13168
J. Phys. Chem. B 2003, 107, 13168-13177
The Thermodynamics of Nonelectrolyte Systems at Constant Activities of Any Number of Components Yu-Feng Hu* State Key Laboratory of HeaVy Oil Processing, UniVersity of Petroleum, Beijing 102200, China, and High-Pressure Fluid Phase BehaVior & Property Research Laboratory, UniVersity of Petroleum, Beijing 102200, China ReceiVed: June 1, 2003; In Final Form: September 28, 2003
The novel linear concentration relations and the simple predictive equations for the thermodynamic properties (chemical potential, activities of all BJ (J ) 1, 2, ..., j), Gibbs free energy, thermal properties, and volumetric properties) have been proposed for the multicomponent system B1-B2-...-Bj-C1-C2-...-Ck-A1-A2...-Ai in terms of the concentration and the properties of its subsystems BJ-C1-C2-...-Ck-A1-A2-...-Ai (J ) 1, 2, ..., j) of equal activities of all C1, C2, ..., Ck and A1, A2, ..., Ai. It has been shown that the process of mixing the nonideal systems BJ-C1-C2-...-Ck-A1-A2-...-Ai (J ) 1, 2, ..., j) of equal activities of all C1, C2, ..., Ck and A1, A2, ..., Ai shows ideal mixing behavior and that the newly established equations are the same in form as those for mixing the ideal systems BJ-C1-C2-...-Ck-A1-A2-...Ai (J ) 1, 2, ..., j) of equal mole fractions of all C1, C2, ..., Ck and A1, A2, ..., Ai. The thermodynamic basis for the ideal mixing behavior has been presented. It has been shown that, while Raoult’s law can describe the ideal behavior of mixtures based on similar components, the present equations describe a new type of “ideal” behavior of nonideal mixtures B1-B2-...-Bj-C1-C2-...-Ck-A1-A2-...-Ai at constant activities of all C1, C2, ..., Ck and A1, A2, ..., Ai. The linear concentration relations have been tested by comparing with the isopiestic measurements for the saturated systems glucose-sorbitol-sucrose(sat)-water, glucose-mannitol-sucrose(sat)water, sorbitol-glucose-mannitol(sat)-water, sorbitol-sucrose-mannitol(sat)-water, mannitol-sorbitolsucrose(sat)-water, mannitol-sucrose-urea(sat)-water, and sucrose-urea-mannitol(sat)-water at 298.15 K. As expected, the former six systems conform to the new relations very well, but large deviation is observed in the last system.
1. Introduction Scatchard1,2 introduced the concept of the semi-ideality to interpret the deviations of the osmotic coefficient of aqueous solutions of sucrose from the ideal behavior. The semi-ideality means that solute-solute interactions can be neglected and the solute-solvent interactions can be simply described by a hydration number. Since then, Sarolea-Mathot3 and McGlashan and Rastogi4 have successfully treated acetone-chloroform mixtures and dioxane-chloroform mixtures by the assumption of semi-ideal behavior. Later, Stokes and Robinson5,6 improved this model by replacing the fixed hydration number by a series of hydration equilibria (they used the model also for electrolyte solutions).7,8 They have shown that mixed aqueous nonelectrolyte solutions B1-B2-...-Bj-A1(H2O) conforming to this model obey the simple linear isopiestic relation j
∑
mBJ
J)1m° B
)1
J
(aA1 ) const and 0 e (mBJ/m°BJ) e 1)
(1)
where B1, B2, ..., Bj denote the nonelectrolyte solutes present below their solubility limits and A1 represents water. The quantities m and a are the molality (mol (kg H2O)-1) and * E-mail:
[email protected]. Fax: 86-10-69744849.
activity, respectively. The superscript degree symbol together with the subscript BJ represents the quantity of BJ in the binary solutions BJ-A1(H2O) (J ) 1, 2, ..., j). Note that eq 1 is also known as Zdanovskii’s rule9-11 for mixed electrolyte solutions. Recently,12,13 we have shown that the thermodynamic behavior of the mixed solution conforming to eq 1 is as simple as that of an ideal solution, that is, the constituent binary solution mixed ideally under isopiestic equilibrium. Based on this ideal solution behavior, a new set of simple predictive equations have been developed for thermodynamic properties of the mixed solutions, including water activity,6 activity coefficient of a solute in the mixed solutions,6,14 volumetric properties,12 thermal properties,13 and so on. More recently, we have shown that the mixing behavior of viscosities of electrolyte solutions obeying eq 1 is very simple, and thus, eq 1 has been used together with the Eyring’s absolute rate theory to yield a novel predictive equation for viscosity of the mixed solutions.15 Note that although the new predictive equations for the thermodynamic and transport properties are tested using the data of mixed electrolyte solutions, they are certainly true for the mixed nonelectrolyte solutions. Equation 1 was first proposed for ternary aqueous solutions and was extended to include any number of unsaturated solutes later. In this study, we define A1, A2, ..., Ai as the solvent components, B1, B2, ..., Bj as the solute components for solutes present below their solubility limits, and C1, C2, ..., Ck as the solute components present as saturated solutions. We attempt
10.1021/jp035528f CCC: $25.00 © 2003 American Chemical Society Published on Web 11/01/2003
Thermodynamics of Nonelectrolyte Systems
J. Phys. Chem. B, Vol. 107, No. 47, 2003 13169
