Definitions based on mole fractions

We may choose the reference state of any or all of the components to be the pure component or components in the same state of aggregation as the solution at all temperatures and pressures of interest. Then, according to Equation (8.71), 

the properties of the component in its standard state are identical to those of the component in its reference state. This reference state is primarily used for systems whose properties can be studied over the entire range of composition even though a region of partial immiscibility may exist. 

The reference state of each component in a system may be defined in many other ways. As an example, we may choose the reference state of each component to be that at some composition with the condition that the composition of the reference state is the same at all temperatures and pressures of interest. For convenience and simplicity, we may choose a single solution of fixed composition to be the reference state for all components, and designate xf to be the mole fraction of the /cth component in this solution. If (Afikx) represents the values of the excess chemical potential based on this reference state, then (A/if x ) [T, P, x ] is zero at all temperatures and pressures at the composition of the reference state. That this definition determines the standard state is seen from Equation (8.71), for then 

The question of what the mole fraction of a component is in its standard state for the chemical potential seldom arises the important point is that the value of the chemical potential of the component in its standard state is fixed when the reference state of the component has been defined. It is of interest to discuss this question when the pure components are not the reference states, so that a better understanding of the standard state may be obtained. The mole fraction of the fcth component in its standard state cannot be determined from Equation (8.86), because the absolute values of the chemical potential are unknown. However, according to Equations (8.71) and (8.72), 

The determination of the mole fraction of the fcth component in its standard state for the entropy follows the same argument that was used for the chemical potential. By the use of Equation (8.77) and the condition that [7, P, x] must equal Se[T, P] for the standard state, we find that 

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