Differential molar enthalpies

The molar enthalpy for the transition from a solid to a supercooled liquid is not a constant with respect to temperature. The molar heat capacities of the solid and supercooled liquid forms of the solute inLuence its magnitude at temperatures below the melting point. It is frequently assumed that either the molar heat capacity of the solid at constant prestiiiippnd the molar heat capacity of its liquid form at constant pressure,pi, are nearly constant or that they change at the same rate with a change in temperature. In either case, the molar differential heat capacity, deLned as... 

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. 

Different types of heats of adsorption have been defined in classical thermodynamics but they are numerically similar. Their relationship to experimental determinatiorrs is more or less straightforward [68Cer, 83Cerj. The molar differential heat of adsorption, of a component i from the gas phase (1) on a solid (2) is defined as the difference in enthalpy associated with the transfer of one mole of i to the surface of the substrate at constant T, P and other components nj. Asstrming ideal gas behavior, the differential heat of adsorption is defined as [66Defj... 

An example is the partial molar enthalpy Hi of a constituent of an ideal gas mixture, an ideal condensed-phase mixture, or an ideal-dilute solution. In these ideal mixtures. Hi is independent of composition at constant T and p (Secs. 9.3.3, 9.4.3, and 9.4.7). When a reaction takes place at eonstant T and p in one of these mixtures, the molar differential reaction enthalpy H is eonstant during the proeess, H is a linear function of and Af// and Ai7m(rxn) are equal. Figure 11.6(a) illustrates this linear dependence for a reaction in an ideal gas mixture. 

Recall that AHm(rxn) is a molar integral reaction enthalpy equal to A//(rxn)/A, and that ArH is a molar differential reaction enthalpy defined by and equal to (dH/d )T,p-... 

We can also find the effect of temperature on the molar differential reaction enthalpy Af//. From Eq. 11.3.5, we have (9Ar///97 )p j = AfC. Integration from temperature T to temperature T yields the relation... 

First let us consider a solution process in which solute is transferred from a pure solute phase to a solution. The molar differential enthalpy of solution, A o H, is the rate of change of H with the advancement soi at constant T and p, where soi is the amount of... 

Note that because the values of and are independent of the solution composition, the molar differential and integral enthalpies of solution at infinite dilution are the same. 

The relations between A//(sol) and the molar integral and differential enthalpies of solution are illustrated in Fig. 11.9 on the next page with data for the solution of crystalline sodium acetate in water. The curve shows A//(sol) as a function of fsou with fsoi defined as the amount of solute dissolved in one kilogram of water. Thus at any point along the curve, the molality is /Mb = soi/(l kg) and the ratio A//(sol)/ soi is the molar integral enthalpy of solution A//m(sol, /mb) for the solution process that produces solution of this molality. The slope of the curve is the molar differential enthalpy of solution ... 

La can be related to molar differential and integral enthalpies of solution as follows. The enthalpy change to form a solution from amounts a and b of pure solvent and solute is given, from the additivity rule, by A//(sol) = (haHa + n H ) — (haH + We... 

We see that Lb is equal to the difference between the molar differential enthalpies of solution at the molality of interest and at infinite dilution. 

The third method assumes we measure the integral enthalpy of solution A//(sol) for varying amounts soi of solute transferred at constant T and p from a pure solute phase to a fixed amount of solvent. From Eq. 11.4.5, the molar differential enthalpy of solution is given by Asoi/f = dA//(sol)/ d soi when a is held constant. We make the substitution... 

Most reactions cause a change in the composition of one or more phases, in which case A//m(rxn) is not the same as the molar differential reaction enthalpy, Af/f = dH/d )r,p, unless the phase or phases can be treated as ideal mixtures (see Sec. 11.2.2). Corrections, usually small, are needed to obtain the standard molar reaction enthalpy Ar f ° from Ai/m(rxn). 