to establish the linear relations for the systems B1-B2-...Bj-C1-C2-...-Ck-A1-A2-...-Ai in which activities of any number of components (C1, C2, ..., Ck and A1, A2, ..., Ai) are constant. A new set of simple predictive equations has been proposed for thermodynamic properties of the systems B1-B2...-Bj-C1-C2-...-Ck-A1-A2-...-Ai in terms of the properties of the subsystems BJ-C1-C2-...-Ck-A1-A2-...Ai (J ) 1, 2, ..., j), including chemical potential, activities of all BJ, Gibbs free energy, thermal properties, volumetric properties, and so on. The process of mixing the nonideal solutions BJC1-C2-...-Ck-A1-A2-...-Ai (J ) 1, 2, ..., j) at constant activities of all C1, C2, ..., Ck and A1, A2, ..., Ai has been studied to see whether it is as simple as that of mixing the ideal solutions BJ-C1-C2-...-Ck-A1-A2-...Ai (J ) 1, 2, ..., j) at constant mole fractions of all C1, C2, ‚‚‚, Ck and A1, A2, ..., Ai. The thermodynamic basis for the ideal mixing behavior of mixing the nonideal solutions BJ-C1-C2-...-Ck-A1-A2-...Ai (J ) 1, 2, ..., j) at constant activities of all C1, C2, ..., Ck and A1, A2, ..., Ai has been explored. The novel linear relations have been tested by comparing with the isopiestic results of the saturated solutions B1-B2-C1-A1(H2O) at 298.15 K. The conditions under which the new linear relations hold have been examined by studying the effects of solute-solute and solute-solvent interactions on the isopiestic behavior of the saturated solutions B1-B2-C1-A1(H2O) at 298.15 K. Equation 1 is in excellent agreement with the isopiestic behavior of the unsaturated system mannitol(B1)-sucrose(B2)water(A1) of which the water activities are greater than those of the binary saturated solutions BJ(sat)-water (J ) 1 or 2).6 Robinson and Stokes empirically extended eq 1 to describe the isopiestic behavior of the unsaturated system mannitol(B1)sucrose(B2)-water(A1) of which the water activities are less than that of the binary saturated solution mannitol(sat)-water.6 They calculated mBsupersat by using eq 1 together with the values 1 of m°B2, where mBsupersat is the molality of the supersaturated 1 solution of mannitol. However, the resulting value of mBsupersat 1 is not a constant for a given water activity. Therefore, in this study, efforts have also been made to see whether it is due to the breakdown of eq 1 when applied to the cases being examined by Stokes and Robinson. This is necessarily important because it is related to whether the simple predictive equations6,9,11-15 for the thermodynamic and transport properties are applicable to the mixtures of which the water activities are less than those of the binary saturated solutions of its constituent components. 2. Theoretical Section Solvation Equilibrium and Average Hydration Numbers for Unsaturated and Saturated Solutes. For the semi-ideal mixture of nonelectrolyte solutes, B1-B2-A1(H2O), the solutesolute interactions can be neglected and the solute-solvent interactions can be described by the stepwise hydration equilibrium.6 Such equilibrium
BJ(l-1) + A1 ) BJ(l) (J ) 1, 2; l ) 1, 2, ..., n)
(2)
has the equilibrium constant Kl and the average hydration number hhBJ-A1. According to the semi-ideal solution model,1-6,16 the resulting species form the ideal mixture based on the mole fraction (x), and the activity coefficients of the species are unity. This leads Kl and hhBJ-A1 to
Kl ) xBJ(l)/(xBJ(l-1)aA1) (l ) 1, 2, ..., n)
(3)
hhBJ-A1 ) σ/Σ
(4)
and
with n
Σ)1+
K1...KlaA l ∑ l)1
(5)
1
and n
σ ) dΣ/d ln aA1 )
lK1...KlaA l ∑ l)1 1
(6)
It is clear from eqs 3-6 that hhBJ-A1 depends only on water activity, aA1. For nonelectrolyte solution B1-B2-C1-A1(H2O), the interactions between the unsaturated solute and the solvent (B1-A1 and B2-A1) can be described by eqs 2-6, and the corresponding average hydration number is only a water activity-dependent quantity. So let us now consider the stepwise hydration equilibrium for the interactions between the saturated solute and the solvent (C1-A1) and the average hydration number (hhC1-A1) for the saturated solute. Such equilibrium
C1(s) + A1 a C1(1)
(7)
and
C1(l-1) + A1 a C1(l)
(l ) 2, ..., n)
(8)
have equilibrium constants K′1 and K′l, which are, respectively, given by
K′1 ) xC1(1)/aA1
(9)
K′l ) xC1(1)/(xC1(l-1)aA1)
(10)
and
Here, K′1 is equal to the equilibrium constant K′′0 for C1(s) a C1(aq) times the equilibrium constant K′′1 for C1(aq) + A1 a C1(1), where K′′0 ) aC1(aq)/aC1(s) and K′′1 ) aC1(1)/(aC1(aq)aA1). Because aC1(aq) must be equal to that in binary saturated system C1-A1(H2O), aC1(s) is, by definition, unity, and aC1(1) ) xC1(1),1-6,16 we can reach the conclusion from eqs 4-6, 9 and 10, that the average hydration number hhC1-A1 also depends only on water activity, aA1. However, it can be seen from eqs 3 and 9 that the equilibrium constant K′1 for the saturated solutes is different from K1 (K1 ) xBJ(1)/(xBJ(0)aA1), see eq 3) for unsaturated solutes, and thus the average hydration number hhC1-A1 differs from hhB1-A1 for a given water activity, aA1. The Linear Concentration Relation for the System B1B2-...-Bj-C1-C2-...-Ck-A1-A2-...-Ai at Constant Activities of All C1, C2, ..., Ck and A1, A2, ..., Ai. For the semiideal mixtures B1-B2-C1-A1(H2O), the solute-solvent interactions can be described by the binding equilibria, and the values of hhB1-A1, I ∈ (1, 2, ..., i), and hhC1-A1 depend only on the activity of A1. If we denote the free quantity with the superscript f, then the amount nAf(B1 J°) of the free solvent A1 and the total J°) amount n(B of species in the ternary subsystems BJ-C1-A1 t
13170 J. Phys. Chem. B, Vol. 107, No. 47, 2003
Hu
(J ) 1, 2) can be expressed as
nAf(B1 J°) ) nA(B1J°) - hhBJ-A1nB(BJJ°) - hhC1-A1nC(B1J°)
(11)
define the quantities KBJ ) hhBJ-A1 + aA1 - hhBJ-A1aA1 and K ) (hhC1-A1aA1 - aA1 - hhC1-A1)/(1 - aA1) and then rewrite eqs 14 and 18 as
and J°) n(B t
)
nAf(B1 J°)
nC(B1J°)
+
+
KBJ )
nB(BJJ°)
J
)
[
nAf(B1 J°) J°) n(B t
