Differential molar entropies

Also, the differential molar entropy of adsorption, AJ, is defined as... 

The differential molar entropies can be plotted as a function of the coverage. Adsorption is always exothermic and takes place with a decrease in both free energy (AG < 0) and entropy (AS < 0). With respect to the adsorbate, the gas-solid interaction results in a decrease in entropy of the system. The cooperative orientation of surface-adsorbate bonds provides a further entropy decrease. The integral molar entropy of adsorption 5 and the differential molar entropy are related by the formula = d(n S )ldn for the particular adsorbed amount n. The quantity can be calculated from... 

In the case of adsorption in micropores, when the condition of temperature invariance of the characteristic curve is fulfilled, the net differential molar heat of adsorption, q, and the differential molar entropy of adsorption, AS, may be expressed (2) as... 

Figure 4. Dependence of differential molar entropy of adsorption, AS, on degree of filling, 0, for zeolite L (AS, e.u.)...

Note that the first right-hand side term in Eq. (63) is the same as the last in Eq. (S4). Differentiating Eq. (63) with respect to n we obtain the differential molar entropy of the adsorbate... 

S being the differential molar entropy of component 1, and the differential molar volume of component 1, we know that ... 

Fig. 9. Differential molar entropy of the Xe adlayer on Ih(lll) as a function of Xe coverage. Data points (circles) and theoretical curves assuming a completely mobile 2D ideal gas (sohd line), a completely mobile Volmer gas (dashed line) and a localized adlayer (dash-dotted line) [88K1]. The coverage is given in substrate...

It should be noted that these methods of evaluation of the entropy of adsorbed hydrogen and OH species provide integral molar entropies, which are defined as the entropy of the adlayer divided by the number of moles of adsorbed species. Conversely, differential molar entropies are defined as the derivative of the entropy of the adlayer with the number of moles of adsotbed species. Clearly, with the available experimental data, the evaluation of integral molar entropies involves a smaller uncertainty, in comparison to differential molar entropies. 

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) ... 

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)... 

For an ideal gas we will show later that the molar entropy is a function of the independent variables, molar volume Vm and temperature T. The total differential dSra is given by the equation... 

For a given uptake and temperature T, dSs/dT = Cp where Cp is the differential molar heat capacity of sorbed fluid. This expression can be approximated by Tm ASs/AT = Cp where Tm is the mean temperature corresponding with the interval AT over which ASs is the entropy change, and where Cp refers to the temperature Tm. For classical oscillators Cp should be 24.9 J/mole/deg, and thus it is interesting to compare Cp calculated as above with this value. A5S/AT did not vary much with amount sorbed, so that Cp found for one uptake is typical. Several values of Cp are given below. All are near but a little below the classical oscillator value. 

The molar differential heat capacity at the melting point has proved to be negligible only for benzene and rigid, polyaromatic hydrocarbons, and a compilation of literature data indicates that ACpon the average is 80% of the molar entropy of fusion (Neau and Flynn, 1990). Although the Lrst assumption has been applied in many studies, the second assumption has gained favor in recent pharmaceutical literature (Subrahmanyam et al., 1992 Claramonte et al., 1993 Yu et al., 1994). 

Similar arguments and definitions can be applied to the other partial molar thermodynamic functions and properties of the components in solution. By differentiation of Equation (8.71), the following expressions for the partial molar entropy, enthalpy, volume, and heat capacity of the kth component are obtained ... 

Once the species present in a solution have been chosen and the values of the various equilibrium constants have been determined to give the best fit to the experimental data, other thermodynamic quantities can be evaluated by use of the usual relations. Thus, the excess molar Gibbs energies can be calculated when the values of the excess chemical potentials have been determined. The molar change of enthalpy on mixing and excess molar entropy can be calculated by the appropriate differentiation of the excess Gibbs energy with respect to temperature. These functions depend upon the temperature dependence of the equilibrium constants. 

And the ideal partial molar entropy is obtained by differentiation with respect to ni or n2... 

The entropy of a pure substance is a function of temperature Tand pressure p. Equations 2.13 and 3.9 yield the total differential of the molar entropy ds of a pure substance ... 

In the general case, the integral molar entropy of adsorption is not equal to the mean differential entropy of adsorption over the range of surface excess concentration from 0 to r, because of the extra term of the right-hand side of Equation (2.67). 

Differential and integral molar entropies of adsorption follow immediately from the measured heats as T (3q/3n). and T" (Aq/3n). respectively. We also have, for equilibrium... 

The above expressions furnish four interrelations between the molar entropy Ss and the various differential entropies (dSs/dris). Note that it is only when the three intensive quantities T, P,(j) are held fixed that the molar entropy of the adsorbed phase is equal to its partial molal counterpart. The terms involving 1 are usually small and are generally neglected. 

Tliis equation defines the partial molar property of species i in solution, where the generic symbol Mt may standfor the partial molar internal energy t/, the partial molar enthalpy //, the partial molar entropy 5,, the partial molar Gibbs energy G,, etc. It is a response function, representing the change of total prope ity n M due to additionat constant T and f of a differential amount of species i to a finite amount of solution. 

The differential molar variations of the entropy and heat capacity can also be expressed via the Jjj values. Clearly the expressions (3)-(4) are valid within the limit of negligible adsorption, i.e. the Henry region. 

Rychlicki, G., Terzyk, A.P., and Zawadzki,. (1993). Low-coverage adsorption of methanol, ethanol and carbon-tetrachloride on homo and heterogeneous surface — differential heat and integral molar entropy. Polish J. Chem., 67, 2019—28. 

We end this section by considering the phenomenological characterization of SI solutions in terms of their partial molar entropies and enthalpies. If one assumes that equation (5.35) is valid for a finite interval of temperatures and pressures, then, by differentiation, we obtain... 

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