Thermodynamics molar quantities

Relationships which exist between ordinary thermodynamic variables also apply to the corresponding partial molar quantities. Two such relationships are... 

As noted above, all of the partial molar quantities are concentration dependent. It is convenient to define a thermodynamic concentration called the activity aj in terms of which the chemical potential is correctly given by the relationship... 

Perhaps the most significant of the partial molar properties, because of its appHcation to equiHbrium thermodynamics, is the chemical potential, ]1. This fundamental property, and related properties such as fugacity and activity, are essential to mathematical solutions of phase equihbrium problems. The natural logarithm of the Hquid-phase activity coefficient, Iny, is also defined as a partial molar quantity. For Hquid mixtures, the activity coefficient, y, describes nonideal Hquid-phase behavior. 

From the definition of a partial molar quantity and some thermodynamic substitutions involving exact differentials, it is possible to derive the simple, yet powerful, Duhem data testing relation (2,3,18). Stated in words, the Duhem equation is a mole-fraction-weighted summation of the partial derivatives of a set of partial molar quantities, with respect to the composition of one of the components (2,3). For example, in an / -component system, there are n partial molar quantities, Af, representing any extensive molar property. At a specified temperature and pressure, only n — 1) of these properties are independent. Many experiments, however, measure quantities for every chemical in a multicomponent system. It is this redundance in reported data that makes thermodynamic consistency tests possible. 

In some cases, reported data do not satisfy a consistency check, but these may be the only available data. In that case, it may be possible to smooth the data in order to obtain a set of partial molar quantities that is thermodynamically consistent. The procedure is simply to reconstmct the total molar property by a weighted mole fraction average of the n measured partial molar values and then recalculate normalised partial molar quantities. The new set should always be consistent. 

For pure substances, n is usually held constant. We will usually be working with molar quantities so that n = 1. The number of moles n will become a variable when we work with solutions. Then, the number of moles will be used to express the effect of concentration (usually mole fraction, molality, or molarity) on the other thermodynamic properties. 

So far our discussion of chemical thermodynamics has been limited to systems in which the chemical composition does not change. We have dealt with pure substances, often in molar quantities, but always with a fixed number of moles, n. The Gibbs equations... 

As chemists, we are most often concerned with reactions proceeding under conditions in which the temperature and pressure are the variables we control. Therefore, it is useful to have a set of properties that describe the effect of a change in concentration on the various thermodynamic quantities under conditions of constant temperature and pressure. We refer to these properties as the partial molar quantities. 

Most thermochemical calculations are made for closed thermodynamic systems, and the stoichiometry is most conveniently represented in terms of the molar quantities as determined from statistical calculations. In dealing with compressible flow problems in which it is essential to work with open thermodynamic systems, it is best to employ mass quantities. Throughout this text uppercase symbols will be used for molar quantities and lowercase symbols for mass quantities. 

In connection with the development of the thermodynamic concept of partial molar quantities, it is desirable to be familiar with a mathematical relationship known as Euler s theorem. As this theorem is stated with reference to homogeneous functions, we will consider briefly the namre of these functions. 

In this chapter, we shall consider the methods by which values of partial molar quantities and excess molar quantities can be obtained from experimental data. Most of the methods are applicable to any thermodynamic property J, but special emphasis will be placed on the partial molar volume and the partial molar enthalpy, which are needed to determine the pressure and temperature coefficients of the chemical potential, and on the excess molar volume and the excess molar enthalpy, which are needed to determine the pressure and temperature coefficients of the excess Gibbs function. Furthermore, the volume is tangible and easy to visualize hence, it serves well in an initial exposition of partial molar quantities and excess molar quantities. 

In thermodynamics these quantities are usually expressed in energy per mole (quantities in bold), while we are interested in the rate of energy change or energy flow in energy per time. For these we replace the molar enthalpy H (cal/mole) of the fluid by the rate of enthalpy flow in a flow system... 

This is often called the thermodynamic force for diffusion. It is necessary to divide by Avo-gadro s number NA since p, is a molar quantity. Thermodynamics show that... 

Figure 11. (a) The calculated partial molar entropy of oxygen (sQJ and (b) the calculated partial molar enthalpy of oxygen (fto2) as a function of 8 for La02Sr08Fe0 55Tio4503 s. Symbols are calculated by the Gibbs-Helmholtz equation. Lines correspond to the partial molar quantities calculated by statistical thermodynamics. 

