Differential heat, of solution

If dn moles of pure solid i, with molar enthalpy H°, are added at constant T and p to a solution in which the partial molar enthalpy is Hi, then the heat absorbed is 4q = dH = Hi - m) dn. (The system contains both solid and solution.) The differential heat of solution is defined as 4q/dn  

The differential heat of solution is a more generally useful quantity than the integral heat of solution defined in Section 7.22. 

1 What is the importance of the chemical potential What is its interpretation  

2 How can the quantity —dG/d be viewed as a driving force towards chemical equdibrium. Discuss. 

3 Sketch G versus for a reaction for which AG° 0. What are the roles of both AG° and the mixing Gibbs energy in determining the equilibrium position  

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The solubihty of the ammonium haUdes in water also increases with increasing formula weight. For ammonium chloride, the integral heat of solution to saturation is 15.7 kj /mol (3.75 kcal/mol) at saturation, the differential heat of solution is 15.2 kj /mol (3.63 kcal/mol). The solubihty of all three salts is given in Table 1 (7). 

The latent heats at 25 C are 7656 kcal/kmol for acetone and 10,490 kcal/kmol for water, and the differential heat of solution of acetone vapor in pure water is given as 2500 kcal/kmol. The specific heat of air is 7.0 kcal/(kmol-K). 

The heat absorbed when unit mass of solute is dissolved in an infinite amount of solvent is the differential heat of solution for zero concentration, Lo, and this is evidently equal to the integral heat of solution for concentration s plus the integral heat of dilution for concentration s ... 

As discussed further in Chapter 9, energy relationships are also influenced by surface properties, which must be taken into account once the crystals become smaller than ca. 1pm (Tab. 8.5). Langmuir (1971) calculated AGr(298) in liquid water as a function of particle size, using differential heat of solution values (as a function of... 

Figure 3.15 Plot of integral heat of solution Aifsoln(n) versus n (= moles H20/moles acid), showing the infinite-dilution limit A/fsoln(oo), the heat of dilution AHdn(ti, n2) from nx to n2, and the differential heat of solution (slope of tangent line) 8H(n ), 8H(n2) for representative concentrations...

Furthermore, the slope of the A/fsoln curve at n = n is called the differential heat of solution 8H(ni) ... 

How are partial molar quantities determined experimentally Sidebar 6.3 illustrates the general procedure for the special case of the partial molar volumes VA, Vr of a binary solution (analogous to the graphical procedure previously employed in Section 3.6.7 for finding differential heats of solution). As indicated in Sidebar 6.3, each partial molar... 

At 15-40°C the solubility of allopurinol (53) in the presence of polyvinylpyrrolidone increased with increasing concentration of the polyvinylpyrrolidone while differential heats of solution decreased and free energy of partitioning increased. This indicates that allopurinol forms a complex with polyvinylpyrrolidone (74JPP84). 

Heats of solution are not constant but generally vary with concentration of the components. For example, when HCl is dissolved in water AH/m changes from —17.9 to —17.4 kcal/mol as one proceeds from unit molality to infinite dilution. To handle such cases one distinguishes between integral heats of solution, AH/m, and differential heats of solution (dAH/dm)T,p that pertain to the addition of an infinitesimal amount of solvent to a solution of molality m. 

By the differential heat of solution is meant the heat which is evolved when 1 mol of the solute is dissolved in a very large quantity of the saturated solution. For very sparingly soluble substances whose saturated solutions may be regarded as infinitely dilute, the three kinds of heat of solution become identical. 

The differential heat of solution is of theoretical importance in the calculation of the variation of solubility with pressure. If a saturated solution containing solid solute is subjected to a pressure greater than the vapour pressure of the solution, the gaseous phase disappears and the system becomes divariant (two phases and two components). The concentration of the saturated solution (i.e. the solubility) is then a function of the pressure as well as of the temperature. When a condensed system of this kind is subjected to a further change in pressure the solid solute and the solution will not remain in equilibrium unless the temperature is changed simultaneously. As in the analogous case of the variation with pressure of the melting point of a pure substance (p. 221), the Clausius equation assumes the form — L/ ... 

