Gibbs free energy derivative

Fig. 12.2. Free energy data for electron transfer between the protein cytochrome c and the small acceptor microperoxidase-8 (MP8), from recent simulations [47]. Top Gibbs free energy derivative versus the coupling parameter A. The data correspond to solvated cytochrome c the MP8 contribution is not shown (adapted from [47]) Bottom the Marcus diabatic free energy curves. The simulation data correspond to cyt c and MP8, infinitely separated in aqueous solution. The curves intersect at 77 = 0, as they should. The reaction free energy is decomposed into a static and relaxation component, using the two steps shown by arrows a static, vertical step, then relaxation into the product state. All free energies in kcalmol-1. Adapted with permission from reference [88]...

Calculate the number of moles of cyt c(Fe ) formed from cyt c(Fe ) with the Gibbs free energy derived from the oxidation of 1 mole of glucose. 

Starting with the expression for changes in the internal energy of a polymer, Eq. (12.27), and the definitions of the enthalpy and the Gibbs free energy, derive the expressions for / and S given in Eq. (12.28). Show that these expressions imply... 

Of these the last eondition, minimum Gibbs free energy at eonstant temperahire, pressure and eomposition, is probably the one of greatest praetieal importanee in ehemieal systems. (This list does not exhaust the mathematieal possibilities thus one ean also derive other apparently ununportant eonditions sueh as tliat at eonstant U, S and Uj, Fisa minimum.) However, an experimentalist will wonder how one ean hold the entropy eonstant and release a eonstraint so that some other state fiinetion seeks a minimum. 

P rtl IMol r Properties. The properties of individual components in a mixture or solution play an important role in solution thermodynamics. These properties, which represent molar derivatives of such extensive quantities as Gibbs free energy and entropy, are called partial molar properties. For example, in a Hquid mixture of ethanol and water, the partial molar volume of ethanol and the partial molar volume of water have values that are, in general, quite different from the volumes of pure ethanol and pure water at the same temperature and pressure (21). If the mixture is an ideal solution, the partial molar volume of a component in solution is the same as the molar volume of the pure material at the same temperature and pressure. 

The same relations (11) and (12) hold for the Gibbs free energy in the (N, p,T) ensemble. Equation (11) is also valid for a quanmm mechanical system. Note that for a linear coupling scheme such as Eq. (10), the first term on the right of Eq. (12) is zero the matrix of second derivatives can then be shown to be definite negative, so that the free energy is a concave function of the Xi. 

The importance of the Gibbs free energy and the chemical potential is very great in chemical thermodynamics. Any thermodynamic discussion of chemical equilibria involves the properties of these quantities. It is therefore worthwhile considering the derivation of equation 20.180 in some detail, since it forms a prime link between the thermodynamics of a reaction (AG and AG ) and its chemistry. 

The partial molar entropy of a component may be measured from the temperature dependence of the activity at constant composition the partial molar enthalpy is then determined as a difference between the partial molar Gibbs free energy and the product of temperature and partial molar entropy. As a consequence, entropy and enthalpy data derived from equilibrium measurements generally have much larger errors than do the data for the free energy. Calorimetric techniques should be used whenever possible to measure the enthalpy of solution. Such techniques are relatively easy for liquid metallic solutions, but decidedly difficult for solid solutions. The most accurate data on solid metallic solutions have been obtained by the indirect method of measuring the heats of dissolution of both the alloy and the mechanical mixture of the components into a liquid metal solvent.05... 

In most applications, thermodynamics is concerned with five fundamental properties of matter volume (V), pressure (/ ), temperature (T), internal energy (U) and entropy (5). In addition, three derived properties that are combinations of the fundamental properties are commonly encountered. The derived properties are enthalpy (//). Helmholtz free energy (A) and Gibbs free energy ) ... 

The challenge that we now face is to justify these remarks and to derive a quantitative relation between the Gibbs free energy and the maximum nonexpansion work that a system can do. 

When we consider a measurable change in the Gibbs free energy, the equation we have derived becomes... 