nA(B1J°) - hhBJ-A1nB(BJJ°) - hhC1-A1nC(B1J°) (nA(B1J°) - hhBJ-A1nB(BJJ°) - hhC1-A1nC(B1J°)) + nC(B1J°) + nB(BJJ°) (13)
where J ) 1 or 2 and xAf(B1 J°) is the mole fraction of the free solvent A1 in ternary mixture BJ-C1-A1 (J ) 1, 2). Equation 13 can be rewritten as
nB(BJJ°) )
J
1
2
∑ J)1
nBJ
] [
nA(B1J°) - nA1 + K
nB(BJJ°)
hhBJ-A1 + aA1 - hhBJ-A1aA1
2
∑hhB -A nB - hhC -A nC J
1
J
1
1
2
nt ) (nA1 -
(15)
1
2
∑hhB -A nB - hhC -A nC ) + nC + J)1 ∑ nB J)1 J
1
J
1
1
nBJ
]
nC(B1J°) - nC1 ) 0
nB(BJJ°)
(21)
Because hhC1-A1 depends only on aA1, K is constant at a given water activity. Because eq 21 holds for the quaternary system B1-B2-C1-A1 and its two ternary subsystems BJ-C1-A1 (J ) 1, 2) of equal aA1, irrespective of the actual values of the variable, that is, eq 21 is an identical equation,
nBJ
2
∑ J)1
nB(BJJ°)
nC(B1J°) (14)
Similarly, the amount nAf 1 of the free solvent A1 and the total amount nt of species in the mixture B1-B2-C1-A1(H2O) are
J)1
(20)
nA(B1J°) ) nA1
(22)
nC(B1J°) ) nC1
(23)
and
hhC1-A1aA1 - aA1 - hhC1-A1
nAf 1 ) nA1 -
2
∑ J)1
(aA1 ) const and aC1 ) const)
1 - a A1
n(BJ°) + hhBJ-A1 + aA1 - hhBJ-A1aA1 A1
+ K(1 - aA1)nC1
1
at constant activities of A1 and C1. Combining eqs 19 and 20 yields
aA1 ) xAf(B1 J°) )
(19)
2
∑KB nB ) (1 - aA )nA J)1
) (nA(B1J°) - hhBJ-A1nB(BJJ°) - hhC1-A1nC(B1J°)) + nC(B1J°) + nB(BJJ°) (12) where the superscript (BJ°) denotes the quantity in the ternary subsystems BJ-C1-A1 (J ) 1, 2). The activity of A1 is1-6,16
(1 - aA1) K(1 - aA1) nA(B1J°) + nC(B1J°) (BJ°) nB J nB(BJJ°)
1
1
J
2
∑ J)1
nBJ
nB(BJJ°)
at constant activities of A1 and C1 and within the range 0 e J°) [nBJ/n(B BJ ] e 1. Equations 22 and 23 can be generalized to the mixtures B1-B2-...-Bj-C1-C2-...-Ck-A1-A2-...-Ai in terms of the concentrations of its subsystems BJ-C1-C2-...Ck-A1-A2-...Ai (J ) 1, 2, ..., j) of equal activities of all C1, C2, ..., Ck and A1, A2, ..., Ai:
(16) j
∑ J)1
nB J
nA(BIJ°) ) nAI
(I ) 1, 2, ..., i)
(24)
nC(BKJ°) ) nCK
(K ) 1, 2, ..., k)
(25)
nB(BJJ°)
The activity of A1 is 2
nA1 a A1 )
∑hhB -A nB - hhC -A nC J)1 J
1
J
1
1
and 1
2
(nA1 -
∑ J)1
hhBJ-A1nBJ - hhC1-A1nC1) + nC1 +
(17)
2
∑ J)1
nBJ
Equation 17 is equivalent to 2
∑(hhB -A J)1 J
1
+ aA1 - hhBJ-A1aA1)nBJ ) (1 - aA1)nA1 + (hhC1-A1aA1 - aA1 - hhC1-A1)nC1 (18)
As shown previously, both hhBJ-A1 (J ) 1, 2) and hhC1-A1 depend only on the water activity, that is, they have the same values in multicomponent mixtures, B1-B2-C1-A1, as in their ternary subsystems, BJ-C1-A1 (J ) 1, 2), of equal aA1. So we can
j
∑ J)1
nBJ
nB(BJJ°)
where 0 e [nBJ/nB(BJJ°)] e 1, aA1 ) const (I ) 1, 2, ..., i), and aCK ) const (K ) 1, 2, ..., k). Thermodynamic Properties of the System B1-B2-...-BjC1-C2-...-Ck-A1-A2-...-Ai at Constant Activities of All C1, C2, ..., Ck and A1, A2, ..., Ai. The thermodynamic relations for the system B1-B2-...-Bj-C1-C2-...-Ck-A1-A2-...Ai at constant temperature and pressure can be expressed as j
dG )
k
∑ µB dnB + K∑) 1µC J)1 J
J
i
dnCK + K
µA dnA ∑ I)1 I
I
(26)
Thermodynamics of Nonelectrolyte Systems
J. Phys. Chem. B, Vol. 107, No. 47, 2003 13171
and j
k
i
∑ nC ∑nB dµB + K)1 J)1 J
dµCK + K
J
µBJ - µB(BJJ°) ) RT ln
nA dµA ) 0 ∑ I)1 I
(27)
I
k
∑
K)1
i
µCKnCK -
µA nA ) ) ∑ I)1 I
I
k
-
∑
K)1
i
nCK dµCK -
∑ I)1
j
nAI dµAI +
∑µB dnB
J)1
xB(BJJ°),d
(µAI ) const and µCK ) const)
where G and µ are Gibbs free energy and chemical potential, respectively. Equation 26 can be rewritten as
d(G -
xBd J
J
J
It can be shown that if the system B1-B2-...-Bj-C1-C2...-Ck-A1-A2-...-Ai and its subsystems BJ-C1-C2-...-CkA1-A2-...Ai (J ) 1, 2, ..., j) of equal mole fractions of all C1, C2, ..., Ck and A1, A2, ..., Ai are all ideal mixtures, then eqs 24 and 25 and eq 31 also hold, that is, ((∂µBJ/∂µAI′)nideal B1,nB2,...,nBn (BJ°) ideal J°) /∂µ ) ) 0). Therefore, we can obtain (∂µ(B BJ AI ′ nBJ
(28) (µBJ -
For the systems BJ-C1-C2-...-Ck-A1-A2-...Ai (J ) 1, 2, ‚‚‚ j), we can thus obtain
(29)
where I′ ∈ (1, 2, ..., i) and K ) 1, 2, ..., k. In the literature, a similar equation has been given for the system BJ-A1 (J ) 1, 2, ..., j).17 For the system B1-B2-...-Bj-C1-C2-...-Ck-A1A2-...-Ai, we reach
(∂µBJ/∂µAI′)nB1,nB2,...,nBJ ) -(∂nAI′/∂nBJ)µAI′,nBj*BJ (µAI*I′ ) const, µCK ) const)
(30)
Combination of eqs 24, 29, and 30 yields
(∂µBJ/∂µAI′)nB1,nB2,...,nBJ - (∂µB(BJJ°)/∂µA(BI′J°))nBJ ) 0 (µAI*I′ ) const, µCK ) const)
(31)
µB(BJJ°))ideal
) RT ln
xBd J xB(BJJ°),d
(µAI ) const and µCK ) const)
(∂µB(BJJ°)/∂µA(BI′J°))nBJ ) -(nA(BI′J°)/nB(BJJ°))µAI′ (µAI*I′ ) const, µCK ) const)
aBJ/aB(BJJ°) ) (aBJ/aB(BJJ°))ideal ) xBJ/xB(BJJ°)
(35)
Because the ideal system B1-B2-...-Bj-C1-C2-...-Ck-A1A2-...-Ai is prepared by mixing its ideal subsystems BJ-C1C2-...-Ck-A1-A2-...Ai (J ) 1, 2, ..., j) of equal mole fractions of all C1, C2, ..., Ck and A1, A2, ..., Ai, the relations xCK ) xC(BKJ°) and xAI ) xA(BIJ°) hold for all C1, C2, ..., Ck and A1, A2, ..., Ai. Therefore,
xB(BJJ°) ) 1 -
i
∑ I)1
xA(BIJ°) -
k
xC(B °) ) ∑ K)1 J
K
1-
A boundary condition is now introduced to solve eq 31, where
∑ I)1
k
xAI -
∑
K)1
j
xCK )
∑xB
J)1
J
j
∑xB /(1 - J)1 ∑ xB ) f 0
J)1
(34)
where superscript d denotes the same infinite dilute conditions as those mentioned in eqs 32 and 33. From eqs 33 and 34, we J°) obtain µBJ - µ(B ) (µBJ - µB(BJJ°))ideal, that is, BJ
i
j
(33)
J
and eq 35 can be rewritten as
J
j
(xBJ ) nBJ/(
∑ J)1
k
nBJ +
∑ K)1
i
nCK +
nA )) ∑ I)1 I
for the system B1-B2-...-Bj-C1-C2-...-Ck-A1-A2-...Ai and
xB(BJJ°)/(1 - xB(BJJ°)) f 0 (xB(BJJ°) ) nB(BJJ°)/(nB(BJJ°) +
k
i
nA(B °))) ∑ nC(B °) + ∑ K)1 I)1 J
K
J
I
for its subsystems BJ-C1-C2-...-Ck-A1-A2-...-Ai (J ) 1, 2, ..., j), so that under these conditions
