Moving thermodynamic information into coordinate systems where temperature and/or pressure are independent variables is the function of the Legendre transform. Enthalpy is one such transform of the energy -- given here for a fixed-mass system by the formal expression

H(S,P) = U(S,V) - ( U/ V)(V)

Substituting the (negative) thermodynamic pressure for the U-V derivative yields the conventional formula H = U + PV and a new variable set involving enthalpy, entropy, and pressure.

Understanding this as a complete data transformation rather than as a rote formula assures us that no thermodynamic information is lost as it passes from one set of coordinates (USV) to another (HSP) and that events seen in the parent system (e.g., critical points or pairs of coexisting states) may also be observed in the transformed system. Other thermodynamic potentials (the Helmholtz energy, the Gibbs energy, and other un-named forms for mixtures) are also generated through Legendre transformation and may similarly be regarded as data rearrangements.

To see how the thermodynamic surface transforms geometrically as we move from USV space to HSP space, we must modify the enthalpy function so as to magnify its curvature. Instead of plotting the enthalpy directly, we plot the difference (for each value of S and P) between the absolute enthalpy and a reference value taken from a tangent plane fixed at the critical point.

The result is shown below over the same physical range as the USV model and with the same stability-based color scheme. The enthalpy variable (now magnified 20X) is plotted positive-downward and is termed Hd(difference). Four white isobars and two blue isotherms are drawn across the surface, with the upper curves of each representing critical values and intersecting at the critical point.

Legendre transformation causes an interchange of curvatures. The blue and yellow regions of the enthalpy surface are now saddle-shaped, and the red region is biconvex -- just the opposite of the USV function. The three subcritical isobars follow that curvature and show the "depression" in the unstable (red) zone. As noted earlier, ethylene cannot exist as a homogeneous (single-phase) substance with any of the property combinations given by points in the unstable region. Such conditions are in violation of the Second Law of Thermodynamics.

But even with this reversal of curvatures, phase-change processes may still be interpreted through Gibbs' tangent methods. To show this, a tangent line has been drawn along one of the isobars (and bridging the unstable region) so that it touches the surface at two points. As with the tangent plane on the USV surface, it may be shown that these points also share the same temperature, pressure, and chemical potential and, in the same way as before, represent liquid and vapor states that can coexist in two-phase equilibrium. As the tangent line moves from isobar to isobar, yielding successive pairs of coexisting states, it generates a ruled tangent surface that begins at the critical point and extends downward in pressure. That surface is shown in the figure with a transparent color, and the tangent points it defines are connected with a heavy black curve -- the coexistence curve in HdSP space. The vapor portion of the curve (with higher values of S) lies to the left of the critical point, and the liquid portion lies to the right.

Enthalpy is one of the "workhorse" functions of classical thermodynamics because it simplifies the solution of an important class of everyday problems (e.g., in fluid flow and heat transfer) that would otherwise require constant recalculation of the pressure-volume product. But we rarely appreciate its more fundamental role in the overall family of thermodynamic variables. Enthalpy is more than just the "U + PV" that we learned in the basic course. It is a complete fundamental form in itself that conveys the full spectrum of thermodynamic information about an equilibrium system modeled with the independent variables S and P.

The Helmholtz energy A is another Legendre transformof the energy, and it is derived from the USV function in a similar manner. A(T,V) = U(S,V) - ( U/ S)(S) = U - TS

Like the enthalpy, A is also a fundamental form, but its characteristic variables are temperature and volume instead of entropy and pressure. To the casual observer the Helmholtz energy may not seem as useful as the enthalpy because the TS term in its definition doesn't seem to have the physical meaning that the PV term has for the enthalpy. Indeed TS isn't even a measurable quantity because the absolute entropy contains an arbitrary constant!

But practical calculations aside, fundamental functions play a vital role in classical thermodynamics because they are extremum-seeking quantities for systems moving toward equilibrium. we show the A function below