The thermodynamic description of mixtures
3.9 Ideal solutions
Because in biochemistry we are concerned primarily with liquids, we need expressions for the chemical potentials of the substances in a liquid solution.
We can anticipate that the chemical potential of a species ought to increase with concentration, because the higher its concentration, the greater its chemical
“punch.” In the following, we use J to denote a substance in general, A to denote a solvent, and B a solute. This is where we implement the strategy described at the beginning of Section 3.2, to transform equations that work for gases into equations that work for liquids.
The key to setting up an expression for the chemical potential of a solute is the work done by the French chemist François Raoult (1830–1901), who spent most of his life measuring the vapor pressures of solutions. He measured the par- tial vapor pressure,pJ, of each component in the mixture, the partial pressure of the vapor of each component in dynamic equilibrium with the liquid mixture, and established what is now called Raoult’s law:
The partial vapor pressure of a substance in a liquid mixture is proportional to its mole fraction in the mixture and its vapor pressure when pure:
pJxJpJ* (3.14)
In this expression, pJ* is the vapor pressure of the pure substance. For example, when the mole fraction of water in an aqueous solution is 0.90, then, provided Raoult’s law is obeyed, the partial vapor pressure of the water in the solution is 90%
that of pure water. This conclusion is approximately true whatever the identity of the solute and the solvent (Fig. 3.21).
The molecular origin of Raoult’s law is the effect of the solute on the entropy of the solution. In the pure solvent, the molecules have some entropy due to their pB*
pB
pA*
pA
Pressure
p=pA+pB
0 1
Mole fraction of A, xA Fig. 3.21 The partial vapor pressures of the two
components of an ideal binary mixture are proportional to the mole fractions of the
components in the liquid. The total pressure of the vapor is the sum of the two partial vapor pressures.
random motion; the vapor pressure then represents the tendency of the system and its surroundings to reach a higher entropy. When a solute is present, the molecules in the solution are more dispersed than in the pure solvent, so we cannot be sure that a molecule chosen at random will be a solvent molecule (Fig. 3.22). Because the entropy of the solution is higher than that of the pure solvent, the solution has a lower tendency to acquire an even higher entropy by the solvent vaporizing. In other words, the vapor pressure of the solvent in the solution is lower than that of the pure solvent.
A hypothetical solution of a solute B in a solvent A that obeys Raoult’s law throughout the composition range from pure A to pure B is called an ideal solu- tion. The law is most reliable when the components of a mixture have similar mo- lecular shapes and are held together in the liquid by similar types and strengths of intermolecular forces. An example is a mixture of two structurally similar hydro- carbons. A mixture of benzene and methylbenzene (toluene) is a good approxima- tion to an ideal solution, for the partial vapor pressure of each component satisfies Raoult’s law reasonably well throughout the composition range from pure benzene to pure methylbenzene (Fig. 3.23).
No mixture is perfectly ideal, and all real mixtures show deviations from Raoult’s law. However, the deviations are small for the component of the mixture that is in large excess (the solvent) and become smaller as the concentration of solute decreases (Fig. 3.24). We can usually be confident that Raoult’s law is reli- able for the solvent when the solution is very dilute. More formally, Raoult’s law is a limiting law (like the perfect gas law) and is strictly valid only at the limit of zero concentration of solute.
The theoretical importance of Raoult’s law is that, because it relates vapor pres- sure to composition and we know how to relate pressure to chemical potential, we can use the law to relate chemical potential to the composition of a solution. As we show in the following Derivation, the chemical potential of a solvent A present in solution at a mole fraction xAis
AA*RTlnxA (3.15)
whereA* is the chemical potential of pure A.5This expression is valid through- out the concentration range for either component of a binary ideal solution. It is valid for the solvent of a real solution the closer the composition approaches pure solvent (pure A).
(a) (b)
Fig. 3.22 (a) In a pure liquid, we can be confident that any molecule selected from the sample is a solvent molecule. (b) When a solute is present, we cannot be sure that blind selection will give a solvent molecule, so the entropy of the system is greater than in the absence of the solute.
5If the pressure is 1 bar, A* can be identified with the standard chemical potential of A,A両.
Mole fraction of methylbenzene, x(C6H5CH3)
0 1
Pressure,p/Torr
80 60 40 20 0
Total
Methylbenzene Benzene
Fig. 3.23 Two similar substances, in this case benzene and methylbenzene (toluene), behave almost ideally and have vapor pressures that closely resemble those for the ideal case depicted in Fig. 3.21.