aBd J/aB(BJJ°),d ) xBd J/xB(BJJ°),d (µCK ) µCd K, µAI ) µAd I)
(32)
where the superscript d denotes the infinite dilute behavior mentioned above. Integration of eq 31 from µAd I′ to µAI′ gives
aBJ/aB(BJJ°) ) xBJ/
j
∑xB J)1
(36)
J
Note that eq 36 reduces to the equations obtained by Stokes and Robinson6 for aqueous solutions of nonelectrolytes B1B2-...-Bj-A1(H2O) at constant activity of A1, namely,
aBJ/aB(BJJ°) ) mBJ/
j
∑mB J)1
J
(BJ°) (BJ°) (BJ°) J°) Now let ∆mixG(B BJ , ∆mixHBJ , ∆mixSBJ , and ∆mixVBJ denote the changes of Gibbs free energy, enthalpy, entropy, and volume accompanying the process of preparing the subsystems BJ-C1-C2-...-Ck-A1-A2-...Ai (J ) 1, 2, ..., j). Let ∆mixG, ∆mixH, ∆mixS, and ∆mixV represent the corresponding properties accompanying preparing the system B1-B2-...-Bj-C1-C2...-Ck-A1-A2-...-Ai having the same activities of all C1, C2, ..., Ck and A1, A2, ..., Ai as those of the subsystems BJ-C1C2-...-Ck-A1-A2-...-Ai (J ) 1, 2, ..., j). Then, we obtain
13172 J. Phys. Chem. B, Vol. 107, No. 47, 2003 i
∆mixG ) RT
Hu
j
I
I
∑
∆mixS )
J
J)1n(BJ°) BJ
k
ln aBJ + RT ∆mixGB(BJJ°) ) RT
i
∑ I)1
k
∑
nA(BIJ°) ln aA(BIJ°) + RT
nBJ
j
nA ln aA + RT∑nB × ∑ I)1 J)1
K)1
∑ nC K)1
I
I
J
J
J
k
∑ nC (Hh C K
K)1
∆mixHB(BJJ°)
i
)
∑ I)1
- HCK) (39)
nB(BJJ°)(H h BJ
- HAI) +
∑
K)1 i
nC(BKJ°)(H h CK - HCK) (40)
I
I
J
J
J
k
∑ nC (Vh C K)1 K
K
- VCK) (41)
i
nA(B °)(V h A - VA ) + nB(B °)(V h B - VB ) + ∑ I)1 J
I
I
J
k
∑ K)1
J
∑ J)1
(BJ°) cP,B J
j
∑
nBJ
nC(BKJ°)(V h CK - VCK) (42)
∆mixGB(BJJ°) + RT
J)1n(BJ°) BJ
xB J
j
∑
J)1
nBJ ln
∑i niMi)/dsolution
∆mixH )
∑ J)1
nBJ
∆mixHB(BJJ°)
nB(BJJ°)
where i denotes the solutes, we can express ∆mixV and J°) ∆mixV(B as BJ j k 1 i ∆mixV ) ( nAIMAI + nBJMBJ + nCKMCK) d I)1 J)1 K)1 i nA M A j nB M B k nC MC I I J J K K (48) I)1 dA J)1 dB K)1 dC I J K
∑
∑
∑
∆mixVB(BJJ°) )
∑
∑
∑ xB J)1
i
1
k
∑nA(B °)MA + nB(B °)MB + K)1 ∑ nC(B °)MC ) (B °) I)1
dBJ
J
(
∑ I)1
J
I
J
i
I
nA(BIJ°)MAI
J
J
-
J
nB(BJJ°)MBJ
dA I
-
dBJ
K
K
k
nC(BKJ°)MCK
K)1
dCK
∑
(49)
J°) where dAI, dBJ, dCK, d, and d(B denote the densities of AI, the BJ solutes BJ and CK, the system B1-B2-...-Bj-C1-C2-...-CkA1-A2-...-Ai, and its subsystems BJ-C1-C2-...-Ck-A1A2-...Ai (J ) 1, 2, ..., j) of equal activities of all C1, C2, ..., Ck and A1, A2, ..., Ai, respectively. M is molar mass. From eqs 45, 48, and 49, there follows
(43)
j
i
∑ J)1
J
j
(47)
nB(BJJ°)
J
where H and H h denote molar enthalpy and the partial molar enthalpy of pure component and V and V h denote molar volume and partial molar volume of pure component, respectively. Combining of eqs 24 and 25 with eqs 36-38, 39, and 40 and with eqs 41 and 42, respectively, and remembering the fact that aAI ) aA(BIJ°) for all A1, A2, ..., Ai and aCK ) aC(BKJ°) for all C1, C2, ..., Ck, we obtain
∆mixG )
J
and
J
I
∑xB
J°) where cP and c(B P,BJ are the specific heat capacities. With the help of
∑
j
I
)
cP )
- HBJ) +
nA (V h A - VA ) + ∑nB (V h B - VB ) + ∑ I)1 J)1
∆mixVB(BJJ°)
nBJ
j
Vsolution ) (nsolventMsolvent + nA(BIJ°)(H h AI
k
∆mixV )
K
(46)
j
J)1
j
I
∑
J)1
nBJ ln
Differentiation of eq 44 with respective to temperature gives the following linear relation
nC(BKJ°) ln aC(BKJ°) +
nA (H h A - HA ) + ∑nB (H h B - HB ) + ∑ I)1 J)1
∆mixH )
xB J
j
ln aCK (37) K
RTnB(BJJ°) ln aB(BJJ°) (38) i
∆mixSB(BJJ°) + R
(
nBJ
i
∑
nB(BJJ°) I)1
nA(BIJ°)MAI + nBJMBJ + i
(
(44)
nB J
k
∑ nC(B °)MC J
nB(BJJ°)K)1
j
K
K
k
)
/dB(BJJ°) )
nA MA + ∑nB MB + ∑ nC MC )/d ∑ I)1 J)1 K)1 I
I
J
J
K
K
(50)
If we define j
∆mixV )
∑ J)1
nBJ
∆mixVB(BJJ°)
nB(BJJ°)
Combination of eqs 43 and 44 yields
(45)
YB J )
since
nB J
i
∑
nB(BJJ°) I)1
nA(BIJ°)MAI +
nBJ
k
∑ nC(B °)MC J
nB(BJJ°)K)1
K
K
+ nBJMBJ
Thermodynamics of Nonelectrolyte Systems i
j
k
nA MA + ∑nB MB + ∑ nC MC ∑ I)1 J)1 K)1 i
I
I
j
nBJ
∑ ∑ I)1 J)1
J
nB(BJJ°) j
J
K
nA(BIJ°)MAI +
∑ J)1
(
J. Phys. Chem. B, Vol. 107, No. 47, 2003 13173
nBJ
j
∑
J)1
TABLE 1: Isopiestic Results for the Saturated Systems B1-B2-C1-A1(H2O) at 298.15 K Taking Aqueous NaCl as the Reference Solution
) K k
nBJMBJ +
j
∑∑
nBJ
K ) 1J)1n(BJ°) BJ
i
∑nA(B °)MA + nB MB +
nBJ
J
nB(BJJ°) I)1
I
I
J
J
nC(BKJ°)MCK ) k
∑ nC(B °)MC J
nB(BJJ°)K)1
K
)
K
then eq 50 can be rearranged into j
d)
j
(YB /dB(B °)) ∑ ∑ J)1 J)1 J
YBJ/
J
J
(51)
3. Comparisons with Isopiestic Measurements For the saturated solution B1-B2-...-Bj-C1-A1(H2O), eqs 24 and 25 can be written as the following linear relations (note (BJ°) (BJ°) (BJ°) J°) that nC(B1J°)/nB(BJJ°) ) (nC(B1J°)/wA(B1J°))/(n(B BJ /wA1 ) ) mC1 /mBJ ):
mBJ
j
∑
J)1m(BJ°) BJ
j
∑ J)1
)1
( ) mBJ
mB(BJJ°)
mC(B1J°) ) mC1
(52)
(53)
at constant activities of A1(H2O) and C1 and within the range 0 J°) e [mBJ/m(B BJ ] e 1, where m is molality. In this study, the experimental deviations from eqs 52 and 53 are defined by j
∆0 )
j
∆1 )
∑ J)1
∑
mBJ
-1
(54)
mC(B1J°) - mCexptl 1
(55)
J)1m(BJ°) BJ
( ) mBJ
mB(BJJ°)
at constant activities of water and C1 and within the range 0 e J°) [mBJ/m(B BJ ] e 1. Table 1 shows the isopiestic results of the saturated systems glucose-sorbitol-sucrose(sat)-water, glucose-mannitol-sucrose(sat)-water, sorbitol-glucose-mannitol(sat)-water, sorbitol-sucrose-mannitol(sat)-water, and mannitol-sorbitolsucrose(sat)-water at 298.15 K. It is clear that |∆0| e 0.0012 and |∆1| e 0.0036 for all of these systems, indicating that eqs 52 and 53 are in good agreement with the experimental results. The unsaturated systems mannitol-sorbitol-water, mannitol-sucrose-water, sorbitol-sucrose-water, glucose-sorbitol-water, glucose-mannitol-water, and glucose-sucrosewater obey eq 1. These indicate that the solute-solute interactions in the above systems and those in the systems shown in Table 1 are all negligible compared to the solute-solvent interactions.6 Therefore, the results reported in Table 1 confirm that if the interactions between the unsaturated solutes (B-B interactions) are negligible, then the saturated systems obey eqs 52 and 53. The water activities of the saturated systems reported in Table 1 are all less than those of the binary saturated solutions C1A1(H2O). Therefore, we calculate mCsupersat by using eq 56 1
