Gibbs Condition of Critical Point

Gibbs defined a critical point “as one at which the distinction between coexistent phases vanishes.” He used two equations to determine the critical point: U=0 and V=0, where U is the determinant of Gibbs energy Hessian, and V is the determinant formed from U by substituting any one of its rows with the derivatives of U with respect to molar fractions. This condition is rarely mentioned in thermodynamic books. To help understand this condition, a geometric interpretation of the two equations will be presented. In the compositional space, U=0 represents the spinodal surface and V=0 is equivalent to that an eigenvector of Gibbs energy Hessian is tangent to the spinodal surface, where the tangent point is the critical point.