?Commonly, if \(M^{E}\) for a binary system has a single sign, then the partial

Chapter 10, Problem 10.42

(choose chapter or problem)

Commonly, if \(M^{E}\) for a binary system has a single sign, then the partial properties \(\bar{M}_{1}^{E}\) and \(\bar{M}_{2}^{E}\) have the same sign as \(M^{E}\) over the entire composition range. There are occasions, however, where the \(\bar{M}_{i}^{E}\) may change sign even though \(M^{E}\) has a single sign. In fact, it is the shape of the \(M^{E}\) vs. \(x_{1}\) curve that determines whether the \(M^{E}\) changes sign. Show that a sufficient condition for \(\bar{M}_{1}^{E}\) and \(\bar{M}_{2}^{E}\) to have single signs is that the curvature of \(M^{E}\) vs. \(x_{1}\) has a single sign over the entire composition range.

Text Transcription:

M^E

barM_1^E

barM_2^E

barM_i^E

x_1