Molar quantities partial

Partial molar quantities are used to describe the change in properties of a multi-component system when one component is added at constant temperature, pres- 

It follows from equation (1.3.4) that, at constant temperature and pressure, a change in the volume of a two-component solution is given by 

This expression may be integrated under conditions that the relative amounts of Ha and ng, that is, the composition of the solution, do not change  

The resulting equation states that the volume of the solution may be calculated given the number of moles of each component and their partial molar volumes. In terms of the molar volume, this equation becomes 

It can be shown that there is a relationship between the partial molar volumes for a given solution composition. Taking the total derivative of the volume on the basis of equation (1.4.4), one obtains at constant temperature and pressure 

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. 

Considering any extensive property K, the partial molar quantity is defined by 

Once again, integrating as in the Gibbs-Duhem equation, yields 

This is identical with the definition of t, in terms of G, but this is only true for G because only the pressure P and temperature T hold constant for G. Therefore, in general 

The relationship between partial molar quantities is analogous to the relationships between the extensive variables. As an example, 

This equation can be obtained by differentiating Eq. (1.47) with respect to rii at constant temperature, pressure and n. Next, we will define the fugacity, which has all the features of chemical potential, but unlike the chemical potential has an absolute value. 

The fugacity, often represented by the symbol f, has the units of pressure. It is defined from the following relationship  

Another relationship is needed to complete the definition of fugacity  

Let us derive the expression for calculating cp or f. Subtract RTd In XiP from both sides of Eq. (1.94)  

Chemical potential /x, can therefore be expressed by any of the alternative definitions 

Although each definition (6.11a-c) is fully equivalent to (6.8), the G-based equation (6.11c) is most convenient for defining /x under the usual laboratory conditions of constant T and P. 

The variations (d/drij) with respect to chemical species (usually at constant T, P) are referred to as partial molar quantities. For any molar extensive property X, the partial molar derivative Xt with respect to chemical component i is given by 

11c) shows that chemical potential ixt could also be described as partial molar Gibbs energy  

However, the implications and usefulness of partial molar quantities go far beyond this example. 

If xi contains an entropy contribution, the form of the chemical potential is unaltered but its resolution into entropy and heat contributions must be carried out according to operations like those applied above to the free energy of mixing. 

From the chemical potential we may at once set down expressions for the activity ai of the solvent and for the osmotic pressure tz of the solution, using standard relations of thermodynamics. For the activity 

Since the pure solvent has been chosen as the standard state, ai = Pi/Pij to the approximation that the vapor may be regarded as an ideal gas. For the osmotic pressure itVi=—(mi—Mi) where Vi is the molar volume of the solvent. Thus, according to Eq. (26) 

The osmotic method is most useful, of course, in dilute solutions where the theory which we have developed above is invalid. Disregarding this deficiency for the moment, we may expand the logarithmic term in series with the retention only of terms in low powers of V2. Then 

It is more convenient to use the concentration c in grams per ml. Now V2 = cv, where v is the (partial) specific volume of the polymer, and since X is the ratio of the molar volumes of polymer and solvent, we have V2/XY1 = cv/xYi = c/M. Hence 

If is of interest to note that for ideal systems we always have 

We can also establish, by somewhat long but elementary algebra, that 

These inequalities are important in the study of the stability of thermodynamic systems. 

Starting from the definition (7.1) we shall now establish several important properties of partial molar quantities for an ideal system. Taking first the partial molar enthalpy, we have, from (6.32), 

The partial molar volumes of the components of an ideal system possess this same property  

A binary solution becomes more dilute as any of the solute composition variables becomes smaller. In the limit of infinite dilution, the expressions for b/ a become  

The rule of thumb that the molarity and molality values of a dilute aqueous solution are approximately equal is explained by the relation Macb/pX = MaWb (fromEq. 9.1.14), or cb/pX = niB, and the fact that the density pX of water is approximately 1 kg Hence, if the solvent is water and the solution is dilute, the numerical value of cb expressed in molL 4s approximately equal to the numerical value of wb expressed in mol kg T 

We can describe the composition of a phase with the amounts of each species, or with any of the composition variables defined earlier mole fraction, mass fraction, concentration, or molality. If we use mole fractions or mass fractions to describe the composition, we need the values for all but one of the species, since the sum of all fractions is unity. 

Other composition variables are sometimes used, such as volume fraction, mole ratio, and mole percent. To describe the composition of a gas mixture, partial pressures can be used (Sec. 9.3.1). 

When the composition of a mixture is said to be fixed or constant during changes of temperature, pressure, or volume, this means there is no ehange in the relative amounts or masses of the various species. A mixture of fixed composition has fixed values of mole fractions, mass fractions, and molalities, but not necessarily of concentrations and partial pressures. Concentrations will change if the volume changes, and partial pressures in a gas mixture will change if the pressure changes. 

