Gibbs-Helmholtz relation

By combining Eq. (1) with the Gibbs-Helmholtz relation we obtain... 

The AT driving force is the isobaric equivalent of the isothermal Ag in Equation (3.10). The Gibbs-Helmholtz relation may be applied to... 

Finally, once E and H are determined by integration of (1.18.13) or via heat capacities, F and G may be found by the Gibbs-Helmholtz relation (1.18.33), thus closing the loop. The reader is well advised to ponder the methodology of thermodynamics, because it is through this general approach that the theory is particularly powerful in the analysis of phenomena. Other aspects of this structure will be pointed out in later sections. 

Standard Gibbs energies of adsorption are often encountered. When A j G is accurately known as a function of temperature, standard enthalpies and entropies of adsorption can also be obtained, using the appropriate Gibbs Helmholtz relations (sec. 1.2.15). 

For the implied infinitesimal changes of y with T the factor RTF remains constant. Generally, however, the derivative dy/dT also depends on T. From (4.3.22) S° will eventually be obtained as a function of x and T. As, for each T, F is accessible as a function of x, it is also possible to derive the surface excess entropy as a function of the monolayer composition. Accurate data are, as before, a prerequisite. From S the surface excess enthalpy = TS is obtainable. Alternatively, one can differentiate y /T with respect to the temperature, obtaining the enthalpy directly using the appropriate Gibbs-Helmholtz relation. 

If the c.m.c. is known as a function of T, the micelllzation enthalpy can immediately be derived, using the following Gibbs -Helmholtz relation (see [1.2.15.8a]... 

Equations (4.8) and (4.9) can also be written in terms of the average affinity and average heat of reaction. We start from the Gibbs-Helmholtz relation (4.33) and apply this to one state in which = 1, and another in which T and V have the same values but = 0 ... 

The enthalpy of formation is obtained from enthalpies of combustion, usually made at 298.15 K while the standard entropy at 298.15 K is derived by integration of the heat capacity as a function of temperature from T = 0 K to 298.15 K according to equation fB 1.27.16). The Gibbs-Helmholtz relation gives the variation of the Gibbs energy with temperature... 

An alternative approach that may be applied to determine AH (Hupu) involves combination of equation 4 with the Gibbs-Helmholtz relation which leads to the following formula 18,19) ... 

This results in the following Gibbs-Helmholtz relation ... 

In some textbooks (3.197) is rewritten by using the changes of enthalpy Ah and the Gibbs free energy Ag before and after the reaction, and the Gibbs-Helmholtz relation is given by... 

Derive relations for the physical properties of materials whose magnetization follows the Curie-Weiss law M = AxM-ol T + ), where is a parameter, called the Weiss constant. Check on the expression (5.8.14) by basing your derivation on the Gibbs-Helmholtz relation and solving the first-order differential equation. 

The temperature dependence of the Gibbs energy of formation allows one to determine the entropy of formation according to the Gibbs-Helmholtz relation AS = -(dAG/dT), i.e., the temperature dependence of the cell voltage relates to the entropy of the cell reaction, AS = nq(dE/dT). The enthalpies of formation and cell reaction follow according to AH = AG + TAS. Knowledge of these fundamental thermodynamic quantities allows the comprehensive determination of the thermodynamic properties of the system. 

Instead of measuring the equilibrium cell voltage Aeoo at standard conditions directly, this can be calculated from the reaction free energy AG for one formula conversion. In this context one of the fundamental equations is the GIBBS-HELMHOLTZ relation [7]. 

One can demonstrate that the errtropy S arrd the thermodynamic potential 4> are cormected through the generalized Gibbs-Helmholtz relations... 

From the temperature dependence of the cmc as well as the enthalpy of demicellization it is possible to calculate the demicellization entropy and its temperature dependence using the Gibbs-Helmholtz relation AG AH - TAS. A plot of all three thennodynamic parameters as a function of temperature is shown in Figure 39 for the anionic surfactant sodium lauroyl-alaninate (SLA). It is clearly evident that the A(/-value for demicellization of SLA shows only a weak temperature dependence whereas the two tenns AH and TAS are strongly temperature dependent in a similar way. This type of enthalpy-entropy compensation is quite common for phenomena where changes in "hydrophobic hydration" are involved, because a considerable increase in heat capacity occurs when hydrophobic groups are exposed to water (see below) [ 124-131]. 

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