mC1 mref
a
mB1
mB2
exptl
∆0
calcd
∆1
sucrose(B1)-sorbitol(B2)-mannitol(sat)(C1)-H2O(A1) 1.8082 2.0869 0 0.9482 0 2.4090 1.0860 0.3682 1.9855 1.0628 1.0624 0.0006 -0.0004 0.7537 1.5414 1.0386 1.0372 0.0010 -0.0014 1.3136 0.8918 1.0009 0.9989 -0.0004 -0.0020 1.6392 0.5185 0.9776 0.9786 0.0007 0.0010 1.9056 2.1786 2.1486 1.1932 0.9593 0 1.0754 0 1.0293 1.1302 0.2253 0.7876 1.1178 1.1170 0.0000 -0.0008 0.4609 0.5343 1.1034 1.1018 -0.0005 -0.0016 0.5611 0.4272 1.0960 1.0981 -0.0001 0.0021 0.7568 0.2176 1.0871 1.0856 0.0003 -0.0015 sorbitol(B1)-glucose(B2)-mannitol(sat)(C1)-H2O(A1) 0.9888 0.6550 0 1.1546 0 0.6520 1.1520 0.1386 0.5136 1.1542 1.1518 -0.0007 -0.0024 0.2784 0.3752 1.1522 1.1536 0.0005 0.0014 0.4276 0.2269 1.1548 1.1540 0.0008 -0.0008 0.5825 0.0726 1.1534 1.1550 0.0006 0.0016 1.6692 1.8990 0 1.0850 0 1.8562 1.0912 0.3785 1.4869 1.0912 1.0904 0.0003 -0.0008 0.7150 1.1591 1.0875 1.0899 0.0009 0.0014 0.9909 0.8885 1.0847 1.0868 0.0006 0.0021 1.5332 0.3565 1.0836 1.0856 -0.0005 0.0020
mCsupersat 1
4.0172 3.9474 3.8836 3.8596
2.1486 2.1515 2.1506 2.1194
1.8219 1.8184 1.8182 1.8122 2.8842 2.8786 2.8689 2.8628
glucose(B1)-sorbitol(B2)-sucrose(sat)(C1)-H2O(A1) 4.4429 1.0032 0 5.8242 0 1.0150 5.8201 0.2556 0.7560 5.8225 5.8189 -0.0004 -0.0036 0.4880 0.5219 5.8267 5.8258 0.0006 -0.0009 0.6589 0.3480 5.8237 5.8207 -0.0003 -0.0030 0.8048 0.2014 5.8252 5.8272 0.0006 0.0020 4.8508 2.3620 0 5.5460 0 2.4252 5.5209 0.5855 1.8252 5.5324 5.5298 0.0005 -0.0026 1.0087 1.3916 5.5330 5.5365 0.0009 0.0035 1.4408 0.9474 5.5422 5.5397 0.0006 -0.0025 1.9189 0.4542 5.5413 5.5396 -0.0003 -0.0017
7.3996 7.3852 7.3806 7.3595
mannitol(B1)-sorbitol(B2)-sucrose(sat)(C1)-H2O(A1) 4.2712 0.5036 0 5.9427 0 0.5029 5.9365 0.1017 0.4012 5.9337 5.9361 -0.0003 0.0024 0.2036 0.2999 5.9396 5.9427 0.0006 0.0031 0.3405 0.1626 5.9402 5.9386 -0.0005 -0.0016 0.4078 0.0957 5.9425 5.9419 0.0001 -0.0006
b b b b
mannitol(B1)-glucose(B2)-sucrose(sat)(C1)-H2O(A1) 4.2348 0.5649 0 5.9268 0 0.5607 5.9212 0.1808 0.3814 5.9230 5.9246 0.0003 0.0016 0.2851 0.2782 5.9259 5.9281 0.0006 -0.0022 0.3870 0.1768 5.9301 5.9276 0.0004 -0.0025 0.4276 0.1360 5.9243 5.9225 -0.0005 0.0021
b b b b
a
Molality (mol kg-1). b Not calculated.
(
mCsupersat ) mC1/ 1 1
mB1 m°B1
-
)
mB 2 m°B2
6.5720 6.5972 6.5948 6.5978
(56)
is the molality of super-saturated solution where mCsupersat 1 C1(supersat)-A1(H2O)6 and m°B1 and m°B2 are the molalities of the unsaturated binary solutions B1-A1(H2O) and B2-A1(H2O), which are in isopiestic equilibrium with the saturated solution B1-B2-C1-A1(H2O). Because eq 1 is based on the fact that the average hydration number hhBJ-A1 (J ) 1, 2) is the same in
13174 J. Phys. Chem. B, Vol. 107, No. 47, 2003
Hu
TABLE 2: Isopiestic Results for the Saturated Systems Sucrose(B1)-Mannitol(B2)-Urea(sat)(C1)-H2O(A1) and Sucrose(B1)-Urea (B2)-Mannitol(sat)(C1)-H2O(A1) at 298.15 K Taking Aqueous CaCl2 and Aqueous NaCl as the Reference Solutions, Respectively
aBJ/aB(BJJ°) ) mBJ/
3.0276a
1.7296
a
mB1
mB2
exptl
calcd
∆0
∆1
sucrose(B1)-mannitol(B2)-urea(sat)(C1)-H2O(A1) 0.8260 0 20.4256 0 0.9428 20.2189 0.6591 0.1901 20.3776 20.3751 -0.0004 -0.0025 0.5568 0.3067 20.3450 20.3462 -0.0006 0.0012 0.4026 0.4835 20.3203 20.3246 0.0002 0.0043 0.2129 0.6989 20.2508 20.2530 -0.0009 0.0022 sucrose(B1)-urea(B2)-mannitol(sat)(C1)-H2O(A1) 1.9608 0 0.9685 0 2.3260 1.2206 1.6215 0.4237 1.0087 1.0229 0.0091 1.2689 0.8603 1.0582 1.0779 0.0170 0.9426 1.2588 1.1029 1.1259 0.0219 0.4366 1.8490 1.1665 1.1858 0.0176
YBJ )
nBJ
i
∑
nB(BJJ°) I)1
j
∆mixG ) 0.0142 0.0197 0.0230 0.0192
the mixed solution as in its binary subsystems under isopiestic equilibrium6 but the average hydration number of the saturated solute (hhC1-A1) is different from that (hhBJ-A1 (J ) 1, 2)) of the unsaturated solute, the isopiestic behavior of the saturated systems B1-B2-C1-A1(H2O) strictly do not obey eq 1, that ) 1. It can be seen from is, mB1/m°B1 + mB2/m°B2 + mC1/mCsupersat 1 at a given water the eighth column that the values of mCsupersat 1 activity are indeed not constant. It has been found6 that the system sucrose-urea-water shows large deviation from eq 1 due to very strong solute-solute interactions in binary solution urea-water.18 To study the effect of the C-C and C-A interactions on the isopiestic behavior of mixed saturated solutions, we also measured the isopiestic behavior of the saturated solutions sucrose-urea-mannitol(sat)water and sucrose-mannitol-urea(sat)-water, and the results are shown in Table 2. It is clear that the system sucrose-ureamannitol(sat)-water shows large deviations from eqs 52 and 53 with 0.0091 e |∆0| e 0.0219 and 0.0142 e |∆1| e 0.0230. However, the system sucrose-mannitol-urea(sat)-water conforms to eqs 52 and 53 very well with 0.0002 e |∆0| e 0.0009 and 0.0012 e |∆1| e 0.0043. These results, together with those reported in Table 1, suggest that the effects of the saturated and unsaturated solutes are different. For eqs 52 and 53 to hold, it is necessary that the interactions between the unsaturated solutes are negligible. However, the C1-C1, C1-A1, and A1A1 interactions do not affect the linear behavior symbolized by eqs 52 and 53.
