Partial molal derivative

Interesting is a comparison of the volumes occupied by individual complexes in solution and in the solid state. The partial molal volumes can be obtained from precise measurements of the solution densities of the complexes as a function of concentration [177]. These values may be subsequently compared with the unit cell volumes per complex molecule derived from the crystal structure. For Fe[HB(pz)3]2, the apparent molal volume in tetrahydrofuran solution was determined as 340.9 em mol Taking into account that the complex in solution forms an equilibrium between 86% LS and 14% HS isomers and employing the volume difference between the two spin states AF° = 23.6 cm mol S the volume of the LS isomer was calculated as 337.6 cm mol This value agrees closely with the volume of 337.3 cm mol for the completely LS complex in solid Fe[HB(pz)3]2 [105]. 

The pressure-volume-temperature (PVT) properties of aqueous electrolyte and mixed electrolyte solutions are frequently needed to make practical engineering calculations. For example precise PVT properties of natural waters like seawater are required to determine the vertical stability, the circulation, and the mixing of waters in the oceans. Besides the practical interest, the PVT properties of aqueous electrolyte solutions can also yield information on the structure of solutions and the ionic interactions that occur in solution. The derived partial molal volumes of electrolytes yield information on ion-water and ion-ion interactions (1,2 ). The effect of pressure on chemical equilibria can also be derived from partial molal volume data (3). 

Partial molal volumes can be related to the corresponding Gibbs free energy terms through the partial derivatives on P (see equations 2.28 and 2.33). 

Before the derivation of a general expression for the partial molal quantities, a simple example of partial molal quantity is treated. Consider liquids metal A and metal B, which form a complete solution of composition A Bj Let the volume of 1 g atom of liquid A and liquid B be and Pg, respectively. The volume of the solution, F(x), the mole fraction of which is x (1 — x), generally satisfies the following relation. 

The mean ionic activity coefficients of hydrobromic acid at round molalities (calculated by means of Equation 2) are summarized in Tables XI, XII, and XIII for x = 10, 30, and 50 mass percent monoglyme. Values of —logio 7 at round molalities from 0.005 to 0.1 mol-kg-1 were obtained by interpolating a least squares fit to a power series in m which was derived by means of a computer. These values at 298.15° K are compared in Figure 2 with those for hydrochloric acid in the same mixed solvent (I) and that for hydrobromic acid in water (21). The relative partial molal enthalpy (H2 — Hj>) can be calculated from the change in the activity coefficient with temperature, but we have used instead the following equations ... 

In the event that the molar volumes of solute and solvent are not comparable, and the thermal agitation is not adequate to achieve maximum entropy of mixing, a nonideal entropy of mixing exists (Bustamante et al., 1989). Two equations that account for the nonideal entropy of mixing have been derived by considering the partial molal volurfiA,and the volume fraction, occupied by each solution component. The L rst was developed by Flory and Huggins (Hildebrand, 1949 Kertes, 1965) ... 

The events outlined in Fig. 2 are expressed in Eq. (5.3), derived from a series of substitutions and differentiations of AGmix with respect to the solvent s partial molal content (Cowie, 1973, 1991) ... 

The partial molal free energy of a spray of finely dispersed droplets is not so simply related to escaping tendency as it is in systems usually treated by thermodynamic methods. Even the definition of the partial molal free energy is not entirely devoid of difficulty. To discuss equilibrium we wish to think of the transfer of a small amount of adsorbent from each droplet in the spray to another phase— e.g., the gas phase (16). If a little material leaves each droplet, there must be a concomitant decrease in surface area, A, and an increase in specific area, quantity related to escaping tendency of material is not the derivative of free energy with respect to n, the number of moles in the spray at either constant area or constant specific area. The condition is that the number of droplets, v, be constant. (This means that area divided by mass to the 2/3 power is nearly invariant.)... 

The derivative of each of the curves 1, 2, 3, etc., where it crosses the Vt line is the derivative of the volume with respect to n, at constant v—i.e., it is the partial molal volume corresponding to G of Equation 8. The functions represented by curves 1, 2, and 3 are not homogeneous. The slope of V is the derivative of the volume at constant specific area. V, of course, was defined so that it is a homogeneous function of n. 

