SOLUTION PROPERTIES

An appreciation for thermodynamics of multicomponent mixtures is extremely important in food systems since we are often interested in separating components, optimizing fragrance of products, or controlling gases or water in foods. In this section, we will consider the behavior of ideal multicomponent systems with ultimate application to real food systems.

A. Partial Molar Quantities

For pure substances, it is acceptable to use the ordinary thermodynamic functions just described. However, for solutions the thermodynamic theory is expressed in terms of partial molar functions. This can best be explained by considering a solution containing n_Amol of A and n_Bmol of B. If we have a very large volume of solution so that adding 1 mol of A or B does not change the concentration appreciably, then we can measure the increase in volume when 1 mol is added at constant T and P. The increase in volume is called the partial molar volume of the component in the solution at the specified temperature, pressure, and composition. It is denoted by the symbol V_Aand is written

VA¼ ›V

›nA

 

T;P;nB

ð33Þ

These partial molar functions are necessary because adding pure components together to form solutions does not result in properties of the solution equal to the sum of the properties in pure state. For example, if 100 cm³ of ethanol are mixed at 258C with 100 cm³of water, the volume is 190 cm³. It can be shown that a property of partial molar functions is that the total value of the function is the sum of the product of moles of the component and its partial

molar function. For example, the total volume of a mixture of n_Amol of A and n_B mol of B is

V ¼ nAVAþ nBVB ð34Þ

Partial molar functions can be defined for all state functions. For example, SA ¼ ›ð ÞS

In addition, all thermodynamic relations that applied to state functions for pure components also apply to partial molar functions. For example,

›GA To illustrate its application, consider the formation of a binary solution from n_A mol of A and nBmol of B:

n_Aþ n_B solution ð37Þ

At a given T and P, the DG, for the solution process is

DG ¼ G_soln2 nAGA8þ n_BGB8 ð38Þ

Where GA8 and GB8 denote the molar free energies of the pure components. By analogy to Eq. (34),

Exactly analogous equations can be written for any extensive thermodynamic state variable (U, H, S, V, A, C_v, C_p, etc.) ultimately defining quantities such as enthalpy of solution, entropy of solution, etc.

B. Chemical Potential

As we have seen, the partial molar Gibbs free energy G_A is equivalent to the chemical potentialmA. Instead of chemical potential, it is often convenient to use a related function, the absolute activity l_A, defined by

mA¼ RT ln lA ð41Þ

Relations expressed for mA also apply to l_A. For example, the condition for equilibrium of component A between gas and liquid phases is

l^g_A¼l¹_A ð42Þ

For solutions, it is convenient also to express the difference between the value of mAin the solution and its value in some reference state. That is,

mA2m^þ_A ¼ RT ln lA

l^þ_A ^{¼ RT ln aA ð43Þ}

where the ratio of absolute activity to the activity in some reference state defines a relative activity a_A.

For nonelectrolyte solutions of liquids, a convenient reference state is the pure liquid at P ¼ 1 atm and the temperature specified for the solution. If the reference state is identified atm⁸_A andl⁸_A, Eq. (43) becomes

Defined in this way, the relative activity is called simply activity. Furthermore, the ratio of the activity to the mole fraction x_Ais called the activity coefficient:

gA¼aA

x_A ð45Þ

where g_Ais the activity coefficient. For an ideal solution g_A¼ 1.0. The activity coefficient for a homologous series of aromatic compounds given in Table 1.1.

C. Ideal Mixtures

An ideal gas is one in which there are no cohesive forces acting on the molecules and hence the internal pressure is zero [(›U/›V)T¼ 0]. For an ideal solution the cohesive forces are equal and uniform between all species. For example, if there are two components A and B, the intermolecular forces between A and A, B and B, and A and B are all the same. Just as the ideal gas law has served a useful purpose in treating practical cases even when large deviations from ideality occurs, the concept of an ideal solution serves the same purpose.

An important property of solutions is the vapor pressure of a component above the solution. Since this partial vapor pressure is a measure of the given species to escape from the solution into the vapor phase, its study provides information on the physical state within the solution.

