Phase-Boundary Derivatives in Multicomponent Systems

The phase-boundary conditions are closely associated with the singularity conditions for the full (c + 2)-dimensional metric M(c+2) of a c-component system. In each case, a metric singularity is associated with a null eigenvector of M(c+2), 

Each such null vector may be considered an invariant or symmetry of the thermodynamic system, because it corresponds to an operation (change of extensive variables Xt) that produces no response in any intensive state variable and thus leaves the thermodynamic state unaltered (Sidebar 7.2). As described in Sidebar 10.3, these invariants also play a role somewhat analogous to overall rotations and translations ( null eigenmodes of the Hessian matrix) in the theory of molecular vibrations. 

Of course, any linear combination of null eigenvectors also satisfies (12.54a, b), so we are able to choose individual vectors of this null manifold with considerable freedom. We know in general that a / -dimensional manifold of linearly independent null eigenvectors must exist, in order that the rank of the metric matrix, 

In accordance with (12.55), it is always possible to choose a nonsingular principal submatrix M = of order /from M(c+2)  

Thep extensities Xk(k = 1,2./ ) thereby deleted from consideration may be regarded as scale factors having fixed values in all thermodynamic derivatives, and the corresponding 

---