For the majority of solids, solubilization is temperature dependent and increases in elevated temperatures as heat is absorbed during solubilizing. The degree to which temperature can influence solubility is dictated by the differential heat of solution, AHs, which represents the rate of heat change of a solution per mole of solute dissolved. The higher the heat of solution, the greater the influence of temperature on solubility, as shown by Eq. (7). 

By plotting the logarithm of the solubility in moles per liter vs. the reciprocal of the absolute temperature, the differential heat of solution can be calculated as the slope (-A/7s/2.3037 ). A positive heat of solution indicates an endothermic solubilization process (i.e., absorbs heat). Therefore, an increase in temperature increases solubility. A negative value indicates an exothermic solubilization process (i.e., emits heat) and a differential heat of solution near zero indicates that solubility is not significantly influenced by temperature. 

Solution calorimetry can be used on one level to merely obtain the enthalpy of solution for a given solute, or can be used in a deeper sense to obtain a full thermodynamic description of a system. The determination of solubility data over a defined temperature range can be used to calculate the differential heat of solution of a given polymorphic form. One can subtract the differential heats of solution obtained for the two polymorphs to deduce the heat of transition (A//Trans) between the two forms ... 

The absorption coefficient (of solubility) a of a gas is the volume of gas reduced to 0 C and 1 atm. pressure which will be dissolved by unit volume of solvent at the experimental temperature under a partial pressure of the gas of 1 atm. Show that for a dilute solution, Ns in equation (36.23) may be replaced by a. The absorption coefficient of nitrogen is 0.01685 at 15" and 0.01256 at 35 C. Determine the mean differential heat of solution per mole of nitrogen in the saturated solution in this temperature range. 

Another expression for the differential heat of solution of the solute may be obtained by adding and subtracting the term B% at the right-hand side of equation (44.8) thus,... 

Alternatively, the differential heat of solution d(fiJI)/dn2 ]ni, which is equal to L2 — L2, may be obtained if the observed heat change AH, when m moles of solute (equivalent to 712) are added to a definite quantity, e.g., 1(X)0 g. of solvent (equivalent to constant ni), is plotted against m, and the slope determined at any required composition (e.g., C, Fig. 36). If L% at this composition is available, then Ls may be derived immediately. 

It may be remarked that in dilute solution the heat of solution A/f is usually a linear function of the molality (see Fig. 36) the integral heat of solution per mole, i.e., AH/m, is then equal to the differential heat of solution d(AH)/dm. Provided the solution is sufficiently dilute, therefore, the experimental value of the former may be identified with — L2, since this is equivalent to the differential heat of solution at infinite dilution. 

Problem The integral heat of solution at infinite dilution per mole of KCl was found by extrapolation to be 4,120 cal. at 25 C. The integral heat of solution of 1 mole of KCl in 55.5 moles of water, to make a 1 molal solution, is 4,012 cal. From the slope of the plot of the heat of solution per mole of solute against the number of moles of solvent, the differential heat of dilution of 1 molal KCl solution is found to be 3.57 cal. mole What is the differential heat of solution of KCl in this solution ... 

Show that for a solution condsting of nt moles of solvent and n% moles of solute the following relationship holds tit (integral heat of solution — differential heat of solution) ni (differential heat of dilution). 

Since Hi is equal to the partial molar heat content of the solute at infinite dilution, it follows from equation (44.8) that AFT in this case is equal to the differential heat of solution of the solid salt in the infinitely dilute solution. In dilute solution the total heat of solution usually varies in a linear manner with the molality, and so the differential heat of solution is then equal to the integral heat of solution per mole (cf. 44h). 

For a sparingly soluble salt the differential heat of solution in the saturated solution is virtually the same as that at infinite dilution the former can be derived from the solubility of the salt at two temperatures ( 36f), and the resulting value can be used for Aff of the process under consideration. Alternatively, the experimentally determined heat of precipitation of a sparingly soluble salt may be taken as approximately equal, but of opposite sign, to the differential heat of solution. 

Fig. 9.9. Integral heat of solution of NaCl in water as a function of moles of salt added to 1 kg water. Slopes of tangents are differential heats of solution of salt at that concentration.

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