Partial Legendre transform

In similar vein the enthalpy H = U(S, P, Nj) is the partial Legendre transform of U that replaces the volume by pressure as an independent variable. Recall that dU/dV — —P, so that the appropriate Legendre transform is... 

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,... 

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,... 

Obviously, the partly inverted Legendre-transformed representations for reactive systems would similarly generate descriptors of the partially relaxed (electronically or geometrically) reactive systems. 

The fractional saturation of tetramer YT and the fractional saturation of dimer YD are functions only of [02] at specified T, P, pH, etc., as shown by equations 7.1-18 and 7.3-6. However, since the tetramer form is partially dissociated into dimers, the fractional saturation of heme Y is a function of both [02] and [heme]. Ackers and Halvorson (1974) derived an expression for the function Y([02], [heme]). When Legendre transforms are used, a simpler form of this function is obtained, and it can be used to derive limiting forms at high and low [heme]. These limiting forms are of interest because they show that if data can be obtained in regions where Y is linear in some function of [heme], extrapolations can be made to obtain YT and YD. These fractional saturations can be analyzed separately to obtain the Adair constants for the tetramer and the dimmer (Alberty, 1997a). 

But notice now that H is the Legendre transform of E and hence, subject to Eq. (1.26.8). Thus, the sign of the second partial derivative of H with respect to P is opposite that of E with respect to V. Therefore, contrary to naive preconceptions,... 

In cumbersome language the second partial derivative of the function of state Y with respect to the extensive variable Xk has a sign opposite to the second partial derivative of the partially inverted Legendre transform Z taken with respect to its conjugate variable pk. 

For T, V = constant, the system is better described by the free energy, F, which represents the following partial Legendre transform U[T oi U [8.2] ... 

For T, P = constant, the Gibbs energy or Gibbs function, G, representing another partial Legendre transform of U, is properly used ... 

Introducing the enthalpy as another partial Legendre transform of U... 

The thermodynamic potential of the canonical ensemble, the Helmholtz free energy, is the first thermodynamic potential g=F, which is a function of the variables of state u 1 = T, x2=V, x3=N, and x4=z. It is obtained from the fundamental thermodynamic potential / =E (the energy) by the Legendre transform (Eq. (7)), exchanging the variable of state x1 =S of the fundamental thermodynamic potential with its conjugate variable u 1 = / . In the canonical ensemble, the first partial derivatives (Eq. (1)) of the fundamental thermodynamic potential are defined asu2=-p, u3=p, and u 4 = - S. The entropy (Eq. (46)) for the Tsallis and Boltzmann-Gibbs statistics in the canonical ensemble can be rewritten as... 

In conclusion, let us summarize the main principles of the equilibrium statistical mechanics based on the generalized statistical entropy. The basic idea is that in the thermodynamic equilibrium, there exists a universal function called thermodynamic potential that completely describes the properties and states of the thermodynamic system. The fundamental thermodynamic potential, its arguments (variables of state), and its first partial derivatives with respect to the variables of state determine the complete set of physical quantities characterizing the properties of the thermodynamic system. The physical system can be prepared in many ways given by the different sets of the variables of state and their appropriate thermodynamic potentials. The first thermodynamic potential is obtained from the fundamental thermodynamic potential by the Legendre transform. The second thermodynamic potential is obtained by the substitution of one variable of state with the fundamental thermodynamic potential. Then the complete set of physical quantities and the appropriate thermodynamic potential determine the physical properties of the given system and their dependences. In the equilibrium thermodynamics, the thermodynamic potential of the physical system is given a priori, and it is a multivariate function of several variables of state. However, in the equilibrium... 

When the ideal gas law is written in terms of energy, a partial differential equation is obtained. The situation becomes more simple, if we want to get the free energy as fundamental form. Once we have an expression for the free energy, we can use the Legendre transformation to get the ordinary energy. We illustrate now this type of... 

For changes of state at constant composition, we need Cj, and the volumetric equation of state to be able to integrate (3.5.11) and (3.5.13) for Ali and AS. With values for AU and AS, we can then apply the defining Legendre transforms (3.2.9) for H, (3.2.11) for A, and (3.2.13) for G to obtain changes in the other conceptuals. If the change of state includes a change in composition, then we will also need values for the partial molar volume, enthalpy, and entropy as shown in 3.4.3, these partial molar quantities are simply related to the chemical potential. 

Consider a binary mixture of 1 and 2 having molar volume v. Show that the partial molar volumes can be written in the form of Legendre transforms. 

For stability, we then have 8F > 0, which taken at lowest order is thus a constraint on the second derivatives. To find the constraints one commonly carries out a series of linear transformations on the independent variables, which reduces the quadratic to a sum of squares in the transformed variables. From the resulting relations, by elimination of variables and Legendre transformation, the canonical, all the various semigrand canonical, and grand canonical relations can be derived (Valdeavella, Perkyns, and Pettitt 1994). This yields equations that are expressible in terms of the compressibility, the partial molar volumes, and derivatives of the chemical potential, which are directly calculable from the cofactors of a density weighting of the matrix of zeroth moments of the distribution. 

Therefore, A and G are the first and second partial Legendre transformations of U. The total Legendre transformation of U is... 

Hint make a Legendre transform of U whose total differential has the independent variables needed for the partial derivative, and write a reciprocity relation.)... 

If we have an algebraic expression for a state function as a function of independent variables, then a Legendre transform preserves all the information contained in that expression. To illustrate this, we can use the state function / and its Legendre transform /2 described above. Suppose we have an expression for f x, y, z)—this is / expressed as a function of the independent variables x, y, and z. Then hy taking partial derivatives of this expression, we can find according to Eq. F.2.3 expressions for the functions a x, y, z), b x, y, z), and c(x, y, z). 

The other elements of the matrices above will be determined in terms of a, b, and c, parameters considering the appropriate Legendre transformations and employing the partial derivatives properties in relation with the associated Jacobians determinants. 

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