Adsorption isotherms for the system BaS04-H20 at three temperatures have been obtained. Thermodynamic study of these data reveals that part of the free energy decrease in the adsorption process involves changes in the partial molal free energy of the adsorbent. From the three isotherms differential and integral heats of adsorption were derived and compared with new calorimetric determinations of the same thermodynamic functions. In both kinds of measurements exactly the same system and exactly the same materials were used. 

Notice that partial molal properties of B are properly specified only when the partial derivatives are taken while holding only the intensive variables fixed thus one should note the difference between (1.25.10a) and (1.25.10b) with regard to the quantity labeled as B. 

Because of their rigid cell walls, large hydrostatic pressures can exist in plant cells, whereas hydrostatic pressures in animal cells generally are relatively small. Hydrostatic pressures are involved in plant support and also are important for the movement of water and solutes in the xylem and in the phloem. The effect of pressure on the chemical potential of water is expressed by the term VWP (see Eq. 2.4), where Vw is the partial molal volume of water and P is the hydrostatic pressure in the aqueous solution in excess of the ambient atmospheric pressure. The density of water is about 1000 kg m-3 (1 g cm-3) therefore, when 1 mol or 18.0 x 10-3 kg of water is added to water, the volume increases by 18.0 x 10-6 m3. Using the definition ofV,., as a partial derivative (see Eq. 2.6), we need to add only an infinitesimally small amount of water (dnw) and then observe the infinitesimal change in volume of the system (dV). We thus find that Vw for pure water is 18.0 x 10-6 m3 mol-1 (18.0 cm3 mol-1). Although Vw can be influenced by the solutes present, it is generally close to 18.0 x 10-6 m3 mol-1 for a dilute solution, a value that we will use for calculations in this book. 

The concept of free energy is introduced in Chapter 2 (Section 2.2A,B) in presenting the chemical potential of species /, fij. The chemical potential is actually the partial molal Gibbs free energy with respect to that species that is, fij equals (8G/drij)TPEh (Eq. IV.9 in Appendix IV), where the subscripts on the partial derivative indicate the variables that are held constant. We must consider the Gibbs free energy of an entire system to determine the chemical potential of species j. In turn, G depends on each of the species present, an appropriate expression being... 

Using Equation IV.2 we can readily determine the pressure dependence of the Gibbs free energy as needed in the last bracket of Equation IV. 11—namely, dG/dP)TEhn.n is equal to Fby Equation IV.12. Next, we have to consider the partial derivative of this V with respect to rij (see the last equality of Eq. IV. 11). Equation2.6 indicates that dV/dnj)TPEkn. is Vj, the partial molal volume of species j. Substituting these partial derivatives into Equation IV. 11 leads to the following useful expression ... 

The preceding derivations are given in more detail by Marek and Standart (5) and are applied by Marek (4) to several binary systems. The terms and 02 are evaluated from partial molal volume data or approximated from generalized correlations. According to Marek and Standart, the products and T2y2 must satisfy all criteria of thermodynamic consistency, as should yi and y2 for nonassociating systems. 

One of the diflSculties in applying the Born equation is that the effective radius of the ion is not known further, the calculations assume the dielectric constant of the solvent to be constant in the neighborhood of the ion. The treatment has been modified by Webb who allowed for the variation of dielectric constant and also for the work required to compress the solvent in the vicinity of the ion further, by expressing the effective ionic radius as a function of the partial molal volume of the ion, it was possible to derive values of the free energy of solvation without making any other assumptions concerning the effective ionic radius. 

A distinction must be made between the functional dependence of V, on the mole numbers, and the parametric dependence of V) on T and P for, doubling T and P obviously does not double the volume of any mixture. These two quantities are held fixed in specifying the partial derivatives in Eq. (1.19.1). However, as T and P are assigned different sets of values (Pi, Pi), (P2, P2), and so forth, Vi and V change in a manner prescribed by experiment. Thus, the fact that these parameters are held fixed in a given set of partial differentiations does not preclude the partial molal volumes from changing as one passes from one set of experimental conditions to a different set. 

