Chemical Potentials and Standard States

The choice of the standard state is largely arbitrary and is based primarily on experimental convenience and reproducibility. The temperature of the standard state is the same as that of the system under investigation. In some cases, the standard state may represent a hypothetical condition that cannot be achieved experimentally, but that is susceptible 

Introduction to Chemical Engineering Kinetics and Reactor Design, Second Edition. Charles G. Hill, Jr. and Thatcher W. Root. 2014 John Wiley Sons, Inc. Published 2014 by John Wiley Sons, Inc. 

A° and ArG° are both constant quantities at a constant temperature since the standard chemical potentials are constant in these conditions (see Sect. 2.1). Both quantities can be determined. They are equal (in absolute values) to the useful work, or the maximum work available. Such is the case when the chemical reaction takes place reversibly and when the initial and final states are the standard ones. They are described in Fig. 2.3. 

By introducing these activities of unit value (which, in fact, characterize the standard states of species A, B, M, and N— see Sect. 2.1) in Eqs. (2.9)-(2.11), we 

It is interesting to note that the standard molar reaction Gibbs function A G° does not have the same mathematical status as ArG at each extent value. A G°, indeed, is not a derivative, unlike ArG. ArG is simply the system free enthalpy change AGsyst between two particular states, which happen to be the standard ones. 

The fact that the standard molar reaction Gibbs function change A G° and the standard chemical affinity A° are constant at a given temperature is of utmost importance. Its major outcome is the mass law. Indeed, the equilibrium condition ArG = 0 introduced into Eq. (2.9) gives 

Since A G° is constant, the term enclosed within brackets is also a constant. By definition, it is the thermodynamic constant K° of reaction (2.5)  

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Chemical Potentials and Standard States for Completely Ionized Solutions... 

This diagram is worth careful thought. It illustrates several things that are useful in understanding activities, chemical potentials, and standard states, such as the absolute nature of chemical potentials and the necessity of using differences, the equality of chemical potentials in each phase, and the arbitrary nature of the standard state. To further illustrate the la.st point, suppose we choose a new energy level for the standard state more or less at random, such that (/xa — when Xa is 0.5 is 5000 cal mol" . This implies a value of oa of 10 and this in turn defines the physical... 

This diagram is worth careful thought. It illustrates several things that are useful in understanding activities, chemical potentials, and standard states, such as the absolute nature of chemical potentials and the necessity of using differences, the equality of chemical potentials in each phase, and the arbitrary nature of the standard state. 

In our models, we will consider only minerals of fixed composition. Each mineral, then, exists in its standard state, so that its chemical potential and standard potential are the same... 

The algebraic formulation of the relationship among chemical potential p,, standard state chemical potential and thermodynamic activity a, is... 

We can then derive the calculated Gibbs free energy of mixing with respect to the molar amount of the component of interest, thus obtaining the difference between the chemical potential of the component in the mixture and its chemical potential at standard state ... 

The chemical potential can also be expressed in terms of a chemical potential under standard state conditions /a and the activity a for a solute in a given phase. Recognizing that a phase that has an activity equal to unity (i.e., (2=1 defines the standard state at a given temperature and pressure), the equation for the chemical potential fi for an activity other than (2 = 1 is found according to... 

However, the chemical potential is given by Eq. (4-341) for gas-phase reactions and standard states as the pure ideal gases at T°, this equation becomes... 

The isopiestic method is based upon the equality of the solvent chemical potentials and fugacities when solutions of different solutes, but the same solvent, are allowed to come to equilibrium together. A method in which a solute is allowed to establish an equilibrium distribution between two solvents has also been developed to determine activities of the solute, usually based on the Henry s law standard state. In this case, one brings together two immiscible solvents, A and B, adds a solute, and shakes the mixture to obtain two phases that are in equilibrium, a solution of the solute in A with composition. vA, and a solution of the solute in B with composition, a . 

