Heat content, partial molar

The energy of a system can be changed by means of thermal energy or work energy, but a further possibility is to add or subtract moles of various substances to or from the system. The free energy of a pure substance depends upon its chemical nature, its quantity (AG is an extensive property), its state (solid, liquid or gas), and temperature and pressure. Gibbs called the partial molar free heat content (free energy) of the component of a system its chemical potential... 

Along the three-phase line liquid-clathrate-gas the variation of the composition with temperature is considerable (cf. CD in Fig. 3), because when applying Eq. 27 to this equilibrium, the relatively small quantity AH = 0.16 kcal/mole has to be replaced by the much larger difference/ —//ql between the partial molar heat functions of / -hydroquinone and the liquid phase, which amounts to about —6 kcal/mole. The argon content of the solid reaches a minimum at the quadruple point. 

H = qp, heat content F = wp, work function G/N = /r, partial molar free enthalpy... 

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... 

When applying the Eirchhoff equation, in order to determine the variation with temperature of the heat content change accompanying a reaction in solution, tiie heat capacity to be employed is a special quantity, caUed the partial molar heat capacity." This quantity will be described in Chapter XVIII, in connection with a general discussion of the properties of substances in solution. 

The equations derived in 30c, 30d thus also give the variation with pressure and temperature of the fugacity of a constituent of a liquid (or solid) solution. In equation (30.17), Vi is now the partial molar volume of the particular constituent in the solution, and in (30.21), i is the corresponding partial molar heat content. The numerator — fti thus represents the change in heat content, per mole, when the constituent is vaporized from the solution into a vacuum (cf. 29g), and so it is the ideal" heat of vaporization of the constituent i from the given solution, at the specified temperature and total pressure. 

This expression will hold for any solution, but in the special case of an ideal solution, / // is equal to Ni, the mole fraction of that constituent, by equation (34.1). Since the mole fraction, for a mixture of definite composition, is independent of the temperature, it follows that for an ideal solution jH — Hi must be zero, i.e., and Rt are identical. Consequently, the partial molar heat content of any constituent of an ideal solution is equal to the molar heat content of that substance in the pure liquid state H°). As a result, equation (26.6), which for heat contents takes the form... 

These results can be inserted into equation (34.21), and at the same time the partial molar heat content terms Ri and Ri may be replaced by the corresponding molar heat contents Hi and H t, respectively, for the pure substances, since ideal behavior has been postulated for both phases (cf. 30d, 34a) it is then seen that... 

Show that although the partial molar heat content of the constituent of an ideal solution is independent of the composition ( 34a), this is not the case for the partial molar free energy and entropy. Derive expressions for (d/Lii/dN<)r p and (dSi/dNi)T,p for an ideal solution. 

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. 

At infinite dilution, y. is equal to unity at all temperatures, by definition, and at the same time p becomes identical with po, so that, in this case also, the standard partial molar heat content of the solute is equal to its value in the infinitely dilute solution at 1 atm. pressure. 

If the relative partial molar heat content Iri of the solvent (i.e., the differential heat of dilution) is small, as it is for dilute solutions, the activity coefficient of the... 

If the vapor behaves ideally, or if the solute is nonvolatile, as is usually the case with electrolytes, the partial molar heat content ttg of the gaseous solvent may be replaced by its molar heat content Hg. Further, if the pressure is taken as 1 atm., Hg — H ia equal to the molar heat of vaporization AH, of the solvent at its normal boiling point. 

By definition ( 38c), Li is equal to Hi — H i, where Si is the partial molar heat content of the solvent in the solution, and Hu is the molar heat content of the pure liquid solvent, which has been chosen as the standard state for this constituent. Upon differentiation with respect to temperature, at constant (1 atm.) pressure, it is found that... 

The relative partial molar heat contents (Li) in cal. mole and the relative, partial molar heat capacities ( pi — Cpi) in cal. deg. mole, of the water in hydrochloric acid solutions are as follows ... 

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. 

In order to indicate the fact that the value of G as given by equation (42.1) applies to the constituent 2, i.e., the solute, a subscript 2 is sometimes included. However, this is usually omitted, for in the great majority of cases it is understood that the apparent molar property refers to the solute. It i.s seen from equation (42.1) that o is the apparent contribution of 1 mole of the component 2 to the property G of the mixture. If the particular property were strictly additive for the two components, e.g., volume and heat content for ideal gas and liquid solutions, the value of 4>q would be equal to the actual molar contribution, and hence also to the partial molar value. For nonideal systems, however, the quantities are all different. 

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. ... 

IR. Relative Partial Molar Heat Contents.—The partial molar thermal properties, namely, heat content and heat capacity, are of particular interest, as well as of practical importance, as will be seen from some of the examples to be given below. In accordance with the general definition ( 26a), the partial molar heat content of any constituent of a solution is represented by... 

For the solvent, the standard state and the reference state are identical, on the basis of the usual convention, and consequently the molar heat content in the reference state may be represented by The partial molar heat content of the solvent in a solution relative to the heat content in the reference state is then 5"i — /u this quantity is called the relative partial molar heat content of the solvent, and is represented by the symbol ii, so that in any solution... 

For the sake of consistency, a slight modification is sometimes made in this expression. The partial molar heat content of the pure solvent, i.e., 5 , which is the same as in a solution at infinite dilution, is, of course, identical with the molar heat content of the pure solvent, i.e., 5 = hence, equation (44.1) may be written... 

For the solute, the reference state is the infinitely dilute solution, and although this is not the same as the standard state, the partial molar heat contents are the same in both cases ( 37d). The reference value, which is the partial molar heat content of the solute at infinite dilution, can then be represented by the symbol The relative partial molar heat content L2 in any solution is thus given by... 

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