Here it is emphasized that the term partial molal should be reserved for partial derivatives of state functions with respect to composition when the fixed parameters are intensive variables, such as T and P. By contrast, where extensive parameters are held constant, as in Bpi/dT)v,xi, the derivatives are known as differential functions. 

Here also, Hi, being derived from p., (T, F,, ), is the partial molal enthalpy of i in solution at pressure P, while H, being derived from p, (F, 1, xf), is the enthalpy of pure i at unit pressure. 

Since the laws of regular solutions closely approximate those derived by statistical mechanics for lattice models of the liquid state in which the partial molal volume of component i in the mixture is equal to the molal volume of pure liquid /, it is sometimes assumed that Vt =... 

In order to elucidate the influence of substituent ionic groups on the hydration, the partial molal volume and compressibility data of ionic dextran derivatives have been analyzed in terms of three kinds of hydration (22, 28). The partial molal volume of ionic dextran derivatives, V , may be written as the sum... 

Figure 7. Relation between the partial molal compression and the degree of substitution of dextran derivatives at 25°C (22, 2%) (U) CMD (%) DS (O) SPD and (A) DP.

The derivative of Ea with respect to temperature has been shown by LaMer (1(X)) in a pioneering paper to be equal to the difference between the partial molal heat capacity of molecules which react, and that of all molecules. Therefore, dEa/rfT has been termed by LaMer heat capacity of activation. Equation 8 shows that this differs only by R from the specific heat of activation of the transition state theory, as defined by Eq. 6—in fact, the difference between the two is often neglected. It seems to us that dEaldT is the more meaningful quantity. 

Our first objective is to derive the partial molal volumes of both constituents of a binary mixture as a function of ionic strength. Once this is accomplished, a reasonable mixing rule for multicomponent systems can be proposed. 

We next cite flie independently derived results of Section 1.12 we introduce" dS/dT)p = Cp/T, as well as the Maxwell relation dS/dP) = dV/dT)p p derived below. As before, the partial derivatives — 0V/dP)p and dV/dT)p are specified by fiVand by aV, respectively. Also, we introduce dS/dni)jp . and dV/dnijp, to represent partial molal entropies 5, and volumes U, that are again found by independent proceidures developed later. We thus rewrite Eq. (1.10.4b) in the less unwieldy form... 

Thus we thought it necessary to measure the solute activity (or activity coefficient), namely the mean activity (or activity coefficient) of polyelectrolytes. Our subsequent study was extended to the partial molal volume of polyelectrolytes, which is a pressure derivative of the mean activity coefficient. By this volumetric property, the solvait-solute interactions in polyelectrolyte solutions, which could be discussed only qualitatively on the basis of the mean activity coefficient data, were considered in a quantitative way. Furthermore, the catalytic action of polyelectrolytes in ionic reactions was studied and discussed in terms of the activity coefficient data. 

The effect of concentration of free (molecular) ammonia on the activity of the electrolyte was derived mainly from two 80 C data points of Miles and Wilson having 16 to 17 molal free ammonia concentration. Data points below 0.2 ionic strength were fitted by application of Kielland s estimation of ionic activity coefficients(6 2). Details are presented elsewhere(45), together with graphs giving partial pressures of ammonia and hydrogen sulfide for temperatures from 80 to 260 F over a range of liquid concentration. 

His procedure was used for the calculation of the activity coefficients in the aqueous solution of two electrolytes with a common ion from isopiestic data (3). Kelly, Robinson, and Stokes (4) proposed a treatment of isopiestic data of ternary systems with two electrolytes by a procedure based on the assumption that at all values of molal concentrations, mi,m2, the partial derivatives may be expressed by a sum of two functions in their differential form as follows ... 

In the concentration range regarding the ED processes, the effective diffusion coefficient (Z>B) can be predicted via the Gordon relationship (Reid et al, 1987), which accounts for the partial derivative of the natural logarithm of the mean molal activity coefficient (y+) with respect to molality (m) and solvent relative viscosity (rjr) ... 

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