nA(BIJ°)MAi +
∑
mBJ
J)1m(BJ°) BJ
∆mixGB(BJJ°)
j
∆mixS )
∑
mBJ
J)1m(BJ°) BJ
( ) mB(BJJ°)
mC(BKJ°) ) mCK
(K ) 1, 2, ..., k)
(57)
∑ nC(B °)MC
+ nBJMBJ
J
nB(BJJ°)K)1
K
K
mBJ
j
+ RT
∑mB ln J
J)1
∆mixSB(BJJ°) + R
YBJ )
,
j
∑mB
mB J
j
∑
J)1
J
mBJ ln
, and
j
∑mB
J
mBJ mB(BJJ°)
+
mBJ
k
∑ mC(B °)MC J
mB(BJJ°)K)1
K
K
+ mBJMBJ
respectively. Note that under this condition eqs 43-46 refer to the changes in the Gibbs free energies, enthalpy, entropy, and volume of mixing per kilogram of solvent (A1), respectively. It is now clear that as i ) 1 and k ) 0, eq 24 reduces to eq 1 and the linear solubility relation for carbon in ternary alloys at constant activity of carbon19-21 and eqs 36, 43-47, and 51 reduce to the simple predictive equations for activities of all B1, B2, ..., Bj, the Gibbs free energy, thermal properties, and volumetric properties of unsaturated solution B1-B2-...-BjA1(H2O) from the properties of its subsystems BJ-A1(H2O) (J ) 1, 2, ..., j) of equal water activities.12-15 Note that eqs 24 and 25 can also be incorporated into other thermodynamic relations or the transport theories to develop simple predictive equations for other thermodynamic properties such as depression in freezing points and transport properties such as viscosity. In this study, we call the linear relation symbolized by eqs 24, 25, and the related simple equations 31, 36, 43-47, and 51 the iso-a relations. For the joint solubilities of all C1, C2, ..., Ck in the system C1-C2-...-Ck-A1-A2-...-Ai in terms of those in the systems C1-C2-...-Ck-AI (I ) 1, 2, ..., i), eq 25 is equivalent to i
∑ J)1
k
J)1
yCK )
Equations 24, 25, 36, 43-47, and 51 are the novel linear concentration relations and the simple equations for the thermodynamic properties of the multicomponent system B1-B2...-Bj-C1-C2-...-Ck-A1-A2-...-Ai in terms of the concentrations and properties of its subsystems BJ-C1-C2-...Ck-A1-A2-...Ai (J ) 1, 2, ..., j) of equal activities of all C1, C2, ..., Ck and A1, A2, ..., Ai. For the thermodynamic behavior of aqueous saturated solutions B1-B2-...-Bj-C1-C2-...-CkA1(H2O), eqs 24, 25, and 36 are equivalent to eq 52 and
nBJ
J)1
4. Discussion
mBJ
(58)
in eq 51 are reduced to
Molality of aqueous CaCl2.
j
(J ) 1, 2, ..., j)
J
J°) J°) in eqs 43-47 can be replaced by mBJ/m(B While nBJ/n(B BJ BJ , eqs 43 and 46 and
mC1 mref
j
∑mB J)1
nA I
∑ i I)1 nA ∑ I)1
yC(AKI°) ) I
i
x AI
yC(A °) ∑ i I)1 xA ∑ I)1 I
K
(59)
I
with i
yCsatK ) nCK/
nA ∑ I)1
I
I°) and yCsat(A ) nC(AKI°)/nA(AI I°) K
where the superscript (AI°) together with the subscript CK denotes the quantity of the CK component in the system C1C2-...-Ck-AI (I ) 1, 2, ..., i). Furthermore, if the pressure is not too high, then eq 36 gives the simple relation for the pressure of the AIth solvent
Thermodynamics of Nonelectrolyte Systems
P AI )
xAI i
xA ∑ I)1
J. Phys. Chem. B, Vol. 107, No. 47, 2003 13175
PA(AI I°)
(60)
I
For the mixed solvent/solute systems B1-B2-...-Bj-A1-A2...-Ai, eq 24 can be rewritten as j
xBJ
xA(B °) ) xA ∑ (B °) J)1
(I ) 1, 2, ..., i)
J
xBJ
J
I
I
(0 e [xBJ/xB(BJJ°)] e 1, aAI ) const (I ) 1, 2, ..., i)) (61) J°) And nBJ and nBJ/n(B BJ in eqs 43-47 and 51 can be replaced by
J°) xBj and xBJ/x(B BJ , respectively. Note that under this condition eqs 43-46 refer to the changes in the Gibbs free energies, enthalpy, entropy, and volume accompanying the process of mixing per total mole of (solvent + solute), respectively. Now let us consider the case where the multicomponent system B1-B2-...-Bj-C1-C2-...-Ck-A1-A2-...-Ai and its subsystems BJ-C1-C2-...-Ck-A1-A2-...Ai (J ) 1, 2, ..., j) of equal activities of all C1, C2, ..., Ck and A1, A2, ..., Ai are all ideal solutions. Under this condition, it can be shown that the linear relations symbolized by eqs 24 and 25 also hold and that the simple equations 31, 36, 43-47, and 51 also relate the thermodynamic properties of this ideal multicomponent solution B1-B2-...-Bj-C1-C2-...-Ck-A1-A2-...-Ai to those of its ideal subsystems BJ-C1-C2-...-Ck-A1-A2-...Ai (J ) 1, 2, ..., j) of equal activities of all C1, C2, ..., Ck and A1, A2, ..., Ai (i.e., equal mole fractions of all C1, C2, ..., Ck and A1, A2, ..., Ai, xCK and xAI). For convenience here, we call these equations the iso-x relations. For the iso-a relations to hold, it is necessary that the interactions between all B1, B2, ..., Bj solutes be negligible. However, the BJ-AI, CK-CK, and CK-AJ interactions are permitted. Furthermore, it can be seen from the iso-a relations that if all B1, B2, ..., Bj components are sufficiently alike in size and shape and have negligible interchange energy, then there is no excess entropy and excess enthalpy accompanying the process of mixing the nonideal systems BJ-C1-C2-...Ck-A1-A2-...-Ai (J ) 1, 2, ..., j) of equal activities of all C1, C2, ..., Ck and A1, A2, ..., Ai. That is, the mixing process is a process of completely random distributing of molecules of all B1, B2, ..., Bj, resulting in only ideal increase in entropy
nBJ
j
∆mixG ) RT
∑
J)1
nBJ ln
j
∑ nB J)1
J
Because the process of forming the ideal solution is essentially the process of completely random distribution of and zero interchange energies among its constituent components, the process of mixing the systems BJ-C1-C2-...-Ck-A1-A2...Ai (J ) 1, 2, ..., j) of equal activities of all C1, C2, ..., Ck and A1, A2, ..., Ai is an ideal mixing process or at least as simple as that of an ideal mixing process. These comparisons, together with the fact that the iso-a relations are equivalent to the iso-x relations, show that while the classical simple rule such as Raoult’s law describes a kind of ideal behavior, to which many mixtures of very similar components may conform, the present simple relations describe another type of “ideal” behavior of
the mixtures based on similar components B1, B2, ..., Bj and isoactive components C1, C2, ..., Ck and A1, A2, ..., Ai under the special conditions of constant activities of all C1, C2, ..., Ck and A1, A2, ..., Ai. The nonideality of the multicomponent system B1-B2-...-Bj-C1-C2-...-Ck-A1-A2-...-Ai resulting from the addition of isoactive components C1, C2, ..., Ck and A1, A2, ..., Ai has been adequately accounted for in terms of those of its subsystems BJ-C1-C2-...-Ck-A1-A2-...-Ai (J ) 1, 2, ..., j) of equal activities of all C1, C2, ..., Ck and A1, A2, ..., Ai. It is important to note that, although eq 1 can be derived strictly only by using the semi-ideal hydration model for nonelectrolyte mixtures, it is a general linear relation for different kinds of mixtures including aqueous and nonaqueous solutions of electrolytes, nonelectrolytes, (electrolyte + nonelectrolyte) mixtures, aqueous solutions of polyelectrolytes, and watersoluble neutral polymers,22 alloys, and so on. Similarly, the present the iso-a relations are expected to hold for all mixtures mentioned above. The comparisons of these iso-a relations with isopiestic measurements of the solutions NaCl-BaCl2(sat)LaCl3-ThCl4-water and HCl-BaCl2-LaCl3-ThCl4-water at constant aHCl and aH2O will be given elsewhere. 5. Conclusions The thermodynamic behavior of the multicomponent solutions B1-B2-...-Bj-C1-C2-...-Ck-A1-A2-...-Ai at constant activities of all C1, C2, ..., Ck and A1, A2, ..., Ai has been studied by using the semi-ideal hydration model. It has been shown that the average hydration number hhC1-A1 of a saturated solute depends only on water activity and is different from that of an unsaturated solute at a given water activity, and thus, the linear isopiestic relation symbolized by eq 1 and the related simple predictive equations for the thermodynamic and transport properties are strictly not applicable to the mixtures of which the water activities are less than those of their binary saturated subsystems. The novel linear concentration relations and the simple equations for the thermodynamic properties have been proposed for the multicomponent system B1-B2-...