Integral and Differential Heats of Solution

So far we have considered only the volume as a partial molar quantity. But calculations involving solutes will require knowledge of all the thermodynamic properties of dissolved substances, such as H, S, Cp, and of course G, as well as the pressure and temperature derivatives of these. These quantities are for the most part derived from calorimetric measurements, that is, of the amount of heat released or absorbed during the dissolution process, whereas V is the result of volume or density measurements. 

Another difference between V and other properties such as H and G is that absolute values of V are obtainable, whereas they are not for H and G, so that we deal always with values of AH and AG. These deltas or differences can mean more than one thing, and it is important to be clear about the nature of the difference in each case. 

For instance, if we add a mole of NaCl to a kilogram of pure water and measure the heat absorbed, rather than the volume change, we find that about 917 calories are required to keep the temperature constant at 25°C. That is 

If we add another mole of NaCl, we find that this time about 749 cal are required, i.e., 

we expect to find, since 77 is a state variable, that adding 2 moles of NaCl to a kilogram of water gives the same result as the two above operations combined, i.e., 

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Calorimetric data for solutions are handled in a number of different ways, which can be confusing. In addition to integral and differential heats of solution and the partial molar enthalpy of solution, we also have the apparent partial molar enthalpy, the relative partial molar enthalpy, and the relative apparent partial molar enthalpy. To see how these terms cirise, consider the following. 

ZEL Zelikman, S.G. and Mikhailov, N.V., Studies on the stracture and properties of carbochain polymers in dilute solutions. IV. The integral and differential heats of solution and densities of the polymers (Russ.), Vysokomol. Soedin., 1, 1077, 1959. 

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. 

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 ... 

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...

SIDEBAR 6.2 INTEGRAL AND DIFFERENTIAL HEATS AND FREE ENERGIES OF 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). 

The increase of enthalpy that takes place when one mole of solute is dissolved in a sufficiently large volume of solution (which has a particular composition), such that there is no appreciable change in the concentration, is the molar differential heat of solution. When stating a value for this quantity, the specified concentration and temperature must also be quoted. Because the differential heat of solution is almost constant in very dilute solutions, the molar differential and integral heats of solution are equal at infinite dilution. At higher concentrations, the differential heat of solution generally decreases as the concentration increases. 

The partial or differential heat of solution, AH2, is the change in enthalpy when a very small amount of pure solute is added to a large amount of either solution or pure solvent. In the latter instance, the resultant quantity which is properly identified as the partial heat of solution at infinite dilution is sometimes referred to more simply as the heat of solution. For polymer solutions, AH 2 is often expressed as the heat absorbed per unit mass of solute added and can be found as the derivative of the integral heat of mixing ... 

As M is increased in comparison with m, the heat of solution approaches a limiting value, which is evidently a special case of the differential as well as of the integral heat of solution it represents the first stage in the supposed series of small processes when the solute dissolves in initially pure solvent, and is called the heat of solution at infinite dilution ... 

As we have seen in the case above of internal energy [Eq. (24.12)], this relation can be useful for defining various isobaric heat quantities such as integral and differential, molar and specific heats of reaction, transition, solution, mixing, etc. These are all produced similarly at constant p and T and, depending upon the process in question, each one can have various symbols and names. We will be content with only two examples, one integral quantity and one differential quantity ... 

For higher conversion values an integral approach was applied, where the differential equation of plug flow reactor rate=dX d(W/F), was solved numerically with boundary condition Xo(fV/F=0)=0. The solution gives a numerical relationship X=X(W/F) and the predicted conversion is given as Xmodei X(W/F=W/Fexp). In order to determine the activation energy and the heat of adsorption, the Arrhenius and van t Hoff laws were applied, k-=l exp(-Ea/RT), K==K exp(AH/RT). 

The process inputs are defined as the heat input, the product flow rate and the fines flow rate. The steady state operating point is Pj =120 kW, Q =.215 1/s and Q =.8 1/s. The process outputs are defined as the thlrd moment m (t), the (mass based) mean crystal size L Q(tK relative volume of crystals vr (t) in the size range (r.-lO m. In determining the responses of the nonlinear model the method of lines is chosen to transform the partial differential equation in a set of (nonlinear) ordinary differential equations. The time responses are then obtained by using a standard numerical integration technique for sets of coupled ordinary differential equations. It was found that discretization of the population balance with 1001 grid points in the size range 0. to 5 10 m results in very accurate solutions of the crystallizer model. 

In the former case, the solid remains suspended in the liquid in the microcalorimeter cell. Then a mother solution is added, either in one step (to obtain an integral heat. A ffUnt)) or in several steps, leading to differential heats, A H(dlff)l). In the latter case one could also speak of titration calorimetry. some commercial microcalorimeters are especially constructed for such titrations. Since, with these techniques, part of the added adsorptive remains in solution, the enthalpy of dilution A yH must be subtracted it is dependent on composition and can be determined in a blank without adsorbent. The difference between A y H(int) and A y H(dlff) has been discussed before, see sec. 1.3c. 

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