Liquid-State Physical Chemistry: Fundamentals, Modeling, and Applications (2013)
11. Mixing Liquids: Molecular Solutions
11.2. Ideal and Real Solutions
Perfect solutions behave as perfect gas mixtures over the complete concentration range. In practice, perfect solution behavior is obeyed only for a limited range of concentrations, and we label these cases as ideal solutions. For perfect and ideal behavior, a number of consequences follow of which we describe Raoult's and Henry's laws. Thereafter, we discuss how to deal with real solutions.
11.2.1 Raoult's and Henry's Laws
For ideal solutions we have (within the range of validity) the expression
(11.20)
while for the perfect gas the expression
(11.21)
holds. Using the equilibrium condition μ(sol) = μ(vap), we easily obtain
(11.22)
Solving for P_{α} leads to
(11.23)
The parameter K_{H,α} is independent of composition. For the solute we have
(11.24)
known as Henry's law (1803!), and stating that the partial pressure P_{2} of solute 2 is proportional to the mole fraction in the liquid phase with a proportionality constant dependent on the difference in chemical potential, that is, in interaction, between solvent 1 and solute 2. For the solvent, the limiting situation is x_{1} = 1 for which . Hence approximately
(11.25)
known as Raoult's law (1886–1887), and stating that the partial pressure P_{1} of solvent 1 is directly proportional to the vapor pressure of the pure solvent. Obviously, for perfect solutions Raoult's law is valid for the complete composition range.
Some data for Henry's constant are given in Table 11.1. Example 11.1 shows how to use these concepts.
Table 11.1 Henry's constant K_{H} for gases dissolved in H_{2}O and C_{6}H_{6} (in brackets) at 25 °C.^{a)}
Gas |
K_{H} (GPa) |
CH_{4} |
4.185 (0.0569) |
C_{2}H_{2} |
0.135 |
C_{2}H_{4} |
1.155 |
C_{2}H_{6} |
3.06 |
Air |
7.295 |
N_{2} |
8.7650 (0.239) |
O_{2} |
44380 |
H_{2} |
7.16 (0.367) |
He |
12.66 |
CO |
5.79 (0.163) |
CO_{2} |
1.670 (0.0144) |
H_{2} S |
0.055 |
a) Data from Refs [21, 22].
Example 11.1
Assuming that sparkling water contains only H_{2}O (1) and CO_{2} (2), we want to determine for a sealed can the compositions of the vapor and liquid phases and the pressure exerted at 10 °C, knowing that Henry's constant for CO_{2} in H_{2}O at 10 °C is about 990 bar. There is only one degree of freedom according to the phase rule. We use the mole fraction x_{2} of CO_{2} in the liquid and assume it is 0.01. We denote the mole fraction in the gas phase by y. For the solute (2) we have Henry's law, while for the solvent (1) Raoult's law applies.
With K_{H,2} = 990 bar and (steam tables at 10 °C or Antoine's equation), P = 9.912 bar. By Raoult's law, . Therefore, y_{2} = 1 − y_{1} = 0.9988. As expected, the vapor is nearly pure CO_{2}.
11.2.2 Deviations
To describe deviations from perfect solution behavior we have a few options. The most well-known option is introducing the activity a = γ x, where γ is the activity coefficient. The activity coefficient γ_{α} = γ_{α}(T,x) is introduced “to keep up appearances,” that is, to keep the same formal expression for μ_{α} as for the perfect solution. Hence, becomes , with activity a_{α} = γ_{α}x_{α}. Obviously, γ_{ideal} = 1 always. There are two conventions for the activity coefficient.
· Convention I: γ_{α} → 1 for x_{α} → 1 and for x_{α} → 0. This convention is usually employed if the (liquid) solute is fully soluble in the solvent. Because, in a dilute solution of component β in α, molecules of type α are mainly surrounded by molecules of type α, component α behaves largely as if the liquid was pure. The reference state is thus the chemical potential of the pure liquid . This convention is also referred to as Raoult's law convention.
· Convention II: γ_{1} → 1 for x_{1} → 1 and γ_{2} → 1 for x_{2} → 0. This convention is usually employed if the (solid or liquid) solute is only partially soluble in the solvent. For the solvent this convention is identical to convention I. Because, in a dilute solution of component 2 in 1, molecules of type 2 do not interact with each other but mainly with molecules of type 1, the activity coefficient approaches a constant value, characteristic for the 1–2 interaction. Since we would like to have γ_{2} → 1 for x_{2} → 0, the reference state for the solute is taken as that of the pure solute in a hypothetical liquid state extrapolated to infinite dilution using Henry's law. Accordingly, this convention is referred to as Henry's law convention.
From the Gibbs–Duhem equation x_{1}dμ_{1} + x_{2}dμ_{2} = 0, we obtain (using μ = μ* + RT lnγ x) x_{1} d lnγ_{1} + x_{2} d lnγ_{2} = 0. Similarly, as for Eq. (11.7), some manipulation results in
(11.26)
where γ_{1,0} is the limiting value of γ_{1} for x_{1} → 0. Since for x_{1} → 0, γ_{1} → 1, the lower limit in the second integral should be taken as zero. This relation provides a check on the experimental data for γ_{1} and γ_{2} or allows one to calculate γ_{2} if γ_{1} is known.
Another way to introduce deviations from ideal behavior is by introducing the osmotic coefficients g and ϕ, defined for the solvent by
(11.27)
where the molality m_{2} = x_{2}/x_{1}M_{1} and the mole fraction x_{1} = (1 + m_{2}M_{1})^{−1} are used (why g and ϕ are addressed as osmotic coefficients will become clear in the next section). Since for high dilution we should regain ideal behavior, we require that for x_{1} → 1, g → 1, and thus ϕ → 1. We have the relations lnγ_{1} = (g − 1)lnx_{1} and ϕ = −(m_{2}M_{1})^{−1}g ln (1 + m_{2}M_{1})^{−1} ≅ g (1 − ½m_{2}M_{1} + …).
We also can define excess functions by
(11.28)
The partial quantities Z_{α} are directly related to the excess quantities Z^{E}. For example, for G we have Z_{α} = G_{α} = μ_{α} and substitution of Eq. (11.7) in , dividing by x_{1} = 1 − x_{2} and differentiating with respect to x_{2} yields
(11.29)
It is a matter of convenience whether excess or partial quantities are used. Figure 11.3 shows two examples of excess functions.
Figure 11.3 (a) Excess volume V^{E} for a mixture of H_{2}O–C_{2}H_{5}OH; (b) Excess enthalpy H^{E} for a mixture of benzene in cyclohexane. Images obtained from the Dortmund Data Base.
Problem 11.7
Verify Eq. (11.26) and (11.27).