TABLE1.1RelativeVaporPressureofaHomologousSeriesofAromaticCompoundsat258C CompoundNormalboilingpoint (8C)Vaporpressure (mmHg)Activitycoefficient g^{1 i}Berekend b^{1 iw}Experimental b^{1 iw} Butanol117.48.5431516 Hexanol155.51.26803430 Octanol194.50.18110008543 Acetone56.22307.87570 Butan-2-one79.6912710082 Pentan-2-one10237100160110 Heptan-2-one151.54.81600330260 Octan-2-one172.91.96600530330 Nonan-2-one1950.7627000870650 Undecan-2-one2280.1946000037001100 Acetaldehyde219104.2160120 Propanal4829016190130 Butanol759161240200 Pentanal10338250400260 Hexanal131121000500310 Heptanal1534.74200820610 Octanal1771.7170001200910 Nonanal1910.897100027001300 Methylacetate57.517024240200 Methylpropionate79.77095380310 Methylbutyrate102.325390560370 Methylpentanoate1307.31300560560 Methylhexanoate1502.971001200650 Methyloctanoate1930.371300028001000

For a mixture of two ideal components in an ideal mixture (one in which the two components do not interact), the vapor pressures of components 1 and 2 are given by Raoult’s law as

P1¼ x1P⁸₁ ð46Þ

P2¼ x2P⁸₂ ð47Þ

where x1and x2are the mole fractions and P18 and P28 are the vapor pressures of pure liquids, respectively. From Dalton’s law of additive pressure, the total pressure is

P ¼ P1þ P2 ð48Þ

Since most solutions behave nonideally, positive and negative deviations from Raoult’s law occur.

If the solute is volatile, and the solution is dilute and ideal, then Henry’s law applies:

P2¼ kx2 ð49Þ

where k is Henry’s law constant and depends on the nature of both the solute and solvent. Henry’s law can also be written in terms of molality of the solution, and the constant k¹has units atm mol²¹kg²¹of the solvent.

P2¼ k¹m ð50Þ

Values for k¹for several gases are given in Table 1.2.

We defined the chemical potential in terms of thermodynamic potential as

›G

T^ABLE1.2 Henry’s Law Constants for Gases in Water at 298K

Gas k¹(atm mol²¹kg²¹H2O)

[see Eq. (25)]. In a manner analogous to that for free energy, the chemical potential of the ith component in a mixture of ideal gases is

mi¼m^þ_i ð Þ þ RT ln PT i ð51Þ

where m^þ_i ðTÞ is the standard chemical potential of component i and P_i is the partial pressure. Substituting Pi¼ xiP, where P is total pressure, into Eq. (51) yields

mi¼m^þ_i ð Þ þ RT ln P þ RT ln xT _i ð52Þ

The first two terms on the right correspond to the chemical potential of the ith component in the pure state at pressure P. Consequently, Eq. (52) can be written

mi¼m8i;T;P ¼ 1 atmþ RT ln x_i ð53Þ

Equation (53) can be generalized to include condensed phases in which case a two-component system can be represented as

m1¼m81;T;Pþ RT ln x₁ solvent ð54Þ

m₈1¼m₈1;T;Pþ RT ln x2 solute ð55Þ

Usually Eqs. (54) and (55) are written in terms of activity, a₁and a₂, since for an ideal solution a_i¼ x_i [see Eq. (45)]:

m1¼m81;T;Pþ RT ln a1 ð56Þ

D. Nonideal Mixtures

We have just seen for an ideal gas at constant temperature

G ¼ G8 þ RT ln P ð32Þ

and for an ideal mixture or solution, we have

mA¼m^þ_1;T þ RT ln P_A ð51Þ

or for an ideal gas (PV ¼ nRT),

dmA¼ d RT ln Pð AÞ ¼ V dP ð57Þ

Since real systems deviate from ideality, Eqs. (32) and (51) required modification. However, the utility of maintaining the form of these equations

for convenience in thermodynamic analysis was recognized, and a new function called fugacity ( f ) was defined analogously to Eq. (57).

dma ¼ RT ln fA ð58Þ

Integrating Eq. (58) between the given state and some arbitrary reference state yields

mA¼m^þ_A þ RT ln fA

f^þ_A ð59Þ

In solutions, fugacity can be thought of as the true measure of the escaping tendency of a component from the solution or an idealized partial pressure which becomes equal to the partial pressure only when the vapor behaves as an ideal gas. Fugacity and fugacity coefficients for water vapor in equilibrium with liquid water at saturation and 1 atm pressure are given in Table 1.3.

It can be seen from comparison to Eq. (43) that f_A

f^þ_A ¼lA

l^þ_A ^{¼ aA ð60Þ}

indicating that fugacity is related to absolute activity l and to the activity aA. Furthermore, the ratio of fugacity to pressure is an indication of the deviation from nonideality. The fugacity coefficient expresses this ratio:

g^{¼ f}

p ð61Þ

whereg¼ 1.0 for an ideal gas.