Legendre transformation thermodynamics

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... 

A familiar example of Legendre transformation is the relationship that exists between the Lagrangian and Hamiltonian functions of classical mechanics [17]. In thermodynamics the simplest application is to the internal energy function for constant mole number U(S, V), with the differentials... 

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,... 

As in energy representation the fundamental thermodynamic equation in entropy representation (3) may also be subjected to Legendre transformation to generate a series of characteristic functions designated as Massieu-Planck (MP) functions, m. The index m denotes the number of intensive parameters introduced as independent variables, i.e. 

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. 

In 1982, Nalewajski and Parr took the thermodynamic analogy to its logical conclusion by extending the Legendre-transform structure of classical thermodynamics to DFT [8]. One of their results was the Maxwell relation for Equation 18.6,... 

Another example is provided by the minimum energy coordinates (MECs) of the compliant approach in CSA (Nalewajski, 1995 Nalewajski and Korchowiec, 1997 Nalewajski and Michalak, 1995,1996,1998 Nalewajski etal., 1996), in the spirit of the related treatment of nuclear vibrations (Decius, 1963 Jones and Ryan, 1970 Swanson, 1976 Swanson and Satija, 1977). They all allow one to diagnose the molecular electronic and geometrical responses to hypothetical electronic or nuclear displacements (perturbations). The thermodynamical Legendre-transformed approach (Nalewajski, 1995, 1999, 2000, 2002b, 2006a,b Nalewajski and Korchowiec, 1997 Nalewajski and Sikora, 2000 Nalewajski et al., 1996, 2008) provides a versatile theoretical framework for describing diverse equilibrium states of molecules in different chemical environments. 

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 ... 

Equation A2.19 is one of the possible ways to switch from a couple of variables to another conjugate couple. Most transforms adopted in the thermodynamics of systems with two independent variables are based on equation A2.19 and its modifications (Legendre transforms, cf section 2.4). 

This chapter deals with the thermodynamics of one-phase systems, and it is understood that the phase is homogeneous and at uniform temperature. The basic structure of thermodynamics provides the tools for the treatment of more complicated systems in later chapters. This book starts with the fundamentals of thermodynamics, but the reader really needs some prior experience with thermodynamics at the level of undergraduate thermodynamics (Silbey and Alberty, 2001). Legendre transforms play an important role in this chapter, and the best single reference on Legendre transforms is Callen (1960, 1985). Other useful references for basic thermodynamics are Tisza (1966), Beattie and Oppenheim (1979), Bailyn (1994), and Greiner, Neise, and Stocker (1995). 

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. 

---