Maxwell relations Gibbs energy

The other thermodynamic properties for a reaction are related to the Gibbs energy of reaction through Maxwell equations (see Section 2.3). Because of equation 3.1-5, equation 3.1-4 can be written... 

Another important equation, the Gibbs-Helmholtz equation, is derived from the Maxwell relations. A chemist may use this equation to determine the enthalpy change in a reaction, and a pharmaceutical scientist may use it to calculate colligative properties (i.e., freezing point depression and boiling point elevation). The expression for free energy with respect to temperature at constant pressure is given by Equation (1.105) ... 

The chemical potential is the partial molar Gibbs free energy. Partial molar quantities figure importantly in the theory of solutions and are defined at constant temperature and pressure thus, the Gibbs free energy is a natural state function for their derivation. As an example, the partial molar volume is found from the Maxwell relation... 

Equations of State and Maxwell Relations for the Transformed Gibbs Energy... 

The standard formation properties of species are set by convention at zero for the elements in their reference forms at each temperature. The standard formation properties of in aqueous solution at zero ionic strength are also set at zero at each temperature. For other species the properties are determined by measuring equilibrium constants and heats of reaction. Standard transformed Gibbs energies of formation can be calculated from measurements of K, and so it is really these Maxwell relations that make it possible to calculate five transformed thermodynamic properties of a reactant. 

Thus measurements of Af G, ° and Af //, ° at a single temperature yield Af 5, ° at that temperature. In the next chapter we will see that if Af //, ° is known, the standard transformed Gibbs energy of formation can be expressed as a function of temperature, and then all the other thermodynamic properties can be calculated by taking partial derivatives of this function. Note that in equations 3.4-5 to 3.4-9, the only Maxwell relation that does not involve a partial derivative with respect to the temperature is the one that yields Fh (0-... 

One of the Maxwell relations in equation 3.4-14 shows that the average binding of hydrogen ions by a reactant can be calculated by taking the partial derivative of the standard transformed Gibbs energy of formation of a reactant with respect to pH. 

Fundamental Equation and Maxwell Relations for the Transformed Gibbs Energy of a Reactant at Specified T, pH, pMg, and Ionic Strength... 

This fundamental equation shows that S = - BG /dT, Ar G = dG /9f, and f 71n(10)no(H) = BG VdpH. These equations are not directly useful because there is no experimental way to determine G but Maxwell relations for this and other fundamental equations do provide equations for experimental determinable properties. For the system being discussed the standard transformed Gibbs energy of reaction is given by... 

Since temperature, pressure, and composition are the appropriate independent variables for the Gibbs free energy we must now write out the entropy and volume in the form 5 = S(T, P, n, ) and V = V(T, P, , ), take their differential forms, and substitute these in Eq. (1.20.19a). Following the method used in setting up Eq. (1.13.4c), we next introduce the heat capacity at constant pressure, the appropriate Maxwell relation, as well as a and We also introduce the partial molal entropy S, and volume V,- to obtain... 

This type of differential can be converted into other expressions by using the Maxwell relations. The thermal expansion coefficient a originates naturally from the Gibbs free energy, since for the Gibbs free energy the set (T. p, n) are the natural variables. 

The second derivative of the four measures of energy, i.e., internal energy, U, enthalpy, H, Gibbs free energy, G, and Helmholz free energy. A, can be obtained with respect to two independent variables from among temperatnre, pressnre, volnme, and entropy. The order of differentiation does not matter as long as the function is analytic. This property is used to derive the Maxwell relations as follows ... 

Consider the combined first and second laws in terms of the Gibbs free energy, Equation (2.21). How many Maxwell reciprocal relations can be obtained from this equation Write each of them and comment on their physical significance. 

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