Legendre transforms defined

Substituting the integrated fundamental equation for U (equation 2.2-14) in the Legendre transforms defining H, A, and G shows that... 

The calculations in Chapters 3 to 5 have been based on the use of Legendre transforms to introduce pH and pMg as independent intensive variables. But now we need to discuss the reverse process - that is the transformation of Af G ° values calculated from measured apparent equilibrium constants in the literature to Af G° values of species and the transformation of Af° values calculated from calorimetric measurements in the literature to AfH° of species. This is accomplished by use of the inverse Legendre transform defined by (7) ... 

This has been attempted, but it is not possible because and //°(h2o) are not independent variables since h2o contains two hydrogen atoms. However, after the Legendre transformation defining G has been made, the standard transformed chemical potential p °(h2o) can be used in Legendre transform 7.3-1. 

By differentiating the defining equations for H, A and G and combining the results with equation (A2.T25) and equation (A2.T27) for dU and U (which are repeated here) one obtains general expressions for the differentials dH, dA, dG and others. One differentiates the defined quantities on the left-hand side of equation (A2.1.34), equation (A2.1.35), equation (A2.1.36), equation (A2.1.37), equation (A2.1.38) and equation (A2.1.39) and then substitutes die right-hand side of equation (A2.1.33) to obtain the appropriate differential. These are examples of Legendre transformations. ... 

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,... 

Legendre transformation does not affect the essential nature of a function and all of the different potentials defined above still describe the internal energy, not only in terms of different independent variables, but also on the basis of different zero levels. In terms of Euler s equation (2) the internal energy consists of three components... 

In order to focus on the driving force for phase transitions induced by a magnetic field it is advantageous to use the magnetic flux density as an intensive variable. This can be achieved through what is called a Legendre transform [12], A transformed Helmholtz energy is defined as... 

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 ... 

Finally, the remaining (/a, Q) representation describing the equilibrium state of an externally open molecular system with the frozen nuclear framework is examined. The relevant partial Legendre transform of the total electronic energy, which replaces N by /a in the list of independent state-parameters, defines the BO grand-potential ... 

The electronic-nuclear coupling in molecules is also detected in the other partial Legendre-transformed representation H(/r, Q), which defines the combined Hessian G of Equation 30.27. Its first diagonal derivative,... 

Unfortunately, this is mainly formal because neither the integral defining ip nor the Legendre transform are likely to be tractable. However, we show in Section II.C that Eq. (26) is equivalent to the more conventional form of the entropy of mixing as given by the second term in Eq. (5). From a conceptual point of view, it should be noted that the conventional form is normally derived by binning the distribution of particle sizes a and taking the number... 

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... 

The temperature can be introduced as an independent variable by defining the Gibbs energy G with the Legendre transform... 

Legendre transforms are also used in mechanics to obtain more convenient independent variables (Goldstein, 1980). The Lagrangian L is a function of coordinates and velocities, but it is often more convenient to define the Hamiltonian H with a Legendre transform because the Hamiltonian is a function of coordinates and momenta. Quantum mechanics is based on the Hamiltonian rather than the Lagrangian. 

In this section we will consider the Legendre transforms that define the enthalpy H, Helmholtz energy A, and Gibbs energy G. 

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. 

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. 

The Legendre transform that defines the further transformed Gibbs energy G", which provides the criterion for spontaneous change and equilibrium in dilute... 

When the concentrations of ATP and ADP are in a steady state, these concentrations can be made natural variables by use of a Legendre transform that defines a further transformed Gibbs energy G" as follows. 

In order to introduce the chemical potential of molecular oxygen as a natural variable, the following Legendre transform is used to define a further transformed Gibbs energy G" (Alberty, 1996b) ... 

Since coenzymes, and perhaps other reactants, are in steady states in living cells, it is of interest to use a Legendre transform to define a further transformed Gibbs energy G" that provides the criterion for spontaneous change and equilibrium at a specified pH and specified concentrations of coenzymes. This process brings in a further transformed entropy S" and a further transformed enthalpy H", but the relations between these properties have the familiar form. 

In the third formulation, the so-called Hamiltonian formulation, the velocities Lagrangian form are replaced by the so-called generalized momenta pi via a Legendre transformation. The generalized momentum pi, conjugate to the coordinate qi, is defined as... 

The generating function F2 can be obtained by a Legendre transformation from F. Defining F as the Legendre transform of F2 we have... 

Use of a Legendre Transform to Define a Transformed Gibbs Energy at a Specified pH... 

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