Standard partial molar heat capacity

Fig. 17.10. The standard partial molar volume and standard partial molar heat capacity of aqueous NaCl as represented by the HKF model, showing the characteristic inverted-U shape and steep negative slopes at high and low temperatures.

Fig. 17.12. The standard partial molar heat capacity of aqueous HCl as a function of temperature. Squares are the experimental data of Tremaine et al. (1986), and the solvation and non-solvation contributions, which add to the line fitting the data, are from the HKF model. Note how the shapes of the two contributions combine to give the inverted-U shape of the measured heat capacity.

Fig. 17.14. The standard partial molar heat capacities of aqueous HCl, Na" and Cl as a function of temperature. The experimental data for Cl are in fact those for aqueous HCl of Tremaine et al. (1986) from Figure 17.12, illustrating the fact that the heat capacity of is zero by convention (see discussion in text). The heat capacity of aqueous NaCl is a measured quantity, and that of aqueous Na is obtained from the difference of the NaCl and HCl (or Cl ) data.

Hovey, J. K., Tremaine, P. R., Thermodynamics of aqueous aluminium Standard partial molar heat capacities of aluminium(-i-3) from 10 to 55°C, Geochim. Cosmochim. Acta, 50, (1986), 453-459. Cited on page 643... 

Figure 15.11 The standard partial molar heat capacities of aqueous NaCI, Na+, and Cl as a function of temperature. Triangles data from Gardner et al. (1969). Solid squares data from Helgeson et al. (1984, Table 15).

Ciiss, C. M. and Cobble, J. W., 1961, The thermodynamic properties of high temperature aqueous solutions. I. Standard partial molar heat capacities of sodium chloride and barium chloride from 0 to 100 °C. Jr. Amer. Chem. Soc., 83 3223-8. 

Accmate estimations for logic K require knowledge of the standard partial molar heat capacity and volume functions for the reaction, i.e., ACp = -K AGId Pjp and AV = (dAG/dp)r, respectively. The exact differentiation equation yields the expression... 

Similarly, the entropy of ionization, AS°, the standard partial molar heat capacity of ionization, AjC , and the standard partial molar volume of ionization, AV°, can be derived from AG using standard thermodynamic identities (Mesmer et al., 1988) ... 

TABLE 4.5 Ionic Standard Partial Molar Heat Capacities in Non-aqneons Solvents, C (I, S)/J-K -molat 25°C ... 

The properties of ions in solution depend, of course, on the solvent in which they are dissolved. Many properties of ions in water are described in Chapters 2 and 4, including thermodynamic, transport, and some other properties. The thermodynamic properties are mainly for 25°C and include the standard partial molar heat capacities and entropies (Table 2.8) and standard molar volumes, electrostriction volumes, expansibilities, and compressibilities (Table 2.9), the standard molar enthalpies and Gibbs energies of formation (Table 2.8) and of hydration (Table 4.1), the standard molar entropies of hydration (Table 4.1), and the molar surface tension inaements (Table 2.11). The transport properties of aqueous ions include the limiting molar conductivities and diffusion coefficients (Table 2.10) as well as the B-coefficients obtained from viscosities and NMR data (Table 2.10). Some other properties of... 

Information on partial molar heat capacities [1,18] is indeed very scarce, hindering the calculation of the temperature correction terms for reactions in solution. In most practical situations, we can only hope that these temperature corrections are similar to those derived for the standard state reactions. Fortunately, due to the upper limits set by the normal boiling temperatures of the solvents, the temperatures of reactions in solution are not substantially different from 298.15 K, so large ArCp(T - 298.15) corrections are uncommon. 

In equations (18.91) and (18.92), C° 2 and V are the partial molar heat capacity and partial molar volume of the surfactant in the infinitely dilute solution (standard state values). 

To evaluate the integral one needs to have the standard enthalpy change as a function of temperature. This information can be computed from the partial molar heat capacities. The partial molar heat capacity for component j is defined by... 

Hence U2 can be interpreted as the chemical potential of pure solute in a hypothetical liquid state corresponding to extrapolation from infinite dilution (which serves as reference state) to X2 = 1 along a line where Y2 = U that is, along the Henry s law line. In physical terms, it might be regarded as a hypothetical state in which the mole fraction of solute is unity (pure solute), but some thermodynamic properties are those of the solute 2 in the reference state of infinite dilution in solvent 1 (e.g., partial molar heat capacity). Since from the context it should always be clear whether the superscript circle denotes standard state" or "pure substance", no further distinction is introduced. 

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. 