-Bj-C1-C2-...Ck-A1-A2-...-Ai in terms of the concentrations and properties of its subsystems BJ-C1-C2-...-Ck-A1-A2-...Ai (J ) 1, 2, ..., j) of equal activities of all C1, C2, ..., Ck and A1, A2, ..., Ai. The thermodynamic properties being studied include chemical potential, activities of all B1, B2, ..., Bj, Gibbs free energy, thermal properties, and volumetric properties. The process of mixing the nonideal mixtures BJ-C1-C2...-Ck-A1-A2-...-Ai (J ) 1, 2, ..., j) of equal activities of all C1, C2, ..., Ck and A1, A2, ..., Ai is as simple as that of mixing the ideal mixtures BJ-C1-C2-...-Ck-A1-A2-...-Ai (J ) 1, 2, ..., j) of equal mole fractions of all C1, C2, ‚‚‚, Ck and A1, A2, ..., Ai, and the iso-a relations are the same in forms as the iso-x relations. While the classical ideal solution model describes a kind of ideal behavior of many mixtures of very similar components, the present iso-a relations describe another type of “ideal” behavior of the mixtures based on similar components B1, B2, ..., Bj and isoactive components C1, C2, ..., Ck and A1, A2, ..., Ai. The effects of addition of isoactive components C1, C2, ..., Ck and A1, A2, ..., Ai on the thermodynamic behavior of the system B1-B2-...-Bj-C1-C2-...-Ck-A1-A2-...-Ai have been adequately accounted for in terms of those of the subsystems BJ-C1-C2-...-Ck-A1-A2-...Ai (J ) 1, 2, ..., j) of equal activities of all C1, C2, ..., Ck and A1, A2, ..., Ai. The linear concentration relations have been tested by comparing with the isopiestic measurements for the saturated
13176 J. Phys. Chem. B, Vol. 107, No. 47, 2003 systems glucose-sorbitol-sucrose(sat)-water, glucose-mannitol-sucrose(sat)-water, sorbitol-glucose-mannitol(sat)-water, sorbitol-sucrose-mannitol(sat)-water mannitol-sorbitol-sucrose(sat)-water, mannitol-sucrose-urea(sat)-water, and sucroseurea-mannitol(sat)-water at 298.15 K. The relations are in excellent agreement with the experimental results for the former six systems, but the large deviation is observed in the last system. These comparisons confirm that for the iso-a relations to hold it is necessary that the interactions between all B1, B2, ..., Bj solutes be negligible but the B-A, C-A, C-C, and A-A interactions are permitted. From the present iso-a relations, the linear solubility equation for carbon in ternary alloys, Zdanvoskii’s rule for mixed electrolyte solutions, and the linear isopiestic relation for mixed nonelectrolyte solutions, mixed solutions based on electrolytes and nonelectrolytes, and aqueous solutions of polyelectrolytes and water-soluble neutral polymers can be reproduced. Therefore, the present the iso-a relations are expected to hold for all mixtures mentioned above. 6. Experimental Section All of the chemicals used in this study were reagent grade and were recrystallized twice from doubly distilled water. NaCl was dried under vacuum over CaCl2 for 7 days at 423 K. Mannitol, urea, sorbitol, glucose, and sucrose were dehydrated at room temperature under vacuum over CaCl2 to constant weight. Finally, all of the chemicals were stored over P2O5 in a vacuum desiccator before use. The molalities of the NaCl and CaCl2 reference stock solutions were determined by the titration of Cl- with AgNO3. The isopiestic apparatus and the sample cups used in this study are newly constructed and are the same as that used in our previous studies.23 The apparatus was constructed out of copper, and 14 gold-plated silver cups placed in it were 15 mm in diameter and 25 mm in depth. The experimental procedure is similar to those used previously23 and can be described briefly as follows. The following procedure was used to prepare the samples for easy equilibration. First, the binary aqueous solutions of mannitol, urea, sorbitol, glucose, or sucrose were made with nearly the same water activity by direct weighing of water and each of these solutes. Known masses of solid C1 were added until the dissolution rate of the crystals became very slow. Sufficient time was then allowed for the remaining crystals to dissolve to produce nearly saturated solutions. Next, each of the quaternary samples was prepared by mixing these unsaturated ternaries. The reference NaCl or CaCl2 solutions were then prepared to have nearly the same water activities as those of the ternary unsaturated samples by the additivity rule for vapor pressure lowering of ternary aqueous solutions.24 For each equilibration, there were duplicate cups of the reference solution and of the samples investigated. Before each run, an appropriate amount of anhydrous solid mannitol, urea, or sucrose and 2.0-2.5 g of the initial unsaturated solution were weighed into each sample cup to ensure that there was just a small amount of solid at isopiestic equilibrium (to allow a glass ball to roll freely in the sample cup). Preliminary experiments showed that equilibration could be achieved within 6-8 days. The real equilibration period for each run was then chosen as 8 days. Our sample weights are always corrected for buoyancy. After evacuation and degassing through two large vessels placed between the air pump and the apparatus, the apparatus was placed into a thermostat controlled at 298.15 ( 0.001 K. These
Hu physically isolated solutions of three or more components of known initial concentrations and known initial masses were then allowed to equilibrate isothermally by transferring water through a common vapor phase and dissolving some of the solid mannitol, urea, or sucrose in the sample cups. During the whole experimental process, the apparatus was rocked once every 5 s in the thermostat. At equilibrium, a lid was slowly lowered and pressed on to all of the cups within the apparatus. The apparatus was opened and then replaced to the bracket in the same thermostat to avoid possible concentration change in the equilibrium saturated solutions due to the changes in temperatures. At isopiestic equilibrium, all of the test solutions have the same water activity and all of the saturated solutions have the same activities of water and saturated solutes. The molalities of the unsaturated solutes were determined by weighing the sample at isopiestic equilibrium.5,6,10,23 The present saturated and unsaturated solutes contain several vicinal glycol units, and thus, all are capable of being oxidized by periodate to titratable formic acid. Moreover, the solutes containing cis hydroxyl groups are oxidized relatively quickly because of the formation of cyclic ester intermediates.25,26 Therefore, the rates of formation of formic acid from these solutes are faster than those from the solutes containing trans hydroxyl groups.25-27 All of these indicate that we can determine the solubilities of the present saturated solutes by using the following procedure based on the periodate method of oxidation presented by Halsall et al.27 (1) After the equilibrium saturated samples were weighed, the saturated sample of the liquid phase was withdrawn by a pipet fitted with a sintered-glass filter tip preheated slightly above 298.15 K28 and then was weighed and diluted. At the same time, ) of all of the the binary solutions of known molalities (mBbinary J(exptl) solutes were made up by weight and then were diluted. (2) After addition of potassium chloride, all of the above dilute binary and multicomponent solutions were oxidized by sodium periodate for 170 h. (3) All of the solutions were analyzed by titration of liberated formic acid with NaOH after addition of excess of glycol (to destroy the excess of periodate).27 Four to eight titrations were done for a given solution with results agreeing to within (0.05%. (4) Determination of the mole number of formic acid produced by peroxidation of 1 mol of solute i during the given period of oxidation, ki, was accomplished by the relation binary binary ki ) mi(form) /mi(exptl)
