Legendre transforms thermodynamic potentials

As established by the first law, the key feature of energy and other Legendre-transformed thermodynamic potentials is their state character (Section 2.10), i.e., their conservation under cyclic changes of state. For the leading potentials (U, H, A, G) of chemical interest, the differentials of these conserved quantities are given at equilibrium (under the usual conditions of PV-work only) by the expressions... 

To recast the thermodynamic description in terms of independent variables that can be controlled in actual laboratory experiments (i.e., T, /i, and the set of strains or their conjugate stresses), it is sensible to introduce certain auxiliary thermodynamic potentials via Legendre transformations. This chapter is primarily concerned with... 

The enthalpy is useful in considering isentropic and isobaric processes, but often it becomes necessary to rather deal with isothermal and isobaric processes. In such case one needs a thermodynamic function of T and P alone, defining the Gibbs potential G = U(T, P, Nj) as the Legendre transform of U that replaces entropy by temperature and volume by pressure. This transform is equivalent to a partial Legendre transform of the enthalpy,... 

The statistical partition functions are seen to be related by Laplace transformation in the same way that thermodynamic potentials are related by Legendre transformation. It is conjectured that the Laplace transformation of the statistical partition functions reduces asymptotically to the Legendre transformation of MP in the limit of infinitely large systems. 

The inverse of H determines the geometric compliance matrix (Nalewajski, 1993, 1995, 1997, 1999, 2000, 2002b, 2006a,b Nalewajski and Korchowiec, 1997 Nalewajski et al., 1996, 2008) describing the open system in the Qi,F)-representation. The relevant thermodynamic potential is defined by the total Legendre transform of the system BO potential, which replaces the state-parameters (N, Q) with their energy conjugates (/a, F), respectively ... 

Let us now turn to the mixed, partly inverted (N, F)-representation describing the geometrically relaxed, but externally closed molecular system. The relevant thermodynamic potential is now defined by the partial Legendre transformation of W(N, Q) which replaces Q by F in the list of the system parameters of state ... 

The inequalities of the previous paragraph are extremely important, but they are of little direct use to experimenters because there is no convenient way to hold U and S constant except in isolated systems and adiabatic processes. In both of these inequalities, the independent variables (the properties that are held constant) are all extensive variables. There is just one way to define thermodynamic properties that provide criteria of spontaneous change and equilibrium when intensive variables are held constant, and that is by the use of Legendre transforms. That can be illustrated here with equation 2.2-1, but a more complete discussion of Legendre transforms is given in Section 2.5. Since laboratory experiments are usually carried out at constant pressure, rather than constant volume, a new thermodynamic potential, the enthalpy H, can be defined by... 

LEGENDRE TRANSFORMS FOR THE DEFINITION OF ADDITIONAL THERMODYNAMIC POTENTIALS... 

Legendre Transforms for the Definition of Additional Thermodynamic Potentials... 

This shows that the natural variables of G for a one-phase nonreaction system are T, P, and n . The number of natural variables is not changed by a Legendre transform because conjugate variables are interchanged as natural variables. In contrast with the natural variables for U, the natural variables for G are two intensive properties and Ns extensive properties. These are generally much more convenient natural variables than S, V, and k j. Thus thermodynamic potentials can be defined to have the desired set of natural variables. 

The number of Maxwell equations for each of the possible thermodynamic potentials is given by D(D — l)/2, and the number of Maxwell equations for the thermodynamic potentials for a system related by Legendre transforms is [ )(D — 1)/2]2D. Examples are given in the following section. 

These four Legendre transforms introduce the chemical potential as a natural variable. The last thermodynamic potential U T, P, /<] defined in equation 2.6-6 is equal to zero because it is the complete Legendre transform for the system, and this Legendre transform leads to the Gibbs-Duhem equation for the system. 

Now, we proceed to make the Legendre transformation leading from h(n(r, v)) to s(e,n). The thermodynamic potential (5) becomes... 

Legendre transformations are very useful in physics in general. In thermodynamics, e.g., they can be used for switching between thermodynamic potentials. Another application of Legendre transformations is in the theory of canonical transformations to be discussed next. 

There are two ways to derive the Gibbs-Duhem equation for a system (1) Subtract the fundamental equation from the total differential of the thermodynamic potential. (2) Use a complete Legendre transform. As an example of the second method, eonsider fundamental equation 3.3-10 for a single reactant ... 

Wyman (5,6,7) introduced the binding potential, which he represented by the Russian L for linkage. This is a molar thermodynamic property that is defined by a Legendre transform that introduces the chemical potential of the ligand as an independent intensive property. The binding potential is given by... 

Substituting Eq. (6) into the fundamental thermodynamic potential / and using the Legendre transform, we obtain [19]... 

This Legendre transform is always well defined when the fundamental thermodynamic potential f (xv. .., xn) is a convex function of the variables (x]r xm), i.e., the quadrahc form... 

The Legendre transform (Eq. (7)) is involutive [19], i.e., if under the Legendre transformation / is taken to g, then the Legendre transform of g will again be /. The fundamental thermodynamic potential f(xv. .., xn) can be obtained from the first thermodynamic potential g(uy. ..,um,xm+1, xn) by the Legendre back-transformation... 

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