Page 146 - Thermodynamics of Biochemical Reactions

144 Chapter 8 Phase Equilibrium in Aqueous Systems

8.2 TWO-PHASE SYSTEM WITH A CHEMICAL
REACTION AND A SEMIPERMEABLE MEMBRANE

Consider an aqueous two-phase system containing A, B, C, and solvent H,O in
which the reaction A + B = C occurs. The two phases are separated by a
membrane, and the membrane is permeable to all four species. The fundamental
equation for the Gibbs energy of the a phase is
dG, = - S,dT+ KdP + p,&nAa + + + PH2ordnH2Oa (8.2-1)

The corresponding equation for the fl phase is

dG, = - 'fldT+ 'OdP + p.4/JdnAB $- pBBdnBB + k/ldnCB + !-lH20/ld?1H20fl
(8.2-2)

These equations can be used to show that at chemical equilibrium pAa + pBz = pcX
and pAa + pss = pea. Substituting these equilibrium conditions into equation
8.2-1 yields

dGa - S~dT+ KdP + pAidncAa + pBadncBa + pH20rdnHzOz (8.2-3)
where the amount of the A component is ncAn = nAa + n,, and the amount of the
B component is ncBa = nsz + nCz. The corresponding fundamental equation for
the fl phase is

dG, = - bdP + pA/jdnc,4/, f pB/jdncB8 + pH20/jdnH20/j (8.2-4)
The fundamental equation for the Gibbs energy of the system at chemical
equilibrium in each phase is the sum of equations 8.2-3 and 8.2-4:

dc = - SdT+ VdP + ~.dncAa+ pBzdnci5, + pH2OdnHzOa
+ ~Agdn,AiI + iunpdr~cnp + 1-1H~oodnH,Op (8.2-5)
This equation can be used to show that pAa = pA0 = pA, pB, = pBB = p,, and
pHZOa
= ,uHZoB = pHzo. When these phase equilibrium conditions are inserted in
equation 8.2-5, it becomes
dC = - SdT+ VdP + ,uAdncA + ,uBdnCB + pHzOdncHz0 (8.2-6)

where nci represents the amount of component i in the system. The amount of
component A in the system is represented by ncA = ncAz + ncAp, and the amount
of the solvent is represented by ncHzOn = nHzOB + itHzo,. Equation 8.2-6 indicates
that there are D = 5 natural variables, and that they might be taken to be 7; P,
ncA, itcB, and in the criterion of spontaneous change and equilibrium: dG < 0
at constant 7: P, ncA, ncB, and ncllz0.
The Gibbs-Duhem equations for the two phases at equilibrium can be derived
from equations 8.2-3 and 8.2-4:

= - 'ndTf 'zdP - ncAzdblA - ncBadpB - IZHzOzdpH,O (8.2-7)
0 = - S,dT+ VodP - nCApdpA - ncBBdpB - nHzOBdpHzO (8.2-8)
where the phase subscripts on the chemical potentials have been dropped because
of phase equilibrium. dp(H20 can be eliminated between these two equations, and
the resulting equation can be solved for dpB, which is a function of 7: P, and pA.
Thus there are three independent intensive properties, in agreement with
F = C ~ p + 2 = 3 - 2 + 2 = 3. This is in agreement with D = F + p =
3 + 2 = 5. The natural variables for the expression of the criterion for sponta-
neous change and equilibrium based on G might bc taken to be 7: P, ncAlinCH, 17,.
and nB, where it, = nA, + tiBa + igcz + nSz. Then the equilibrium condition would
be dC d 0 at constant '17 P, ncA/ncR, n,, and nil,.
If the phases are both dilute aqueous solutions and the membrane separating
the phases is permeable to all species, the species will have the same concentra-