Partial molar quantities enthalpy, entropy, volume

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. 

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. 

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. 

Extensive thermodynamic quantities Z(p, 7, ni,..., rik) of the type given by Eq. (47), such as volume V, entropy S, enthalpy //, heat capacity C, and Gibbs energy G, yield partial molar quantities ... 

To describe the linkage relationships between partial molar quantities, we first focus on homogeneous functions. Extensive properties such as volume, energy, entropy, enthalpy, and free energy are first-order homogeneous functions of... 

The absolute values of some partial molar quantities, such as the volume, can be determined experimentally. Other absolute values, however, cannot be these include partial molar enthalpies, free enthalpies, and free entropies. However, their changes are measurable. 

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. 

Calculation of A//e -quantities from the dependence of AG on temperature is less reliable than direct calorimetric measurements (Franks and Reid, 1973 Frank, 1973 Reid et al., 1969). However, disagreement between published A//-functions for apolar solutes in aqueous solutions may also stem from practical problems associated with low solubilities (Gill et al., 1975). Calorimetric data have the advantage that, as theory shows, the standard partial molar enthalpy H3 for a solute in solution is equal to the partial molar enthalpy in the infinitely dilute solution, i.e. x3 - 0. A similar identity between X3 and X3 (x3 - 0) occurs for the volumes and heat capacities but not for the chemical potentials and entropies. The design of a flow system for the measurement of the heat capacity of solutions (Picker et al., 1971) has provided valuable information on aqueous solutions. 

Because it applies mostly to electrolytes, it is discussed in Chapter 15. Briefly, Helgeson models the behavior of solutes by developing equations for the standard state partial molar volume (Helgeson and Kirkham 1976) and standard state partial molar heat capacity (Helgeson et al. 1981) as a function of P and T, with adjustable constants such that they can be applied to a wide variety of solutes. If you know these quantities (V°, C°p), you can calculate the variation of the standard state Gibbs energy, and that leads through fundamental relationships to equilibrium constants, enthalpies, and entropies. 

The chemical potential i (i.e. the partial molar free energy) is the most fundamental physical quantity to describe the thermodynamic properties of polymer solutions. If p is represented as functions of absolute temperature (7), pressure (7 ) and the composition (for example, the polymer concentration for a binary mixture), one can determine not only the molecular weight, M, of the solute (the polymer), but also many other thermodynamic quantities, such as the partial molar entropy, the partial molar enthalpy and the partial molar volume of each component. 

Fm is the metal molar volume, which accounts for the steric effects at the interface, and C is a constant fitted to experimental values. This model takes care of the metal-oxygen and the metal-cation pair interactions through the quantities AHq(M) and AH y Neglecting the entropy variations, a partial mixing enthalpy AH Q may be written as a function of the AA, BB and AB pair interactions eAA. and 6ab and as a function of the mean coordination number Z of an atom A dissolved in B, in the following way ... 

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