## Partial molar properties

For simplicity, the theory of nonideal solutions is presented only for the enthalpy, but the same formulas apply to all properties. The enthalpy of a nonideal solution is of the same form as that for an ideal solution (Eq (7.2c)). The sole change is replacement of the molar enthalpy of the pure components (hi) by a quantity called the partial molar enthalpy, denoted by

The partial molar property for species i depends on temperature and on the composition of the solution but the effect of total pressure is negligible. The composition dependence renders hi considerably more difficult to determine than the corresponding molar property of the pure species, hi.

The physical meaning of hi is best shown by explicitly including the mole numbers of each species in solution in writing its total enthalpy. Taking the differential of H(T,p,n1,n2,.. ) at constant T and p yields:

XfHLd'" - dH-Xh dH - X | —I dni or dH -J hidni (7.13)

■ n' ^T,p,nj where the partial derivative with respect to ni is taken with the number of moles of all other components held constant. This partial derivative is the partial molar enthalpy of species i in the solution:

Equation (7.14) has a definite physical meaning: h represents the change in the enthalpy of a solution when a small quantity of species i is added while the amounts of all other components are held constant. Equation (7.13) can be "integrated" in a physical sense by simultaneously adding the pure components to a vessel at rates proportional to their concentrations in the solution (Fig. 7.4). This procedure maintains all concentrations constant during the process, and demonstrates that Eq (7.13) is consistent with Eq (7.12).

A very important relation can be deduced from these two equations. The total differential of Eq (7.12) is dH - ^ ' nidhi + ^ ' hidni, which, when combined with Eq (7.13) yields: Fig. 7.4 Mixing to maintain the solution composition constant. The rates of adding A and B are proportional to their concentration in the solution

The most useful form of the above equations is in terms of mole fractions rather than mole numbers and specialized for an A-B binary solution. Dividing Eq (7.12) by the total moles of A and B gives the molar enthalpy of the solution:

Similar treatment of Eq (7.15) yields:

Taking the total differential of Eq (7.16) and taking Eq (7.15) to account produces the analog of Eq (7.13):

The importance of Eq (7.17) is that it permits hB to be determined if the variation of hA with composition is known. This relation is obtained by integrating Eq (7.17) from xA = 0 (where hB = hB because the solution is pure B) to an arbitrary mole fraction of A:

xA dxA

dx A

If the dependence of hA with xA is determined experimentally, Eq (7.19) permits hB to be calculated. Independent measurement of hB is not needed.

On occasion, it is necessary to invert Eq (7.16) to express the partial molar properties in terms of the molar property. For this purpose, Eq (7.18) is divided by dxA, and since dxB = -dxA- the result is dh/dxA = hA - hB. Multiplying this equation by xB and adding the result to Eq (7.16) gives:

and a similar approach for component B yields:

Partial molar properties cannot be measured directly; only the molar properties are accessible to experiment. However, as shown in Fig. 7.5, Eq (7.20) provides the basis for a graphical method for determination of hA and hBfrom the h vs xA curve.

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### Responses

• Pentti Hakkarainen
What is molar thermodynamics?
8 years ago