For convenience of notation, this book will use Asoi.aT/ to denote the molar enthalpy difference i/A(sln) - H (s). Aso, aH is the molar differential enthalpy of solution of solid A in the solution at constant T and p. The first integral on the right side of Eq. 12.2.3 requires knowledge of Aso1,a7/ over a temperature range, but the only temperature at which it is practical to measure this quantity calorimetrically is at the equilibrium transition temperature Tf. It is usually sufficient to assume Asoi.aT/ is a linear function of T ... 

Here Aso1,a is the molar differential enthalpy of solution of solid or gaseous A in the liquid mixture, and Asoi,a1 is the molar differential volume of solution. Equation 12.3.5 is a relation between changes in the variables T, p, and Xa, only two of which are independent in the equilibrium system. 

If the solubility xb increases with increasing temperature, Asoi,b ° must be positive and the solution process is endothermic. A decrease of solubility with increasing temperature implies an exothermic solution process. These statements refer to a solid of low solubility see page 357 for a discussion of the general relation between the temperature dependence of solubility and the sign of the molar differential enthalpy of solution at saturation. 

The ideal solubility of a solid at a given temperature and pressure is the solubility calculated on the assumptions that (1) the liquid is an ideal liquid mixture, and (2) the molar differential enthalpy of solution equals the molar enthalpy of fusion of the solid (Asoi,B7/=Afus,B )-These were the assumptions used to derive Eq. 12.5.4 for the freezing-point curve of an ideal liquid mixture. In Eq. 12.5.4, we exchange the constituent labels A and B so that the solid phase is now component B ... 

The ideal behavior of the adsorption layer also means that h and hi are independent of the concentration of the mixture, a condition that is rarely met. It is apparent from arguments we present later, however, that molar differential exchange enthalpy is constant in a certain range of composition our assumption therefore has to be accepted. 

It may be noted that some authors [3] use a different procedure to determine the molar differential enthalpy of adsorption by mean of a differential calorimeter coupled with a manometric device. They prefer to work with only one calorimetric vessel connected to the manometric device, the reference cell being maintained under vacuum. In these conditions we obtain IfomEq. 7.48 that the molar heat of adsorption is equal to ... 

Let us consider now the coadsorption of two gases or more, the definition of the calorimetric heat is exactly the same as for the adsorption of a single component. In this case, it corresponds obviously to the differential molar enthalpy of coadsorption. It is not possible to measure directly by calorimetry the differential enthalpy of adsorption of each component present in the mixture. Thus, for the coadsorption of two components A and B, the molar calorimetric coadsorption heat is equal to the molar differential enthalpy of coadsorption ... 

However, if the adsorbed phase can be considered as an ideal solution, in which the molecular interactions are the same as for the adsorption of single components, it is possible to calculate the molar differential coadsorption enthalpy by mean of the relation ... 

In the case of multi-components adsorption, the partial molar differential adsorption enthalpies and entropies of each component i present in the gas mixture cannot be directly measured by experiments. However, it is possible to estimate them by mean of the tangent method based on the well-known Gibbs-Duhem relation. 

It may be noticed that in the case of an ideal adsorbed solution the partial molar differential values ArZ j is constant and equal to the partial molar value of single component ArZ j. If Z represents the enthalpy, we retrieve from the Eq. 7.97 the Eq. 7.60, which allows the prediction of the coadsorption enthalpy from the adsorption enthalpies of single components. 

The molar entropy and the molar enthalpy, also with constants of integration, can be obtained, either by differentiating equation (A2.1.56) or by integrating equation (A2.T42) or equation (A2.1.50) ... 

Similarly, the differential molar enthalpy of adsorption, Ji is defined as... 

Partial differentiation of Eqns (4.17a) and (4.17b) with respect to x allows us easily to relate the variation in the preexponential term with the variation in partial molar entropy of MY, and the variation in activation energy with the variation in partial molar enthalpy by the relationships (Fig. 4.5)... 

The volumetric liquid holdup, 4>L, depends on the gas/vapor and liquid flows and is calculated via empirical correlations (e.g., Ref. 65). For the determination of axial temperature profiles, differential energy balances are formulated, including the product of the liquid molar holdup and the specific enthalpy as energy capacity. The energy balances written for continuous systems are as follows ... 

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