It should be pointed out that the equation of Affm in terms of Awn is not important because it is not feasible to calculate Awn as such. Equation (4.13) in terms of % is important in polymer chemistry A//m here is called the contact energy. A dimensionless quantity, Xi characterizes the interaction energy per solvent molecule divided by kT k being the Boltzman constant). The term A//m is also called the Rory interaction energy. The physical meaning of Xi is 

In the literature of polymer chemistry, the two parameters 82 (the solubility parameter of polymer) and Xi (the interaction energy of the solvent) are frequently investigated. Both are measurable quantities. The quantity AH, on the other hand, is only of secondary importance. 

Once AAm and AH are known, AG can easily be calculated. Thus, according to Hildebrand, 

When we deal with solutions, we need another independent variable to specify the composition of the system, in addition to P, V, and T. This additional independent variable is n, the number of moles of a component in the system. A thermodynamic 

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In experimental work it is usually most convenient to regard temperature and pressure as die independent variables, and for this reason the tenn partial molar quantity (denoted by a bar above the quantity) is always restricted to the derivative with respect to Uj holding T, p, and all the other n.j constant. (Thus iX = [right-hand side of equation (A2.1.44) it is apparent that the chemical potential... 

From cross-differentiation identities one can derive some additional Maxwell relations for partial molar quantities ... 

Neutron magnetic moment Partial molar quantity A... 

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 ... 

Partial molar quantities have per mole units, and for Yj this is understood to mean per mole of component i. The value of this coefficient depends on the overall composition of the mixture. Thus Vj o the same for a water-alcohol mixture that is 10% water as for one that is 90% water. 

For a pure component the partial molar quantity is identical to the molar (superscript °) value of the pure substance. Thus for pure component i... 

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... 

This is converted to a partial molar quantity by differentiation ... 

From the definition of the partial molar quantities [Eq. (8.8)] we write... 

The Gibbs-Duhem equation also follows from the definition of partial molar quantities nid/Hi + r 2d 2 0. With the Gibbs-Duhem equation, d G/dc2 becomes... 

The chemical potential is a partial molar quantity defined by the last term in equation 4 ... 

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. 

The well-known Gibbs-Duhem equation (2,3,18) is a special mathematical redundance test which is expressed in terms of the chemical potential (3,18). The general Duhem test procedure can be appHed to any set of partial molar quantities. It is also possible to perform an overall consistency test over a composition range with the integrated form of the Duhem equation (2). 

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. 

Volume is an extensive property. Usually, we will be working with Vm, the molar volume. In solution, we will work with the partial molar volume V, which is the contribution per mole of component i in the mixture to the total volume. We will give the mathematical definition of partial molar quantities later when we describe how to measure them and use them. Volume is a property of the state of the system, and hence is a state function.1 That is... 

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. 

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 [Pg.213]

A variety of procedures can be used to determine Z, as a function of composition.2 Care must be taken if reliable values are to be obtained, since the determination of a derivative or a slope is often difficult to do with high accuracy. A number of different techniques are employed, depending upon the accuracy of the data that is used to calculate Z, and the nature of the system. We will now consider several examples involving the determination of V,- and Cpj, since these are the properties for which absolute values for the partial molar quantity can be obtained. Only relative values of //, and can be obtained, since absolute values of H and G are not available. For H, and we determine H, — H° or — n°, where H° and are values for H, and in a reference or standard state. We will delay a discussion of these quantities until we have described standard states. 

For a component in a mixture, the fugacity is defined by the same equation as for a pure substance, except that partial molar quantities are substituted for molar quantities. Thus,... 

The integral heat of mixing is, of course, the quantity directly measured in the calorimetric method However, the heat change on diluting a solution of the polymer with an additional amount of solvent may sometimes be measured in preference to the mixing of pure polymer with solvent In either case, the desired partial molar quantity AHi must be derived by a process of differentiation, either graphical or analytical. 

Expressing this equation in terms of relative partial molar quantities, the following relationships are obtained ... 

Partial molar quantity X (varies) X = (dXIdn-o) TtPtn... 

The partial molar quantity of a molar quantity Q, related to the component A, is generally written as 0A and is defined by ... 

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. 

RELATIONSHIPS BETWEEN PARTIAL MOLAR QUANTITIES OF DIFFERENT COMPONENTS... 

From the definition of partial molar quantities [Equation (9.12)], Equation (9.26) can be written as... 

Although the function / is a homogeneous function of the mole numbers of degree 1, the partial molar quantities, and are homogeneous functions of degree 0 that is, the partial molar quantities are intensive variables. This statement can be proved by the following procedure. Let us differentiate both sides of Equation (2.32) with respect to x ... 

Equation (9.33) is one of the most useful relationships between partial molar quantities. When applied to the chemical potential, it becomes... 

This equation is very useful in deriving certain relationships between the partial molar quantity for a solute and that for the solvent. An analogous equation can be written for the derivatives with respect to dn2-... 

CALCULATION OF PARTIAL MOLAR QUANTITIES AND EXCESS MOLAR QUANTITIES FROM EXPERIMENTAL DATA VOLUME AND ENTHALPY... 

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. 

PARTIAL MOLAR QUANTITIES BY DIFFERENTIATION OF J AS A FUNCTION OF COMPOSITION... 

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