Pressure,p/Torr
Total
Propanone Carbon disulfide
Mole fraction of carbon disulfide,
x(CS2)
0 1
500 400 300 200 100 0
Fig. 3.24 Strong deviations from ideality are shown by dissimilar substances, in this case carbon disulfide and acetone (propanone). Note, however, that Raoult’s law is obeyed by propanone when only a small amount of carbon disulfide is present (on the left) and by carbon disulfide when only a small amount of propanone is present (on the right).
DERIVATION 3.5 The chemical potential of a solvent
We have seen that when a liquid A in a mixture is in equilibrium with its va- por at a partial pressure pA, the chemical potentials of the two phases are equal (Fig. 3.25), and we can write A(l)A(g). However, we already have an ex- pression for the chemical potential of a vapor, eqn 3.13, so at equilibrium
A(l)A両(g)RTlnpA
According to Raoult’s law, pAxApA*, so we can use the relation ln x lnyln(x/y) to write
A(l)A両(g)RTlnxApA*A両(g)RTlnpA*RTlnxA
The first two terms on the right, A両(g) and RTlnpA*, are independent of the composition of the mixture. We can write them as the constant A*, the stan- dard chemical potential of pure liquid A. Then eqn 3.15 follows.
Figure 3.26 shows the variation of chemical potential of the solvent predicted by this expression. Note that the chemical potential has its pure value at xA1 (when only A is present). The essential feature of eqn 3.15 is that because xA1 implies that ln xA0, the chemical potential of a solvent is lower in a solution than when it is pure. Provided the solution is almost ideal, a solvent in which a solute is present has less chemical “punch” (including a lower ability to generate a vapor pressure) than when it is pure.
SELF-TEST 3.8 By how much is the chemical potential of benzene reduced at 25°C by a solute that is present at a mole fraction of 0.10?
Answer: 0.26 kJ mol1
Is mixing to form an ideal solution spontaneous? To answer this question, we need to discover whether Gis negative for mixing. The first step is therefore to find an expression for Gwhen two components mix and then to decide whether it is negative. As we see in the following Derivation, when an amount nAof A and nBof B of two gases mingle at a temperature T,
GnRT{xAlnxAxBlnxB} (3.16)
withnnAnBand the xJthe mole fractions in the mixture.
DERIVATION 3.6 The Gibbs energy of mixing
Suppose we have an amount nAof a component A at a certain temperature T and an amount nBof a component B at the same temperature. The two com- ponents are in separate compartments initially. The Gibbs energy of the system (the two unmixed components) is the sum of their individual Gibbs energies:
GinAA*nBB*
where the chemical potentials are those for the two pure components, obtained by the setting the mole fraction to 1 in eqn 3.15. When A and B are mixed, the J(g)
J( l ) Equal J vapor
J liquid
Fig. 3.25 At equilibrium, the chemical potential of a substance in its liquid phase is equal to the chemical potential of the substance in its vapor phase.
−∞
Chemical potential, A
0 A*
Mole fraction of solvent, xA
1
Fig. 3.26 The variation of the chemical potential of the solvent with the composition of the solution.
Note that the chemical potential of the solvent is lower in the mixture than for the pure liquid (for an ideal system). This behavior is likely to be shown by a dilute solution in which the solvent is almost pure (and obeys Raoult’s law).
chemical potentials of A and B fall. Using eqn 3.15, the final Gibbs energy of the system is
GfnAAnBB
nA{A*RTlnxA}nB{B*RTlnxB} nAA*nARTlnxAnBB*nBRTlnxB
where the xJare the mole fractions of the two components in the mixture. The differenceGfGiis the change in Gibbs energy that accompanies mixing. The standard chemical potentials cancel, so
GRT{nAlnxAnBlnxB}
BecausexJnJ/n, we can substitute nAxAnandnBxBninto the expression above and obtain
GnRT{xAlnxAxBlnxB} which is eqn 3.16.
Equation 3.16 tells us the change in Gibbs energy when two components mix at constant temperature and pressure (Fig. 3.27). The crucial feature is that be- causexAandxBare both less than 1, the two logarithms are negative (ln x0 if x1), so G0 at all compositions. Therefore, mixing is spontaneous in all pro- portions. Furthermore, if we compare eqn 3.16 with G HTS, we can con- clude that:
1. Because eqn 3.16 does not have a term that is independent of temperature,
H0 (3.17a)
2. Because G0TSnRT{xAlnxAxBlnxB},
S nR{xAlnxAxBlnxB} (3.17b)
The value of Hindicates that although there are interactions between the molecules, the solute-solute, solvent-solvent, and solute-solvent interactions are all the same, so the solute slips into solution without a change in enthalpy. There is an increase in en- tropy, because the molecules are more dispersed in the mixture than in the unmixed component. The entropy of the surroundings is unchanged because the enthalpy of the system is constant, so no energy escapes as heat into the surroundings. It follows that the increase in entropy of the system is the “driving force” of the mixing.