Temperature dependence of free energy

Figure 2. Temperature dependence of free energy of formation,AG of actinide intermetallics.

Figure 5.16 Enantioselectivity of the redox reaction of ADH from T. ethanolicus with different alcohols (Pham, 1990). Temperature dependence of free energy of activation differences for 2-butanol and 2-pentanol open squares 2-butanol open circles 2-pentanol filled square reduction of 2-butanone filled circle reduction of 2-pentanone.

The equations given above in the shaded boxes are called the Gibbs-Helmholtz equations. These permit us to calculate the change in enthalpy AH and entropy AS from a knowledge of AG. They relate the temperature dependence of free energy, and hence the position of equilibrium to die enthalpy change. 

An analysis of the curves of concentration and temperature dependences of free energies provides a possibility of phase diagram construction and this diagram defines the concentration and temperature regions of all phases realization. The phase diagram corresponds to experimental data of manifestation of phases of chemical reaction (1) in the course of temperature rise. 

Fig. 6. Schematic figures of temperature dependence of free energy change of N5- and N3-or N9- type NaYPSi in the cases assuming N5- (a) and N3- (b) or N9-typ>e (b) NaYPSi as the high temperature-stable phase, where Tc is the crystallization temperature.

FIGURE 1.5 Temperature dependence of free energy (o), internal energy (e), entropy (q), and latent heat (r T). (Redrawn from Shchukin, E.D. et al.. Colloid and Surface Chemistry, Elsevier, Amsterdam, the Netherlands, 2001.)... 

The derived formula (38) determines the dependence of free energy of bcc fullerite on temperature, fullerene concentrations, hydrogen content, its activity, and energetic constants. 

The derived formula (19) shows the dependence of free energy f2 of the PtHx phase on temperature T, concentrations ci, c2, c of < i, 2 fullerenes and hydrogen, degree of ordering r 2 in this phase and energetic constants of pair interaction between 4>n- 4>m(n,m=l,2) fullerenes, fullerenes and platinum atoms, fullerenes and hydrogen atoms. Below we shall study the free energy f2 for hydrofulleride for the purpose of interpretation of calculation results. 

Figure 4. The curve plots of the concentration dependence of free energies f[ (dotted lines) and f2 (full curves) in the region of temperatures of phase transitions I—HI (<5Pt—> PtHx). The intersection and extremal points are marked with circles. r=kTT02/ 0.

The elaborate statistical theory of phase transformations of chemical reaction (1) makes possible the explanation and substantiation of formation of phases of fulleride hydrides and then of fullerite with increase in temperature. The calculation of phases free energies has been performed using the rough simplified assumptions. The dependence of free energies of phases on their composition, temperature, order parameter in fullerenes subsystem, energetic constants has been found. The evaluation of energetic constants has been carried out with the use of experimental data for concentration and temperature ranges of each phase realization. 

On the base of this approach thermodynamics of hydrogen absorbed outside and inside the (10,10) and the (20,20) single-wall carbon nanotubes with diameters 13.56 A and 27.13 A, respectively, was calculated. The dependencies of free energy F and thermodynamical potential H on applied pressure P and temperature T were calculated. The dependencies of content of hydrogen adsorbed on nanotubes m(P,T) surface on pressure and temperature were calculated from these data. For the first time the dependencies of m(P,T) with accounting of quantum effects and van der Waals forces were calculated. 

Figure 15. (a) Temperature dependence of the Gibbs free energy for ice Ih (solid line) and ice Ic (dotted line) at atmospheric pressure, where contributions from the configurational entropy and anharmonic vibrations are omitted, (b) Temperature dependence of the energy which is defined as the sum of the interaction energy at its minimum structure and the vibrational energy for ice Ih (solid line) and ice Ic (dotted line). 

Figure 17. Molecular free energies (a) and internal energies (b) vs surface roughness parameter d at different temperatures for once-folded alkane crystal, using the model in Figure 16. Circles mark the free energy minima. The dashed line shows the temperature dependence of equilibrium energy (after ref 40).

To understand the pressure dependence of free energy, we need to know how pressure affects the thermodynamic functions that comprise free energy, that is, enthalpy and entropy (recaii that G = H - TS). For an ideal gas, enthalpy is not pressure-dependent. However, entropy does depend on pressure because of its dependence on volume. Consider 1 moie of an ideai gas at a given temperature. At a volume of 10.0 L, the gas has many more positions avaiiable for its molecules than if its volume is 1.0 L. The positional entropy is greater in the iarger volume. In summary, at a given temperature for 1 mole of ideal gas... 

Fig. C9.4 The dependence of free energy F(a) on the swelling parameter a in the case where oc(x) is multi-valued function of x, characterizing the solvent quality. As x changes (which can be controlled by, say, temperature change), the shape of F(a) dependence changes such that one minimum is getting deeper on the expense of the other. Deeper minimum corresponds to the more stable state. For this figure, we choose the value y = 0.001.

FIGURE 1.4 Dependence of free energy on molar volume v for a fluid below its critical temperature. L and V are the states of the liquid and vapor phases in equilibrium. 

Based on the dependence of free energy on temperature and composition shown in Figure A.2, the phase diagram can now be constmcted as shown in Figure A.3. The temperatures T,-T(, define unequivocally the compositions of the liquid and solid phases in equilibrium with each other. The curve connecting aU freezing... 

The formation processes of metal-containing nanostructures in carbon or carbon-polymeric shells in nanoreactors can be related to one t) e of reaction series using the terminology of the theory of linear dependencies of free energies (LFE) [16]. Then it is useful to introduce definite critical values for the volume, surface energy of nanoreactor internal walls, as well as the temperature critical value. When the ration Ig k/k is proportional -AAF/RT, the ratio WAV can be transformed into the following expression ... 

The temperature dependence of reaction rates permits evaluation of the enthalpy and entropy components of the free energy of activation. The terms in Eq. (4.4) corresponding to can be expressed as... 

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