Partial molar quantities volume

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... 

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. 

How are partial molar quantities determined experimentally Sidebar 6.3 illustrates the general procedure for the special case of the partial molar volumes VA, Vr of a binary solution (analogous to the graphical procedure previously employed in Section 3.6.7 for finding differential heats of solution). As indicated in Sidebar 6.3, each partial molar... 

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 partial molar properties are not measured directly per se, but are readily derivable from experimental measurements. For example, the volumes or heat capacities of definite quantities of solution of known composition are measured. These data are then expressed in terms of an intensive quantity—such as the specific volume or heat capacity, or the molar volume or heat capacity—as a function of some composition variable. The problem then arises of determining the partial molar quantity from these functions. The intensive quantity must first be converted to an extensive quantity, then the differentiation must be performed. Two general methods are possible (1) the composition variables may be expressed in terms of the mole numbers before the differentiation and reintroduced after the differentiation or (2) expressions for the partial molar quantities may be obtained in terms of the derivatives of the intensive quantity with respect to the composition variables. In the remainder of this section several examples are given with emphasis on the second method. Multicomponent systems are used throughout the section in order to obtain general relations. 

We find from this discussion that, when the reference state of a component in a multicomponent system is taken to be the pure component at all temperatures and pressures of interest, the properties of the standard state of the component are also those of the pure component. When the reference state of a component in a multicomponent system is taken at some fixed concentration of the system at all temperatures and pressures of interest, the system or systems that represent the standard state of the component are different for the chemical potential, the partial molar entropy, and for the partial molar enthalpy, volume, and heat capacity. There is no real state of the system whose properties are those of the standard state of a component. In such cases it may be better to speak of the standard state of a component for each of the thermodynamic quantities. 

Partial molar quantities can be defined as the change of an extensive variable with respect to the mole number of one component at constant temperature, pressure, electric field, and mole numbers of all other components. Then, with Equations (14.73) and (14.74), the change of the partial molar entropy and partial molar volume with the electric field is given by... 

The measurement of partial molar quantities will be illustrated with reference to partial molar volumes. We can measure absolute volumes of solution and, thus, can determine partial molar volumes directly from its definition ... 

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 ... 

Here, is termed the specific chemical enthalpy, B, the specific thermal enthalpy and Bp the specific pressure enthalpy. The combination of the specificrthermal enthalpy, Bp, and the specific pressure enthalpy, Bp, may he named the specific physical enthalpy. When the material species is one of the components in a solution, Equations A-l through A-7 are valid, provided that the specific quantities are changed to the partial molar quantities. Note that superscript 0 refers to the standard state, and subscript 0 refers to the dead state cp is the specific heat, and v is the specific volume. 

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... 

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... 

Equation 41 shows that the chemical potential is a partial molar property. We will need other partial molar quantities (e.g., those for volume, enthalpy, and entropy) in dealing with pressure and temperature effects on energetics of reactions. 

Partial Molar Quantities Equation (1.16) may be simplified by introducing the quantity Vi called the partial molar volume of the component i defined by ... 

All these formulae are of the same structure they are all the sums of partial molar quantities each of which is multiplied by the corresponding stoichiometric coefficient in the reaction under consideration. As a result these equations have certain properties in common. We have already shown that heats of reaction can be added together in the same way as chemical reactions. It is immediately clear that this property is also true of the affinity A, the volume change and the variation of entropy s in general it is true for all quantities which can be put in the form where X is any extensive variable. 

Partial molar quantities are intensive thermodynamic functions. For example, the partial molar volume is the increase of the volume per mole of component i when the number of moles of i are modified with an infinitesimal amount. Thus, the partial molar volume of component i is given by the expression ... 

Another asp>ect of partial molar volumes and heat contents, in particular, arises from the thermod3mamic requirement that for an ideal gas mixture or for an ideal liquid solution, as defined for example in 30a and 34a, respectively, there is no change of volume or of heat content upon mixing the components. This means that the partial molar volume and heat content of each substance in the mixture are equal to the respective molar values for the pure constituents. Any deviation of the partial molar quantity from the molar value then gives an indication of departure from ideal behavior this information is useful in connection with the study of solutions. 

IV. General Methods.—In the methods described above for the determination of partial molar quantities, it has been tacitly assumed that the property G is one which is capable of experimental determination. Such is the case, for example, if 0 represents the volume or the heat capacity. However, if the property under consideration is the heat content then, like the free energj , it cannot be determined directly. In cases of this kind modified methods, which involve measurements of changes in the property, rather than of the property itself, can be used. It should be pointed out that the procedures are quite general and they are frequently adopted for the study of properties susceptible of direct measurement, as vrell as of those which are not. ... 

Any extensive property of a system can be written in terms of its partial molar quantities. Recall that an extensive property is a property that scales proportionally with the size of the system. If the system doubles in size, then value of the extensive property should double. If the size of the system increases by a factor t, then the value of the extensive property should increase by a factor t. For example, taking X to be the volume V, then we expect the total volume of the system V to increase by a factor t if the number of total moles in the system are increased by a factor t, holding the composition of the system fixed. This feature can be expressed mathematically as... 

