Temperature dependence of the free energy

A calculation of the temperature dependence of the free energy for the reactions in Eqs. (15)-(18), and hence the electrochemical potential, showed that with an increase in temperature, formic acid formation became more unfavorable.4 In the case of formaldehyde, methanol, and methane formation, the calculation indicated a positive shift in the reduction potential, but of very small magnitude ca. 30 mV for a temperature change from 300 to 500 K, and ca. 20 mV from 500 to 1200 K.4... 

In terms of the temperature dependence of the free energy (see Fig. 8 where M, L, and C respectively stand for mesophase, liquid, and crystal)... 

Figure 2. The temperature dependence of the free energy. The labeling of asymmetries is as in Fig. 1 [8].

Another approach is to carry out free-energy simulations at several different temperatures, and then construct the equivalent of a van t Hoff plot to separate, say, the enthalpic and entropic contributions to the free energy. This approach is obviously extraordinarily demanding of resources, since every temperature point requires a new free-energy simulation, and unless there are many points, the error in the temperature dependence of the free energy determined by linear regression of the latter on the former may be rather large. 

Fig. 17. Schematic diagrams of the temperature dependence of the free energy surface in the calculations shown in Fig. 14, where TJ

The thermodynamic functions of fc-mers adsorbed in a simple model of quasi-one-dimensional nanotubes s adsorption potential are exactly evaluated. The adsorption sites are assumed to lie in a regular one-dimensional space, and calculations are carried out in the lattice-gas approximation. The coverage and temperature dependance of the free energy, chemical potential and entropy are given. The collective relaxation of density fluctuations is addressed the dependence of chemical diffusion coefficient on coverage and adsorbate size is calculated rigorously and related to features of the configurational entropy. 

If the temperature dependence of the free-energy curves is considered in addition to the (3 particle size, a temperature-composition diagram for the system can be plotted. This is illustrated in Fig. C.7, which shows clearly the way in which the solubility of component B in a increases with decreasing (3 particle size. 

Fig. 18. (a) The temperature dependence of the free energy change for the RNase transition at different pH values. The points represent AF° values calculated from the data. The solid curves are the best fit to a quadratic equation by least squares analysis. The dashed lines indicate the range of relatively high experimental accuracy, (b) The values of AH°, AS", and ACP for the RNase transition from data at pH 2.50. Reproduced from Brandts and Hunt (336). 

The following equation shows the temperature-dependence of the free energy change of a reaction ... 

To determine how the partition function varies with temperature it is necessary to estimate not only the free energy changes associated with each type of interaction, AG°, but also the thermodynamic terms that specify the temperature dependence of the free energy change, namely, the enthalpy change, AH°, the entropy change, AS0, and the... 

Whereas the temperature-dependence of the free energy of a phase is related to its entropy, the pressure dependence is related to its volume. As shown in Section 4.3, (dG/dP)r = V, and the variation in the equilibrium position between phases (as represented by the lines in Fig. 4.3 depends on the volume change associated with the phase transition. Thus as AV for melting is small compared with the volume change associated with vaporization, the melting point is very much less sensitive to pressure than the boiling point. 

Congruent melting of a compound, or any of the polymorphic transformations discussed earlier, is a good example of this type of transformation. To illustrate, consider the melting of a compound. The temperature dependence of the free-energy functions for the liquid is... 

It should first be stressed that the foregoing classification of sterically stabilized dispersions, based on whether they flocculate on heating or cooling, is quite rigorous near to the CFT. It is independent of any temperature dependence of ASp and AHp because it is always possible to write the temperature dependence of the free energy of flocculation as... 

The critical free-energy barrier for secondary nucleation is thus AFc = 4aae/Af, with = 2a/Af and Ic = 2ae/Af. One can see that the chainfolding principle is still applicable to the secondary nucleation. The temperature dependence of the free energy barrier AFc AT has also been well identified in experimental observations [5]. 

Temperature Dependence of the Free Energy Landscape of the src-SH3 Protein Domain. 

A. Cavalli, P. Ferrara, and A. Caflisch, Proteins Struct., Punct., Bioinf., 47, 305 (2002). Weak Temperature Dependence of the Free Energy Surface and Folding Pathways of Structured Peptides. 

To locate the nematic-isotropic transition it is necessary to determine the temperature dependence of the free energy but this is not possible without making further approximations concerning the temperature variation of the segmental interaction parameters X. and This variation with temperature results primarily through their dependence on the orientational order of the system. A rigorous derivation of this dependence is extremely difficult and so we adopt a semi-intuitive approach. As we have seen, the orientational order in a mesophase is characterized by an infinite set of order parameters but the most important of these are the second-rank order parameters, at least close to the nematic-isotropic transition. Indeed for cylindrically symmetric particles both theory and experiment agree that the potential of mean torque is proportional to When the mesophase... 

Eichinger s treatment conforms to the limiting behavior of A required by equation (113) at r= 0 and its analog at 0l. To represent behavior at intermediate temperatures by a continuous function, the temperature dependence of the free energy —RTV A 2 c (see equation 111) is explicitly characterized by a partial molar heat capacity of dilution RTV c d [TA 2 )ldT. This is expanded about 0 in a series in non-negative powers of (T—0u)/ u. Two integrations with respect to T between limits 0 and T yield a complicated series expression for This procedure... 

Like the two-parameter formalism, the Flory-Huggins theory contains nothing specific in its structure to account for the temperature dependence of the free energy of mixing that leads to the... 

As discussed earlier miscibility often arises due to a favorable (negative) heat of mixing. This infers the presence of some specific effect or interaction between the two components. The role of these interactions in mixtures has been reviewed in the literature. Such an interaction can dominate the properties of a system and none of the theories so far discussed were designed to treat such systems. The interaction energy is just included within the interaction parameter and one must question whether the composition and temperature dependencies of the free energy as described by these parameters can be expected to reflect those of the interaction. Semiempirical expansions of the interaction parameter as used by some authors may better describe this behavior. 

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