The Gibbs-Helmholtz Equation

The temperature derivatives of G and A equations (3.35) to (3.38) can be seen to contain S. Because we usually do not know 5 as a function of p, V, and T, these derivatives are of limited value. However, expressions can be derived for the temperature derivatives of G/Tand A/T that do not involve 5, and they can often be used in applications where the original derivative was desired. We derive the equation for (d G/T)/OT)p to illustrate the process. 

Equation (3.42) is called the Gibbs-Helmholtz equation. We will find it to be a very useful relationship. A similar derivation would show that 

The derivatives (dZ/dp)v and (dZ/dV)p are easily obtained by making use of the properties of the exact differential 

Similar expressions for the other derivatives can be as easily derived. However, the equations with p and V as the independent variables are not as useful as those involving p and T or V and T. and we will not take the time to obtain all of these relationships. 

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In modern separation design, a significant part of many phase-equilibrium calculations is the mathematical representation of pure-component and mixture enthalpies. Enthalpy estimates are important not only for determination of heat loads, but also for adiabatic flash and distillation computations. Further, mixture enthalpy data, when available, are useful for extending vapor-liquid equilibria to higher (or lower) temperatures, through the Gibbs-Helmholtz equation. ... 

The partial molar enthalpy for every component i is found from an appropriate form of the Gibbs-Helmholtz equation... 

Convert AS° to kJ/K using the relation 1J = 10-3 kj. Finally, calculate AG° from the Gibbs-Helmholtz equation with T = 298 K. 

The Gibbs-Helmholtz equation can be used to calculate the standard free energy of formation of a compound. This quantity, AGf, is analogous to the enthalpy of formation, AH . It is defined as the free energy change per mole when a compound is formed from the elements in their stable states at 1 atm. 

To find AG° at temperatures other than 25°C, use the Gibbs-Helmholtz equation. 

Strategy The first step is to write a balanced equation for the chemical reaction. Then calculate AH° using Table 8.3 (page 209) and AS° using Table 17.1 (page 456). Finally, use the Gibbs-Helmholtz equation to determine AG° at 230°C (3 sig. fig.)... 

As far as the Gibbs-Helmholtz equation is concerned, there is another reason for ignoring the temperature dependence of AH and AS. These two quantities always change in the same direction as the temperature changes (i.e., if AH becomes more positive, so does AS). Hence the two effects tend to cancel each other. 

From Example 17.5 and the preceding discussion, it should be clear that AG°, unlike AH° and AS°, is strongly dependent on temperature. This comes about, of course, because of the T in the Gibbs-Helmholtz equation ... 

A change in reaction conditions can, and often does, change the direction in which a reaction occurs spontaneously. The Gibbs-Helmholtz equation in the form... 

When the temperature of a reaction system is increased, the sign of AG°, and hence the direction in which the reaction proceeds spontaneously, may or may not change. Whether it does or does not depends on the signs of AH° and AS°. The four possible situations, deduced from the Gibbs-Helmholtz equation, are summarized in Table 17.2 (p. 464). 

Strategy This is another application of the Gibbs-Helmholtz equation. Setting AG° = 0 gives AH° = TAS° solving for T gives T = AH°/AS°. Since you know AH° and AS°, T is readily calculated. 

The relation between reaction free energy, temperature, cell voltage, and reversible heat in a galvanic cell is reflected by the Gibbs-Helmholtz equation [Eq. (31)]. 

There is a very important equation relating to the electromotive forces of reversible cells which was deduced independently by J. Willard Gibbs (1875) and H. von Helmholtz (1882), and is usually called the Gibbs-Helmholtz Equation. 

The Gibbs-Helmholtz equation also links the temperature coefficient of Galvani potential for individual electrodes to energy effects or entropy changes of the electrode reactions occurring at these electrodes. However, since these parameters cannot be determined experimentally for an isolated electrode reaction (this is possible only for the full current-producing reaction), this equation cannot be used to calculate this temperature coefficient. 

The above equation is known as the Gibbs-Helmholtz equation. It enables the evaluation of the AH of a reaction from a knowledge of the free energy change (AG) and of its temperature coefficient [(8AG)/(8T)]P. 

Expressed in this form, the Gibbs-Helmholtz equation is widely used in experimental thermodynamics to determine AH, the enthalpy of a reaction, from the experimentally de-... 

The Gibbs-Helmholtz equation is applicable to closed systems of fixed composition undergoing isobaric (constant pressure) processes. 

The Van t Hoff isotherm establishes the relationship between the standard free energy change and the equilibrium constant. It is of interest to know how the equilibrium constant of a reaction varies with temperature. The Varft Hoff isochore allows one to calculate the effect of temperature on the equilibrium constant. It can be readily obtained by combining the Gibbs-Helmholtz equation with the Varft Hoffisotherm. The relationship that is obtained is... 

Kil fin et al have reported values for the exchange enthalpy of neomycin for three ion-exchange resins. The values of AH were calculated by application of the Gibbs-Helmholtz equation the published results are tabulated below -... 

The temperature dependence of the Gibbs function change is described quantitatively by the Gibbs-Helmholtz equation. 

The value of AGS at 298 K is +5.7 kJmoP1, the positive sign explaining the lack of a spontaneous reaction. Inserting values into the Gibbs-Helmholtz equation, Equation (4.62), yields... 

AG3e73 K has a negative value, implying that the reaction at this new, elevated temperature is now spontaneous. In summary, the Gibbs-Helmholtz equation quantifies a qualitative observation the reaction to denature egg white is not spontaneous at room temperature, but it is spontaneous at elevated temperatures, e.g. when the egg is boiled in water. 

We start with the Gibbs-Helmholtz equation (Equation (4.62)) ... 

The isochore, Equation (4.81), was derived from the integrated form of the Gibbs-Helmholtz equation. It is readily shown that the van t Hoff isochore can be rewritten in a slightly different form, as ... 

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