Finally, the thermodynamic properties of a system considered as variables may be classified as either intensive or extensive variables. The distinction between these two types of variables is best understood in terms of an operation. We consider a system in some fixed state and divide this system into two or more parts without changing any other properties of the system. Those variables whose value remains the same in this operation are called intensive variables. Such variables are the temperature, pressure, concentration variables, and specific and molar quantities. Those variables whose values are changed because of the operation are known as extensive variables. Such variables are the volume and the amount of substance (number of moles) of the components forming the system. 

The subject of partial molar quantities needs to be developed and understood before considering the application of thermodynamics to actual systems. Partial molar quantities apply to any extensive property of a single-phase system such as the volume or the Gibbs energy. These properties are important in the study of the dependence of the extensive property on the composition of the phase at constant temperature and pressure e.g., what effect does changing the composition have on the Helmholtz energy In this chapter partial molar quantities are defined, the mathematical relations that exist between them are derived, and their experimental determination is discussed. 

The quantity (dX/dn j- P is defined as the partial molar quantity of the ith component. It occurs so frequently in the thermodynamics of solutions that it is given a separate symbol, XP Consequently, partial molar quantities are... 

In many processes, we are concerned with mixtures, i.e., gas, liquid or solid solutions, and hence we are concerned with the thermodynamic properties of a component in a solution - partial molar quantities ... 

These equations are of value for solution thermodynamics. For example, if a partial molar quantity of one component in a binary solution has been determined, then the partial molar quantity of the other component is fixed ... 

All the general thermodynamic relations can be applied with minor symbolic modifications to the partial molar quantities ... 

The general thermodynamic relationships, which apply to partial molar quantities, are also valid for the excess quantities. 

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. 

Many relations that hold between extensive thermodynamic variables also hold between the corresponding partial molar quantities. In particular, we have... 

The chemical properties of particles are assumed to correspond to thermodynamic relationships for pure and multicomponent materials. Surface properties may be influenced by microscopic distortions or by molecular layers. Chemical composition as a function of size is a crucial concept, as noted above. Formally the chemical composition can be written in terms of a generalized distribution function. For this case, dN is now the number of particles per unit volume of gas containing molar quantities of each chemical species in the range between ft and ft + / ,-, with i = 1, 2,..., k, where k is the total number of chemical species. Assume that the chemical composition is distributed continuously in each size range. The full size-composition probability density function is... 

An extensive variable may be converted into an intensive variable by expressing it per one mole of a substance, namely, by partially differentiating it with respect to the number of moles of a substance in the system. This partial differential is called in chemical thermodynamics the partial molar quantity. For instance, the volume vi for one mole of a substance i in a homogeneous mixture is given by the derivative (partial differential) of the total volume V with respect to the number of moles of substance i as shown in Eq. 1.3 ... 

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. 

Starting from the definition 5.22 we now establish several important properties of thermodynamic potentials (partial molar quantities of thermodynamic energy functions) for an ideal system of mixture. Differentiating G-H-TS with respect to n, with Tand p constant, we have pt = ht- Tsl and furthermore [d(jWf IT) / dT pn = (1 IT) (dp, / dT) - (p, / T1) = - [(r s, + pt) / T2] = -h,l T2. From this equation we obtain Eq. 5.34 for the partial molar enthalpy hf of a constituent i in an ideal mixture ... 

In conclusion, the partial molar quantity in thermodynamics functions consists of its unitary term and its mixing term as shown above. 

Furthermore, in analogy to the partial molar quantities of thermodynamic functions, the partial molar chemical exergy, echem l, can be defined for a substance i in a gaseous mixture, in a liquid solution, and in a solid solution as shown in Eq. 10.35 ... 

For both hypothetical and pseudo components, physical properties are computed by the same equations, which are based upon the correlations given in the Technical Data Book of the American Petroleum Institute (1). Equivalent molar quantities of these petroleum components are added to the amounts of the discrete (methane, etc.) components, to obtain the complete mixture for the thermodynamic calculations that follow. 

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. 

Similarly, for other surface excess thermodynamic quantities, the corresponding molar quantities are as follows ... 

The difference between a molar surface excess thermodynamic quantity xar r and the corresponding molar quantity x p for the gaseous adsorptive at the same equilibrium T and p is usually called the integral molar quantity of adsorption, and is denoted Aads r,r ... 

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