We now have the foundation for applying thermodynamics to chemical processes. We have defined the potential that moves mass in a chemical process and have developed the criteria for spontaneity and for equilibrium in terms of this chemical potential. We have defined fugacity and activity in terms of the chemical potential and have derived the equations for determining the effect of pressure and temperature on the fugacity and activity. Finally, we have introduced the concept of a standard state, have described the usual choices of standard states for pure substances (solids, liquids, or gases) and for components in solution, and have seen how these choices of standard states reduce the activity to pressure in gaseous systems in the limits of low pressure, to concentration (mole fraction or molality) in solutions in the limit of low concentration of solute, and to a value near unity for pure solids or pure liquids at pressures near ambient. 

Although we cannot determine its absolute value, the chemical potential of acomponent of a solution has a value that is independent of the choice of concentration scale and standard state. The standard chemical potential, the activity, and the activity coefficient have values that do depend on the choice of concentration scale and standard state. To complete the definitions we have given, we must define the standard states we wish to use. 

Here, p and m are the standard chemical potential and concentration (molal scale) of the /-component (z = 1 for solvent, z = 2 for biopolymer) A2 is the second virial coefficient (in molal scale units of cm /mol, i.e., taking the polymer molar mass into account) and m° is the standard-state molality for the polymer. 

In the following sections expressions are developed for the chemical potential of the components or species in solution in terms of the composition —first for an ideal solution and then for real solutions—with special emphasis on reference and standard states. 

These two equations are useful only to change from one reference state to another. As described in Section 8.8 for mole fractions, the compositions of the standard states of the fcth solute for the chemical potential and for the entropy can only be determined by the solution of such equations as... 

The first two of these equations are useful, in addition to showing that the standard state of the kth solute for the chemical potential and the entropy are determined, only in converting from one reference state to another. If it is ever necessary, the composition of the standard state for the chemical potential and the entropy would have to be determined by solution of equations such as ... 

Occasionally the problem arises of converting values of various thermodynamic functions of the components of a solution that have been determined on the basis of one reference state to values based on another reference state. To do so we equate the two relations of the thermodynamic function of interest obtained for the two reference states, because the value of the function at a given temperature, pressure, and composition must be the same irrespective of the reference state. We also equate the relation for the thermodynamic function for the component in the new reference state expressed in terms of the new reference state to that for the same state expressed in terms of the old reference state. The desired relation is obtained when the chemical potentials of the component in the different standard states are eliminated from the two equations. For examples, we use only the chemical potentials and discuss three cases. 

There are four undetermined quantities Apfx, (Apf ), pf, and (pf) and two equations. We must, therefore, define two of the four quantities, which in turn determines the other two quantities and the relationship between them. We can define the reference states for the component and the species. The difference between the standard chemical potential of the component and that of the species is then expressed in terms of the mole fractions in the reference state. The problem is the determination of this difference. The different species may be known from our knowledge of the chemical system, or they may be assumed. However, a definite decision must be made concerning the species, and all calculations must be carried out based upon this decision. Several examples concerning reference and standard states are discussed here and in the following sections. 

When a system exhibits partial immiscibility, we encounter an equilibrium between two phases in the same state of aggregation. In this case the same standard state of a component can be used for both phases. With the use of the condition of equilibrium in terms of the chemical potentials and Equation (10.30), we have... 

Table 5.1. Standard chemical potentials pi , standard molar enthalpy h , and standard molar absolute entropy values s of substances in the standard state of 298 K and...

Table 9.3. Standard chemical potential jj°, standard partial molar enthalpy h°, and standard partial molar entropy s,° for a few hydrated ions Standard state 101.3 kPa, 298 K, unit activity in molality scale.

Both chemical potential and affinity depend on the choice of a standard state. A convenient choice is Ai = 1. The action relative to the standard state (e) is given by ... 

Although activity and fugacity are closely related, they have quite different characteristics in regard to phase equilibria. Consider, for example, the equilibrium between liquid water and water vapor in the interstices of an unsaturated soil. At a given temperature and pressure, the principles of thermodynamic equilibrium demand that the chemical potentials and fugacities of water in the two phases be equal. However, the activities of water in the two phases will not be the same because the Standard State for the two phases is not j the same. Indeed, f° = 1 atm for the water vapor, so its activity is numerically ]... 

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