Partial molar quantities 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... 

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 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... 

This shows that the chemical potential of a component is just its partial molar Gibbs free energy. Note that the definitions of the chemical potential in terms of other thermodynamic variables, given in Chapter 6, Eq. (8), are not partial molar quantities because pressure and temperature are not the variables held constant in these derivatives. 

Partial molar volumes are of interest in part through their thermodynamic connection with other partial molar quantities such as partial molar Gibbs free energy, known also as chemical potential. An important property of chemical potential is that for any given component it is equal for all phases that are in equilibrium with each other. Gonsider a system... 

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... 

The partial molar quantities most commonly encountered in the thermodynamics of polymer lutions are partial molar volume Vi and partial molar Gibbs free energy Gi- The latter quantity is of special significance since it is identical to the quantity called chemical potential, pi, defined by... 

The partial molar quantities of importance for polymer solutions are the partial molar Gibbs free energy (G,), the chemical potential (/i,), and the partial molar volume (F,). 

Let us note here some of the basic principles underlying models, theories, and calculations of hydrophobic effects. The quantity of first interest is the free energy associated with the interaction of the solute with the aqueous environment. This is the chemical potential or partial molar Gibbs free energy of the solute. This quantity provides a driving force for rearranging molecules in thermodynamic systems and many quantities of interest are fundamentally connected to this free energy. Consider first an atomic solute of type A. For example, perhaps A = Ar. The chemical potential at extreme dilution may be expressed as... 

Such quantities Z, are called partial molar quantities and with polymer solutions we are normally concerned with partial molar Gibbs free energies Gi and partial molar volumes V",. In single component systems, such as pure solvent, the partial molar quantities are identical to the corresponding molar quantity which we will denote by a lower case symbol, z,. 

One partial molar quantity that is of particular use in discussion of the equilibrium of multicomponent systems at constant temperature and pressure is the partial molar Gibbs free energy G, of the component /. This is also called the chemical potential, pi of the species i and so it follows that... 

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. 

The composition or the number fraction of component B, Xb, is an example of an order parameter that is conserved in a closed system. Figure 17.6 shows a molar free energy versus composition curve for a binary solution. The molar free energy for a solution at any composition Xb can be written in terms of its partial molar quantities, Fa(Xb) and Fb(Xb) 4... 

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... 

This is called the relative integral molar free energy or the molar free energy of mixing. Hie method of tangential intercepts which we have applied for determination of partial molar quantities from the integral molar quantities can also apply to the relative quantities ... 

The chemical potential is defined as an intensive energy function to represent the energy level of a chemical substance in terms of the partial molar quantity of free enthalpy of the substance. For open systems permeable to heat, work, and chemical substances, the chemical potential can be used more conveniently to describe the state of the systems than the usual extensive energy functions. This chapter discusses the characteristics of the chemical potential of substances in relation with various thermodynamic energy functions. In a mixture of substances the chemical potential of an individual constituent can be expressed in its unitary part and mixing part. 

Equation (1.121) states that the chemical potential of the species i is the change in free energy with respect to the change in number of moles of the species i while the compositions of other species are held constant. Partial molar quantities also follow the same thermodynamic rules. 

The most important partial molar quantity in Physical Chemistry is the partial molar free energy designated as chemical potential and represented as... 

Partial Molar Quantities. — The thermodynamic functions, such as heat content, free energy, etc., encountered in electrochemistry have the property of depending on the temperature, pressure and volume, i.e., the state of the system, and on the amounts of the various constituents present. For a given mass, the temperature, pressure and volume are not independent variables, and so it is, in general, sufficient to express the function in terms of two of these factors, e.g., temperature and pressure. If X represents any such extensive property, i.e., one whose magnitude is determined by the state of the system and the amounts, e.g., number of moles, of the constituents, then the partial molar value of that property, for any constituent i of the system, is defined by... 

In an earlier section the free energy of a phase and the free energy of a total system were discussed generally in terms of the potentials (e.g., equation 48). With the definition of the chemical potential as a function of activity in hand, we will now consider the Gibbs energy of a system. In a similar fashion, the enthalpy and entropy of a system can be computed using the partial molar quantities and the mole numbers of each phase. 

Next we can finally see how the activity coefficient relates to the Margules equations for this case. Recall from Chapter 9 that the partial molar quantity of one component in a binary solution can be obtained graphically from the tangent (as with the chemical potentials /j,b and /ta in the coexisting solutions of Figure 15.3a). From equation (9.6), the partial molar free energy or chemical potential of component A in a solution of A and B is given by... 

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