This standard textbook on chemical thermodynamics contains an Appendix (no. 4) of selected data for aqueous electrolyte solutions. Compiled are activity coefficients, Debye-Huckel parameters, relative partial molar enthalpies, and relative partial molar heat capacities for about 70 of the most common electrolytes in aqueous solution at 25 °C. More recent Debye-HUckel parameters are to be found 1n the pages of Pitaer, Pelper, and Busey and of Bradley and Pitaer (see item [121]) and the paper of Clarke and Glew, Hem [223. 

This table contains standard state thermodynamic properties of positive and negative ions in aqueous solution. It includes enthalpy and Gibbs energy of formation, partial molar entropy, and partial molar heat capacity. The standard state is the hypothetical ideal solution with molality equals 1 mol/kg. 

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. 

The standard state is here a purely hypothetical one, just as is the case with gases ( 30b) it might be regarded as the state in which the mole fraction of the solute is unity, but certain thermodynamic properties, e.g., partial molar heat content and heat capacity, are those of the solute in the reference state, he., infinite dilution (cf. 37d). If the solution behaved ideally over the whole range of compodtion, the activity would always be equal to the mole fraction, even when n = 1, i.e., for the pure solute (cf. Fig. 24,1). In this event, the proposed standard state would represent the pure liquid solute at 1 atm. pressure. For nonideal solutions, however, the standard state has no reality, and so it is preferable to define it in terms of a reference state. 

When the heat of solution of an electrolyte in water to form a dilute solution is measured calorimetrically at several temperatures, the standard partial molar (constant pressure) heat capacity of the electrolyte, Cpe , is obtained from the temperature coefficient of these heats, extrapolated to infinite dilution. Alternatively, the difference between the specific heat of a dilute solution of the electrolyte and that of water is obtained by flow microcalorimetry to yield the same quantity. Accurate density data at the appropriate temperature are required for the use of this technique. A recent description of the methods available for the determination of the heat capacities of aqueous electrolytes is presented by Hakin and Bhuiyan (2010). Such determinations are accurate to 1 to 4 J mol (Hepler and Hovey 1996)... 

AGk, AH, AH thus obtained represent the stoichiometric variations of the Gibbs free energy, enthalpy and entropy, respectively, on the transfer of one mole of solute between the two phases in standard state. AG is the same for the hypothetical ideal state and the real state pro wded that the activity equals unity in both. However AHJ is different in the two cases and reference should be made to the hypothetical ideal state. Because the intermolecular attractions which determine AH are identical in the hypothetical (standard) and reference states, AH refers also to the modification of partial molar enthalpy between the reference states. The same conclusion holds true for the modification of molar heat capacities. A/Sk, like AGk, does not apply to the modification of partial molar entropy between reference states but refers to the hypothetical standard state described above. 

Thermodynamic properties of ions in nonaqueous solvents are described in terms of the transfer from water as the source solvent to nonaqueous solvents as the targets of this transfer. These properties include the standard molar Gibbs energies of transfer (Table 4.2), enthalpies of transfer (Table 4.3), entropies of transfer (Table 4.4) and heat capacities of transfer (Table 4.5) as well as the standard partial molar volumes (Table 4.6) and the solvation numbers of the ions in non-aqueous solvents (Table 4.10). The transfer properties together with the properties of the aqueous ions yield the corresponding properties of ions in the nonaqueous solvents. 

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. 

Both these considerations would be taken into accoimt if the activation process were assumed to occur at a constant pressure, p, such that the partial molar volume of the solvent is independent of the temperature, though this possibility does not appear to have been considered. A full discussion is beyond the scope of this chapter, but the resulting heat capacities of activation are unlikely to differ greatly from those determined at a constant pressme of, say, 1 atm. (see p. 137). Unfortunately, this approach requires the definition of rather clumsy standard states for solutes, e.g., hypothetically ideal, 1 molal, under a pressure such that a given mass of the pure solvent occupies a particular volume. 

The standard state is here also a hypothetical one it is equivalent to a 1 molal solution in which the solute has some of the partial molar properties, e.g., heat content and heat capacity, of the infinitely dilute solution. It has been referred to as the hypothetical ideal 1 molal solution At high dilutions the molality of a solution is directly proportional to its mole fraction ( 32f), and hence dilute solutions in which the activity of the solute is equal to its molality also satisfy Henry s law. Under such conditions, the departure from unity of the activity coefiicient 7 , equal to a2/m, like that of 7n, is a measure of the deviation from Henry s law. 

In this chapter, we discuss the various standard states used for the Gibbs energy and the activity. The standard state used for enthalpy, volume and heat capacity is quite different, and is discussed, along with a more detailed look at partial molar properties, in Chapter 10. 

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