A third transform with the same critical-point placement as in the previous ternary example but with the butane chemical potential held constant instead of the pressure. The tangent line is drawn at a fixed value of

it has a slope equal to the local pressure, and it gives the value of common to the red and green states through its intercept on the (imaginary) wall at VN = 0. While in the previous image (the conventional Gibbs energy, a second transform) the blue and yellow surfaces were biconvex and the red region saddle-shaped, those curvatures are reversed in this drawing. The Legendre transform causes that reversal.

The symbol A++ represents a modified Helmholtz energy with the general independent variable set

Legendre transforms with respect to mole numbers in this set are designated by +, ++, etc.

Iowa State University