Relative Partial Molar Enthalpies

In Chapter 7, we will describe relative partial molar enthalpy in more detail. 

From the Debye-Hiickel expressions for lny , one can derive equations to calculate other thermodynamic properties. For example L2, the relative partial molar enthalpy,q and V2, the partial molar volume are related to j by the equations... 

We described methods for obtaining values for V,-, Cpi, and S but did not apply the methods to //, and G, (or w), since absolute values of Gm and Hm cannot be obtained. We did describe a procedure for obtaining the volume difference V,— Vf using equations (5.40). (5.41) and (5.42),r where V is the volume of the pure substance, and indicated that equations of the same form can be used to obtain //, - Hf. We will return to this method later in this chapter as we describe ways for measuring relative partial molar enthalpies. 

The quantity //, - H° is called the relative partial molar enthalpy and given the symbol L,. It is the difference between the partial molar enthalpy in the solution and the partial molar enthalpy in the standard state. That is,... 

Relative partial molar enthalpies can be used to calculate AH for various processes involving the mixing of solute, solvent, and solution. For example, Table 7.2 gives values for L and L2 for aqueous sulfuric acid solutions7 as a function of molality at 298.15 K. Also tabulated is A, the ratio of moles H2O to moles H2S(V We note from the table that L — L2 — 0 in the infinitely dilute solution. Thus, a Raoult s law standard state has been chosen for H20 and a Henry s law standard state is used for H2SO4. The value L2 = 95,281 Tmol-1 is the extrapolated relative partial molar enthalpy of pure H2SO4. It is the value for 77f- 77°. 

Substitution of relative partial molar enthalpies gives... 

The last term in the calculation is L f, the relative partial molar enthalpy for pure water, which is zero. 

Figure 7.9 (a) Enthalpies of dilution starting with a solution containing 3 moles fEO per 1 mole HC1 (m = 18.50 or w1/2 = 4.30) and (b), relative apparent molar enthalpies (4>L) and relative partial molar enthalpies (L and L >) for the resulting mixtures. 

Figure 7.9(b) shows a graph of values of qL obtained from the series of dilutions of the starting solution as calculated from equation (7.81). Also shown in Figure 7.9(b) are values for L and Lj. These relative partial molar enthalpies can be obtained from 4>L. To find the relationship, we start with the equation... 

In equation (7.70), we defined the relative partial molar enthalpy L, as... 

Besides the partial molar and the relative partial molar free energies of the components, some other important thermodynamic properties are the partial molar and the relative partial molar enthalpies and entropies. The partial molar enthalpy and entropy of the component A are defined by... 

The relative partial molar enthalpy, AHa, and the relative partial molar entropy, A,S a, are defined as... 

The corresponding expressions for the relative partial molar enthalpy and entropy of the component A are... 

The relative values of partial molar enthalpies are used so frequently that it has become customary to use the special symbol to represent them. Thus, L i, which is the relative partial molar enthalpy, is defined by the equation... 

Relative partial molar enthalpies also can be visualized in terms of a diagram, such as Figure 18.4. Although the absolute position of or on the enthalpy scale cannot be specified, the difference between them can be determined. 

Hence, the relative partial molar enthalpies of HCl in aqueous solutions can be expressed by the equation... 

Relative partial molar enthalpies can also be obtained from measurements of enthalpies of dilution. Humphrey et al. [4] have used enthalpy of dilution measurements to calculate relative partial molar enthalpies in aqueous solutions of amino acids. Their data for AH n of aqueous solutions of serine are shown in Table 18.2, where mj is the initial molality of the solution, rrif is the molality after addition of a small amount of solvent, and is equal to the measured AH divided by ti2, which is the number of moles of solute in the solutions. 

Figure 18.6. Relative partial molar enthalpy of water in serine solutions. Data from Ref. 4.

The apparent molar enthalpy d>//2 has been used to derive relative partial molar enthalpies from enthalpy of dilution data. The apparent molar enthalpy is dehned as... 

If Equation (18.37) is applied to relative partial molar enthalpies... 

Table 18.5 lists the relative partial molar enthalpies of glycine and its aqueous solutions at 25°C. [Reprinted with permission from Ref. 9. Copyright 1940 American Chemical Society.]... 

TABLE 19.8. Relative Partial Molar Enthalpies for NaCl(aq) at 25°C ... 

The solubility of pure solid glycine at 25°C in water is 3.33 moles (kg H20). The activity coefficient of glycine in such a saturated solution is 0.729. Data for the relative partial molar enthalpies of glycine in aqueous solution are tabulated in Exercise 15 of Chapter 18. Given AfG and... 

The relative partial molar enthalpies of the species are defined as the partial derivatives, in the last term,... 

But by Eq. (27) the chemical potential difference in parentheses is zero when the solution is in internal equilibrium. Therefore using Eq. (41), defining the relative partial molar enthalpies of the species, Eq. (40) reduces to... 

The relative partial molar enthalpies of the species are obtained by using Eqs. (70) and (75) in Eq. (41). When the interaction coefficients linear functions of T as assumed here, these enthalpies can be written down directly from Eq. (70) since the partial derivatives defining them in Eq. (41) are all taken at constant values for the species mole fractions. Since the concept of excess quantities measures a quantity for a solution relative to its value in an ideal solution, all nonzero enthalpy quantities are excess. The total enthalpy of mixing is then the same as the excess enthalpy of mixing and a relative partial molar enthalpy is the same as the excess relative partial molar enthalpy. Therefore for brevity the adjective excess is not used here in connection with enthalpy quantities. By definition the relation between the relative partial molar entropy of species j, Sj, and the excess relative partial molar entropy sj is... 

Fig. 5. Calculated relative partial molar enthalpies of In and Sb in In-Sb melt at 627°C and experimental points ( ) from Hultgren et al. (1973b).

---