nB
) + ( nB
nA + nB
) ln( nB
nA + nB
)]
+ nAG∗m,A + nBG∗m,B
= gRTnAnB
nA + nB
+ RT [nA ln(
nA
nA + nB
) + nB ln(
nB
nA + nB
)] + nAG∗m,A + nBG∗m,B
= gRTnAnB
nA + nB
+ RT [nA ln nA − nA ln(nA + nB) + nB ln nB − nB ln(nA + nB)]
+ nAG∗m,A + nBG∗m,B
�e algebra used in going to the last line is used tomake it easier to compute the
derivative. In �nding the derivative recall that terms such as nA ln nA require
the application of the product rule.
( ∂G
∂nA
)
nB
= gRTnB
nA + nB
− gRTnAnB
(nA + nB)2
+ RT [ln nA + 1 − ln(nA + nB) −
nA
(nA + nB)
− nB
(nA + nB)
] +G∗m,A
=gRT(xB − xAxB) + RT(1 + ln xA − xA − xB) +G∗m,A
=gRTxB(1 − xA) + RT ln xA +G∗m,A
=gRTx2B + RT ln xA +G∗m,A
On the second line the mole fractions are re-introduced, and in the subsequent
manipulations the relationship xA + xB = 1 is used. G∗m,A is identi�ed as the
chemical potential of pure A, µ∗A, giving the result
µA = µ∗A + gRTx2B + RT ln xA
�is function is plotted in Fig. 5.6.
As g increases the deviation from ideal behaviour (the solid line) increases,
with the e�ect being larger at small xA, corresponding to larger xB.
P5B.8 (a) �e van ’t Ho� equation is [5B.16–163], Π = [B]RT . If the ‘pressure’ P
is expressed as (mass/area), then to transform it to (force/area) requires
multiplication by the acceleration of free fall, g, because force = mass ×
acceleration: Π = gP.
If the ‘concentration’ c is expressed as (mass/volume), then to transform
it to (moles/volume) requires the use of the molar mass, M: [B] = c/M.
With these substitutions the van ’t Ho� equation becomes
gP = cRT
M
hence P = ( c
M
)(R
g
)T hence P = ( c
M
)R′T