(62)
binary denotes the molality of the liberated formic acid where mi(form) in the binary solution of the solute i determined by the titration measurements. (5) The solubilities of mannitol, urea, and sucrose in saturated equilibrium solutions were calculated by eq 63:
mC1 ) nT(1 + mB1MB1 + mB2MB2) - wT(kB1mB1 + kB2mB2)/ (kC1wT - nTMC1) (63) where M and wT represent molar mass and the mass of the saturated equilibrium solution withdrawn by a sintered-glass filter tip prewarmed slightly above 298.15 K and nT denotes the total amount of the liberated formic acid in a mass wT of saturated solution. The values of kB1, kB2, and kC1 during the present period of oxidation are 3.8042, 3.5615, 4.6237, and 0.8425 for mannitol, sorbitol, glucose, and sucrose, respectively. Note that ki ) 0 for urea. The details of derivations of eq 63 are given in the Appendix. The results reported in this study for each sample were the average between the duplicate cups for unsaturated solute and
Thermodynamics of Nonelectrolyte Systems
J. Phys. Chem. B, Vol. 107, No. 47, 2003 13177
TABLE 3: Compositions of the Unsaturated Systems B1-B2-C1-A1(H2O) at 298.15 K Made Up by Weight and Determined by the Periodate Method of Oxidation mC1 system sorbitol(B1)-mannitol(B2)sucrose(C1)-H2O(A1) sorbitol(B1)-glucose(B2)mannitol(C1)-H2O(A1) sorbitol(B1)-glucose(B2)sucrose(C1)-H2O(A1) mannitol(B1)-glucose(B2)sucrose(C1)-H2O(A1)
mB1(exptl) mB2(exptl)
exptl
calcd
2.0282
0.5023
4.2569 4.2552
2.1576
1.5809
0.5836 0.5829
1.8697
1.6736
4.3659 4.3668
0.5812
2.2017
2.6936 2.6922
the mean of four to eight replicate titration measurements for saturated solute in the duplicate cups. The results are reproducible to (0.05% for the molalities of unsaturated solutes and to (0.10% for the molalities of saturated solutes. In the preliminary experiments, the unsaturated solutions B1B2-C1-A1(H2O) of known molalities mB1(exptl), mB2(exptl), and mC1(exptl) were prepared by weight and were analyzed by the above procedure. The molality of C1 (mC1(calcd)) was then calculated from eq 63 by using the values of nT and the above known values of mB1(exptl) and mB2(exptl) and was compared with its value made up by weight (mC1(exptl)). The agreement is good. For illustration, Table 3 shows the systems being used and the values of mB1(exptl), mB2(exptl), mC1(exptl), and mC1(calcd). It is clear that if the values of mB1(exptl) and mB2(exptl) are known, then the value of mC1(exptl) can be well reproduced by the above procedure. At the same time, the solubilities of mannitol in a solution of 2.4090 m sorbitol, 2.0869 m sucrose, or 1.8558 m glucose were determined both by evaporating the equilibrium saturated solutions to dryness28,29 (denoted by EV) and by the above procedure (denoted by OXI). The results are mEV ) 1.0868 and mOXI ) 1.0856, mEV ) 0.9490 and mOXI ) 0.9482, and mEV ) 1.0905 and mOXI ) 1.0916, respectively, indicating that the solubilities obtained from the two methods agree well within the experimental errors. Acknowledgment. The author thanks the Natural Science Foundation of China (Grant Nos. 20276037 and 20006010) for financial support of the theoretical work. Valuable comments from the anonymous referees are also gratefully acknowledged. Appendix Calculation of Solubilities of Mannitol, Sucrose, or Urea in Mixed Aqueous Solutions. Let wT denote the mass of the saturated equilibrium solution withdrawn by a sintered-glass filter tip prewarmed slightly above 298.15 K. Let M and w0 represent molar mass and the mass of water in a mass wT of saturated equilibrium solution. Let the subscripts C1, B1, and B2 denote saturated and unsaturated solutes, respectively. For the mixed equilibrium solutions at molalities mC1 in C1 and mBJ in unsaturated solute BJ, we obtain
wT ) w0(1 + MB1mB1 + MB2mB2 + MC1mC1)
(64)
The total amount of liberated formic acid nT in a mass wT of saturated solution is given by
nT ) w0(kB1mB1 + kB2mB2 + kC1mC1)
(65)
where kB1, kB2, and kC1 are the mole number of formic acid produced by peroxidation of 1 mol of solute during the given period of oxidation (the values of ki during the present period of oxidation are 0, 3.8042, 3.5615, 4.6237, and 0.8425 for urea, mannitol, sorbitol, glucose, and sucrose, respectively). From the above two equations, we obtain
mC1 ) nT(1 + mB1MB1 + mB2MB2) - wT(kB1mB1 + kB2mB2)/ (kC1wT - nTMC1) that is, eq 63. Because the initial masses of solid added are known and the solvent water does not appear in the solid mannitol, sucrose, and urea phase at 298.15 K,28 the values of mB1 and mB2 can be accurately determined simply by weighing the samples at equilibrium.5,6,10,11,22,23 Because the values of nT in a mass wT of equilibrium solution can be determined precisely by titration measurements, the solubility of mannitol or sucrose in the mixed solution is therefore determined accurately from eq 63. References and Notes (1) Scatchard, G. J. Am. Chem. Soc. 1921, 43, 2387. (2) Scatchard, G. J. Am. Chem. Soc. 1921, 43, 2406. (3) Sarolea-Mathot, L. Trans. Faraday Soc. 1953, 49, 8. (4) McGlashan, M. L.; Rastogi, R. P. Trans. Faraday Soc. 1958, 54, 496. (5) Robinson, R. A.; Stokes, R. H. J. Phys. Chem. 1961, 65, 1954. (6) Stokes, R. H.; Robinson, R. A. J. Phys. Chem. 1966, 70, 2126. (7) Stokes, R. H.; Robinson, R. A. J. Am. Chem. Soc. 1948, 70, 1870. (8) Robinson, R. A.; Stoke, R. H. Electrolyte Solutions, 2nd rev. ed.; Butterworth: London, 1965. (9) Zdanovskii, A. B. Tr. Solyanoi Lab. Akad. Nauk SSSR 1936, 6. (10) Robinson, R. A.; Stokes, R. H. J. Phys. Chem. 1962, 66, 506. (11) Robinson, R. A.; Bower, V. E. J. Res. Natl. Bur. Stand. A (Phys. Chem.) 1965, 69A, 19. (12) Hu, Y. F. Phys. Chem. Chem. Phys. 2000, 2, 2380. (13) Hu, Y. F. Bull. Chem. Soc. Jpn. 2001, 74, 47. (14) Hu, Y. F. J. Chem. Soc., Faraday Trans. 1998, 94, 913. (15) Hu, Y. F.; Lee, H. Electrochim. Acta 2003, 48, 1789. (16) Schonert, H. Z. Phys. Chem. N. F. 1986, 150, 163. (17) Gokcen, N. A. J. Phys. Chem. 1960, 64, 401. (18) Stokes, R. H. J. Phys. Chem. 1965, 59, 4012. (19) Turkdogan, E. T.; Leake, L. E. J. Iron Steel Inst. 1955, 179, 39. (20) Smith, R. P. Trans. Am. Inst. Min., Metall. Pet. Eng. 1960, 218, 62. (21) Schenck, H.; Froberg, M. G.; Steinberg, E. Arch. Eisenhuettenwes. 1963, 34, 37. (22) Okubo, T.; Ise, N. J. Phys. Chem. 1970, 74, 4284. (23) Hu, Y. F.; Wang, Z. C. J. Chem. Soc., Faraday Trans. 1998, 94, 3251. (24) Robinson, R. A.; Bower, V. E. J. Res. Natl. Bur. Stand. A (Phys. Chem.) 1965, 69, 365. (25) Loudon, G. M. Organic Chemistry; Addison-Welsey Publishing Co.: San Francisco, CA, 1983. (26) Finar, T. L. Problems and Their Solution in Organic Chemistry; Longman Group Limited: London, 1973; Chapter 19. (27) Halsall, T. G.; Hirst, E. L.; Jones, K. N. J. Chem. Soc. 1947, Part II, 1427. (28) Kelly, F. J.; Robinson, R. A.; Stokes, R. H. J. Phys. Chem. 1961, 65, 1958. (29) Bower, V. E.; Robinson, R. A. J. Phys. Chem. 1963, 67, 1524.
