Partial molar Gibbs free energy

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... 

G Gi, [G] Gibbs free energy, partial molar Gibbs free energy of i, generalized Gibbs free energy... 

The chemical potential is an example of a partial molar quantity /ij is the partial molar Gibbs free energy with respect to component i. Other partial molar quantities exist and share the following features ... 

We divide by Avogadro s number to convert the partial molar Gibbs free energy to a molecular quantity, and the minus sign enters because the force and the gradient are in opposing directions. Recalling the definition of chemical potential [Eq. (8.13)], we write jUj + RT In aj = ii2 + RT In 7jC, where aj... 

P rtl IMol r Properties. The properties of individual components in a mixture or solution play an important role in solution thermodynamics. These properties, which represent molar derivatives of such extensive quantities as Gibbs free energy and entropy, are called partial molar properties. For example, in a Hquid mixture of ethanol and water, the partial molar volume of ethanol and the partial molar volume of water have values that are, in general, quite different from the volumes of pure ethanol and pure water at the same temperature and pressure (21). If the mixture is an ideal solution, the partial molar volume of a component in solution is the same as the molar volume of the pure material at the same temperature and pressure. 

The energy of a system can be changed by means of thermal energy or work energy, but a further possibility is to add or subtract moles of various substances to or from the system. The free energy of a pure substance depends upon its chemical nature, its quantity (AG is an extensive property), its state (solid, liquid or gas), and temperature and pressure. Gibbs called the partial molar free heat content (free energy) of the component of a system its chemical potential... 

The partial molar entropy of a component may be measured from the temperature dependence of the activity at constant composition the partial molar enthalpy is then determined as a difference between the partial molar Gibbs free energy and the product of temperature and partial molar entropy. As a consequence, entropy and enthalpy data derived from equilibrium measurements generally have much larger errors than do the data for the free energy. Calorimetric techniques should be used whenever possible to measure the enthalpy of solution. Such techniques are relatively easy for liquid metallic solutions, but decidedly difficult for solid solutions. The most accurate data on solid metallic solutions have been obtained by the indirect method of measuring the heats of dissolution of both the alloy and the mechanical mixture of the components into a liquid metal solvent.05... 

Hence, for a pure substance, the chemical potential is a measure of its molar Gibbs free energy. We next want to describe the chemical potential for a component in a mixture, but to do so, we first need to define and describe a quantity known as a partial molar property. 

Before leaving our discussion of partial molar properties, we want to emphasize that only the partial molar Gibbs free energy is equal to n,-. The chemical potential can be written as (cM/<9 ,)rv or (dH/dnj)s p H partial molar quantities for fi, into equations such as those given above. 

The value of AG at a particular stage of the reaction is the difference in the molar Gibbs free energies of the products and the reactants at the partial pressures or concentrations that they have at that stage, weighted by the stoichiometric coefficients interpreted as amounts in moles ... 

To find how AG changes with composition, we need to know how the molar Gibbs free energy of each substance varies with its partial pressure, if it is a gas, or with its concentration, if it is a solute. We have already seen (in Section 8.3) that the molar Gibbs free energy of an ideal gas J is related to its partial pressure, P(, by... 

Equation 5 shows how the Gibbs free energy of reaction varies with the activities (the partial pressures of gases or molarities of solutes) of the reactants and products. The expression for Q has the same form as the expression for K, but the activities refer to any stage of the reaction. 

The quantity of primary interest in our thermodynamic construction is the partial molar Gibbs free energy or chemical potential of the solute in solution. This chemical potential reflects the conformational degrees of freedom of the solute and the solution conditions (temperature, pressure, and solvent composition) and provides the driving force for solute conformational transitions in solution. For a simple solute with no internal structure (i.e., no intramolecular degrees of freedom), this chemical potential can be expressed as... 

For instance, figure 6.13 shows the behavior of the partial derivative of the Gibbs free energy of mixing with respect to the molar amount of Si40g component ... 

There have been many attempts to describe the process of mixing and solubility of polymer molecules in thermodynamic terms. By assuming that the sizes of polymer segments are similar to those of solvent molecules, Flory and Huggins derived an expression for the partial molar Gibbs free energy of dilution that included the dimensionless Flory Higgins interaction parameter X = ZAH/RT, where Z is the lattice coordination number. It is now... 

Before discussing all these biopolymer applications, we first take this opportunity to remind the reader that, in general, any thermodynamic variable can be expressed as the sum of two functions, one of which depends only on the temperature and pressure, and another which depends on the system composition (expressed as the mole fraction xt of the /-component). Therefore, for example, the chemical potential fM of the /-component of the system at constant temperature T and pressure p (the general experimental conditions), /. e., partial molar Gibbs free energy (dG/dn TtP may be expressed as (Prigogine and Defay, 1954) ... 

Nagasawa and Takahashi, 1972). More specifically, the value of the second virial coefficient determines the excess chemical potential, juE (also known as the excess partial molar Gibbs free energy), which characterizes the formation of biopolymer-solvent and biopolymer-biopolymer pair contacts ... 

Equation 9.20 gives the pressure dependence of the Gibbs free energy of a pure substance. More generally, for a mixture one should consider the chemical potential /r, which is defined as the partial molar free energy of species k ... 

Gk Partial molar Gibbs free energy of species k J/mol... 

Since ideal conditions simplify calculations, an ideal gas at 1 atm pressure in the gas phase which is infinitely dilute in solution will be utilized. Then the total standard partial molar Gibbs free energy of solution (chemical potential), AG, can be directly related to KD, the distribution constant, by the expression... 

Another expression for the total standard partial molar Gibbs free energy is obtained from Henry s law. Henry s law may be written... 

The standard partial molar Gibbs free energy of solution is related to the enthalpy and entropy functions at the column temperature T by the expression... 

The partial molar excess Gibbs free energy and partial molar excess enthalpy of mixing are defined by the following equations ... 

The partial molar excess Gibbs free energy of the methylene group can be used as a means of characterizing stationary phases. From rearrangement of Equation 11.46, an expression for the standard partial molar Gibbs free energy can be obtained. 

For an ideal solution, the total standard partial molar excess Gibbs free energy, AGT, equals the ideal contribution, AG, and AGp = O. In other words, there is no interaction occurring between the solute and the stationary phase. When a nonpolar mole-, cule such as an alkane is chosen as the solute, AG will increase as the polarity of the stationary phase increases. There-—c... 

Of course AG° will not be the same value for every alkane, especially as the polarity of the stationary phase increases. So it becomes necessary to define a unit of the alkane, the methylene group, on which to base comparisons. Thus, the partial molar excess Gibbs free energy of the methylene group, AGg(cH2>, is defined by... 

The last member of Equation (2) shows that n, is the partial molar quantity associated with the Gibbs free energy, G. Euler s theorem gives then... 

---