Ideal solutions heat capacity

Equations (15) and (16) suggest that for ideal solutions, i.e., for the solutions involving molecules of similar sizes and intermolecular forces, Yj = 1 and the solubility can be estimated based on enthalpy of fusion, heat capacities of solute in solid and liquid states, and triple point temperature. Moreover, Eqs. (15) and (16) imply that solubility increases with temperature. 

This chapter presents quantitative methods for calculation of enthalpies of vapor-phase and liquid-phase mixtures. These methods rely primarily on pure-component data, in particular ideal-vapor heat capacities and vapor-pressure data, both as functions of temperature. Vapor-phase corrections for nonideality are usually relatively small. Liquid-phase excess enthalpies are also usually not important. As indicated in Chapter 4, for mixtures containing noncondensable components, we restrict attention to liquid solutions which are dilute with respect to all noncondensable components. 

Cp is the (total) molar heat capacity of the system at constant pressure, usually approximated, as though for an ideal solution, by1... 

The product FtCP, where Ft is the total molar flow rate and CP is the molar heat capacity of the flowing stream, may replace tic, in these equations, if the stream is an ideal solution. Integration of equation 21.5-8, or its equivalent, may need to take into account the dependence of mcP and FtCP on T and/or /A, and of (—AHRA) on T (see Example 15-7). However, compared to the effect of T on kA, the effect of T on (-AHRA) and cP is usually small. 

The ideal solubility of a non-dissociating solute, assuming the effects of pressure and specific heat capacity change on melting are negligible is [7,8] ... 

The standard state for the heat capacity is the same as that for the enthalpy. For a proof of this statement for the solute in a solution, see Exercise 2 in this chapter. This choice of standard state for components of a solution is different fixjm that used by many thermodynamicists. It seems preferable to the choice of a 1-bar standard state, however, because it is more consistent with the extrapolation procedure by which the standard state is determined experimentally, and it leads to a value of the activity coefficient equal to 1 when the solution is ideal or very dilute whatever the pressure. It is also preferable to a choice of the pressure of the solution, because that choice produces a different standard state for each solution. For an alternative point of view, see Ref. 2. 

Let us consider a system of M ideal monatomic gas molecules in a cubic box kept at a constant temperature T. For a very dilute gas, where the molecules do not interact with one another, the quantum mechanical solution is a number of electronic wave functions with three quantum numbers nx, riy, and for the translational energies in three dimensions. The energy of a molecule for a set of quanmm numbers, the observed average energy, and the heat capacity at constant volume are given by... 

However, two types of systems are sufficienfry important that we can use them almost exclusively (1) liquid aqueous solutions and (2) ideal gas mixtures at atmospheric pressure, hr aqueous solutions we assume that the density is 1 gtcvc , the specific heat is 1 cal/g K, and at any solute concentration, pressure, or temperature there are -55 moles/hter of water, hr gases at one atmosphere and near room temperature we assume that the heat capacity per mole is R, the density is 1/22.4 moles/hter, and aU components obey the ideal gas equation of state. Organic hquid solutions have constant properties within 20%, and nonideal gas solutions seldom have deviations larger than these. 

According to Equation (4.5) the ideal solubility of a compound is only dependent upon the heat of fusion, the difference in heat capacity of the solid and supercooled liquid and the melting point of the compound. Since there are no properties of the solvent included in the ideal solubility equation, the solubility of a compound should be the same in all solvents. This equation overlooks all solute-solvent and solvent-solvent interactions. 

The physical state of each substance is indicated in the column headed State as crystalline solid (c), liquid (liq), gaseous (g), or amorphous (amorp). Solutions in water are listed as aqueous (aq). Solutions in water are designated as aqueous, and the concentration of the solution is expressed in terms of the number of moles of solvent associated with 1 mol of the solute. If no concentration is indicated, the solution is assumed to be dilute. The standard state for a solute in aqueous solution is taken as the hypothetical ideal solution of unit molality (indicated as std state, m = 1). In this state the partial molal enthalpy and the heat capacity of the solute are the same as in the infinitely dilute real solution (aq. m). 

The change of enthalpy on mixing, AHM[T, P, x], at constant temperature and pressure is seen to be zero for an ideal solution. The change of the heat capacity on mixing at constant temperature and pressure is also zero for an ideal solution, as are all higher derivatives of AHM with respect to both the temperature and pressure at constant composition. Differentiation of Equations (8.57), (8.59), and (8.60) with respect to the pressure yields... 

This excess heat capacity cE is the difference between the mean molar heat capacity of the non-ideal binary solution, - C kat + j), and the mean molar heat capacity of... 

These design features eliminate almost all pathways to a serious accident, but elimination of the lithium chemical energy would be even better. Hence, the ideal DT fusion reactor would utilize lithium that is combined with other elements to produce a fluid that is not reactive with air or water, but still retains the low density and high heat capacity of pure lithium. Various lead-lithium solutions have been considered, but none fully satisfy these criteria. 

Activities of Electrolytes.—When the solute is an electrolyte, the standard states for the ions are chosen, in the manner previously indicated, as a hypothetical ideal solution of unit activity in this solution the thermodynamic properties of the solute, e.g., the partial molal heat content, heat capacity, volume, etc., will be those of a real solution at infinite dilution, i.e., when it behaves ideally. With this definition of the standard state the activity of an ion becomes equal to its concentration at infinite dilution. 

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. 

This is the equation for the crystallization curve of a solution provided that the solution is ideal, that no mixed crystals are formed, and that the difference between the heat capacities in the liquid and solid states is small enough to justify neglecting the second term in (22.4). This important equation is due to Schroder and van Laar.j ... 

So iar in our energy calculations, we have been considering each substance to be a completely pure and separate material. The physical properties of an ideal solution or mixture may be computed from the sum of the properties in question for the individual components. For gases, mole fractions can be used as weighting values, or, alternatively, each component can be considered to be independent of the others. In most instances so far in this book we have used the latter procedure. Using the former technique, as we did in a few instances, we could write down, for the heat capacity of an ideal mixture,... 

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. 

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. 

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