Graphical representation for binary systems

The relations derived in this section provide us a means to easily extract the partial molar properties of a system from a graph of the corresponding molar property with repect to composition. A generic plot of the variation of a molar property of a binary mixture with composition is given by the solid line in Fig. 5.1. When. 2 = 0, the system consists only of component 1, and the value of the molar property should be equal to the molar property of pure component 1 Xf. Likewise, when = 1, the system consists only of component 2, and tine value of X should be equal to X. The variation of a molar property of an unmixed system composed of pure 1 and pure 2 is given by the thick dashed line in Fig. 5.1. 

The distance between the thick dashed line and the solid line represents the difference between a molar property of the mixture and that of the unmixed system. This difference is equal to 

Now consider a line that is tangent to X (i.e., the solid line) at the composition x = x. This is denoted by the thin dashed line in Fig. 5.1. The intercept of the tangent line at x-2 = 0 and x-2 = 1 are equal to Xi and X2, respectively that is, the intercepts of the tangent line are equal to the partial molar properties of the system at a given composition. To demonstrate this, let s determine the equation for 

Knowledge of both the value of the tangent line at a point and its slope allows us to write the equation 

From Eq. (5.15), we see that when = 0, the tangent line has a value equal to Xi(a7 ), and when X2 = 1, ithasavalue equal to X2(2 2). This is shown graphically by the thin dashed line in Fig. 5.1. For an ideal solution, the molar volume V of a mixture satisfies 

---