On the basis of the above analysis it has been shown the partial molar quantities are easily obtained from intensive quantities like the molar volume when this quantity is plotted as a function of an intensive composition variable like the mole fraction. The plots in fig. 1.2 show that the molar volume is almost a linear function of the mole fraction of solute. If the curves in fig. 1.2 were actually perfect straight lines, the partial molar volumes would be constant independent of solution composition. Such a situation would arise if the solution were perfectly ideal. In reality, very few solutions are ideal, as will be seen from the discussion in the following section. In order to see more clearly the departure from ideality, one defines and calculates a quantity called the excess molar volume. This quantity is equal to the actual molar volume less the molar volume for the solution if it were ideal. The latter can be considered as the volume of the solution that would be found if the molecules of the two components form a solution without expansion or contraction. Thus, the ideal molar volume can be defined as... 

Water is unique among polar solvents in that it is a small molecule with a low molar volume. As a result, the concentration of water in pure water is 55.5 M. This means that the mole fraction of water in dilute aqueous solutions is close to one, and the partial molar quantities in these solutions are close to the corresponding quantities for the pure solvent. Other solvents have considerably higher molar volumes, and therefore, lower concentrations in the pure solvent. The... 

The simulation is performed in a grand canonical ensemble (GCE) where all microstates have the same volume (V), temperature and chemical potential under the periodic boundary condition to minimize a finite size effect [30, 31]. For thermal equilibrium at a fixed pu, a standard Metropolis algorithm is repetitively employed with single spin-flip dynamics [30, 31]. When equilibrium has been achieved, the lithium content (1 — 5) in the Li, 3 11204 electrode at a given pu is determined from the fraction of occupied sites. The thermodynamic partial molar quantities oflithium ions are theoretically obtained by fluctuation method [32]. The partial molar internal energy Uu at constant Vand T in the GCE is readily given by [32, 33]... 

The approach to the thermodynamics of solubilization in micellar solutions is based on the determination of a given partial molar property of the solute (volume, enthalpy, heat capacity, compressibility) as a function of the surfactant content. The simplest approach is to use the pseudophase model. The partial molar quantity, L will thus be an average value of Y in the micellar and aqueous phases, as described by... 

The addition of solute will also influence the cmc of the surfactant, which in turn means that a correction is needed for the overall partial molar quantity, F This has been taken into account in the models proposed by Roux et al. and DeLisi et al. ° The models have been applied to different thermodynamic properties, mostly volumes and heat capacities, and for different surfactant-solute systems. 

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... 

Partial molar quantities play an important role in the study of non-ideal mixtures but we have to use them to only a limited extent in elementary thermodynamics. They can usually be replaced by the corresponding molar quantities. Thus, in simple calculations involving perfect gases or ideal solutions, Vi can be replaced by the volume of one mole of pure i in the appropriate physical state. 

Experimental values for some of the partial molar quantities can be obtained from laboratory measurements on mixtures. In particular, mixture density measurements can be used to obtain partial molar volumes, and heat-of-mixing data yield information on partial molar enthalpies. Both of these measurements are considered here. In Chapter 10 phase equilibrium measurements that provide information on the partial molar Gibbs energy of a component in a mixture are discussed. Once the partial molar enthalpy and partial molar Gibbs energy are known at the anie temperature, the partial molar entropy can be computed from the relation S-, = (G — H-,)/T. 

So far we have considered only the volume as a partial molar quantity. But calculations involving solutes will require knowledge of all the thermodynamic properties of dissolved substances, such as H, S, Cp, and of course G, as well as the pressure and temperature derivatives of these. These quantities are for the most part derived from calorimetric measurements, that is, of the amount of heat released or absorbed during the dissolution process, whereas V is the result of volume or density measurements. 

For characterization of the interrelation between the composition and extensive properties of the solution the outstanding American physico-chemist Gilbert Newton Lewis (1875-1946) introduced additional intensive parameters under the common mme partial molar quantity. Among them are partial molar volume, partial molar heat capacity, chemical potential, etc. 

For changes of state at constant composition, we need Cj, and the volumetric equation of state to be able to integrate (3.5.11) and (3.5.13) for Ali and AS. With values for AU and AS, we can then apply the defining Legendre transforms (3.2.9) for H, (3.2.11) for A, and (3.2.13) for G to obtain changes in the other conceptuals. If the change of state includes a change in composition, then we will also need values for the partial molar volume, enthalpy, and entropy as shown in 3.4.3, these partial molar quantities are simply related to the chemical potential. 

Because of the simplicity of the functions of state of the ideal gas, they serve well as models for other mixing experiments. Dilute solutions, for example, can be modeled as ideal gases with the empty space between the gas atoms being filled with a second component, the solvent. In this case, the ideal condition can be maintained as long as the overall interaction between solvent and solute is negligible. Deviations from the ideal mixing are treated by evaluation of the partial molar quantities, as illustrated on the example of volume, V, in Fig. 2.25. The first row of equations gives the definitions of the partial molar volumes and Vg and shows the addition... 

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