A note on good practice Equation 5 was derived on the basis of the "molar convention for writing the reaction Gibbs free energy that means that the n must be interpreted as a pure number. That convention keeps the units straight FE° has the units joules per mole, so does RT, so the ratio FE°/RT is a pure number and, with n a pure number, the right hand side is a pure number too (as it must be, if it is to be equal to a logarithm). 

For a reaction that has reached equilibrium, the rates of the forward and reverse reaction are equal and the Gibbs free energy is at its minimum value. If we assume the pressure and temperature to be constant, the derivative of G with respect to the reactants and products will be equal to zero for the reaction in Eq. (1), i.e. 

Here we sketch a heuristic derivation of the quasichemical formula (9.34). Consider a solution of species a and w. The Gibbs free energy is... 

The tools for calculating the equilibrium point of a chemical reaction arise from the definition of the chemical potential. If temperature and pressure are fixed, the equilibrium point of a reaction is the point at which the Gibbs free energy function G is at its minimum (Fig. 3.1). As with any convex-upward function, finding the minimum G is a matter of determining the point at which its derivative vanishes. 

Second-order phase transitions are those for which the second derivatives of the chemical potential and of Gibbs free energy exhibit discontinuous changes at the transition temperature. During second-order transitions (at constant pressure), there is no latent heat of the phase change, but there is a discontinuity in heat capacity (i.e., heat capacity is different in the two... 

The enthalpies of solution and solubilities reviewed here provide much of the experimental information required in the derivation of single-ion hydration and solvation enthalpies, Gibbs free energies, and entropies for scandium, yttrium, and lanthanide 3+ cations. 

As indicated in the previous section, the derivation of Gibbs free energies of transfer, and thence of entropies of transfer, from trichloride... 

The most important property of a liquid-gas interface is its surface energy. Surface tension arises at the boundary because of the grossly unequal attractive forces of the liquid subphase for molecules at its surface relative to their attraction by the molecules of the gas phase. These forces tend to pull the surface molecules into the interior of the liquid phase and, as a consequence, cause liquids to minimize their surface area. If equilibrium thermodynamics apply, the surface tension 7 is the partial derivative of the Helmholtz free energy of the system with respect to the area of the interface—when all other conditions are held constant. For a phase surface, the corresponding relation of 7 to Gibbs free energy G and surface area A is shown in eq. [ 1 ]. 

The X transition in liquid helium shown in Figures 11.5 and 11.6 is a second-order transition. Most phase transitions that follow the Clapeyron equation exhibit a nonzero value of A5m and AYmi that is, they show a discontinuity in 5 and Fm. the first derivatives of the Gibbs free energy Gm- Thus, they are caHA A first-order transitions. In contrast, the X transition shows a zero value of A5m and AVm and exhibits discontinuities in the second derivatives of Gm, such as the heat capacity Cpm-... 

We have defined the chemical potential of a component as the partial derivative of the Gibbs free energy of the system (or, for a homogeneous system, of the phase) with respect to the number of moles of the component at constant P and T—i.e.,... 

The excess thermodynamic properties correlated with phase transitions are conveniently described in terms of a macroscopic order parameter Q. Formal relations between Q and the excess thermodynamic properties associated with a transition are conveniently derived by expanding the Gibbs free energy of transition in terms of a Landau potential ... 

Table 5.59 lists Gibbs free energies of formation from the elements of mica end-members obtained with the procedure of Tardy and Garrels (1974). For comparative purposes, the same table lists Gibbs free energies of formation from the elements derived from the tabulated and S% p values (same sources as in table 5.57) by application of... 

It is evident from equation 5.204 that the intrinsic significance of equation 5.206 is closely connected with the choice of standard state of reference. If the adopted standard state is that of the pure component at the P and T of interest, then AG%i is the Gibbs free energy of reaction between pure components at the P and T of interest. Deriving in P the equilibrium constant, we obtain... 

We can then derive the calculated Gibbs free energy of mixing with respect to the molar amount of the component of interest, thus obtaining the difference between the chemical potential of the component in the mixture and its chemical potential at standard state ... 

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