Molar integral entropy

To characterize the state of the adsorbed phase, it is useful to evaluate its molar entropy, s , defined as the mean molar value for all the molecules adsorbed over the complete range of surface coverage up to the given amount adsorbed. The molar integral entropy of adsorption. As, is then defined as... 

UHV surface analysis, apparatus designs, 36 4-14 see also Ultrahigh vacuum surface analysis mechanisms, 32 313, 319-320 Modified Raney nickel catalyst defined, 32 215-217 hydrogenation, 32 224-229 Modifying technique of catalysts, 32 262-264 Modulated-beam mass spectrometry, in detection of surface-generated gas-phase radicals, 35 148-149 MojFe S CpjfCOlj, 38 352 Molar integral entropy of adsorption, 38 158, 160-161... 

The partial molar entropy of adsorption AI2 may be determined from q j or qsi through Eq. XVII-118, and hence is obtainable either from calorimetric heats plus an adsorption isotherm or from adsorption isotherms at more than one temperature. The integral entropy of adsorption can be obtained from isotherm data at more than one temperature, through Eqs. XVII-110 and XVII-119, in which case complete isotherms are needed. Alternatively, AS2 can be obtained from the calorimetric plus a single complete adsorption isotherm, using Eq. XVII-115. This last approach has been recommended by Jura and Hill [121] as giving more accurate integral entropy values (see also Ref. 124). 

The phenomenology described above can be applied to any thermodynamic extensive function, Yt, for a solution. The integral molar enthalpy, entropy and... 

Let Sg° be the standard entropy of the gas at the standard pressure (p0 = 1 atm) and at the experimental temperature. The partial molar entropy Ss and the integral entropy Ss of the adsorbed phase are given by ... 

Substitution of this into Eq. 5.21 and application of the Stirling s approximation yield the integral molar configurational entropy as ... 

Fig. 5.2 Molar configurational entropies, differential and integral, as the function of the fractional coverage. Also shown are the /9 approximation of the two entropies and the accurate difference between them.

In contrast. Fig. 11.6(b) shows the nonlinearity of the entropy as a function of during the same reaction. The nonUnearity is a eonsequenee of the dependence of the partial molar entropy 5, on the mixture eomposition (Eq. 11.1.24). In the figure, the slope of the eurve at each value of equals Af5 at that point its value ehanges as the reaction advances and the composition of the reaetion mixture ehanges. Consequently, the molar integral reaction entropy A5m(rxn) = A5(rxn)/A approaehes the value of Ar5 only in the limit as A approaches zero. 

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) ... 

Corresponding to the integral heat and entropy of formation of the solution are the partial molar heats A//, and entropies AS, of solution of the components where... 

The relationships presented thus far for partial, integral and relative partial molar free energies are applicable in a similar manner to entropy, enthalpy and also volume. 

The variation of the entropy of formation of an ideal solution with composition is shown inFigure3.10. It is again a characteristic ofan ideal solution that the partial (ASA,ld, A. Sg1, ld) and the integral molar (ASM,ld) entropies of its formation are independent of temperature. 

The same type of polynomial formalism may also be applied to the partial molar enthalpy and entropy of the solute and converted into integral thermodynamic properties through use of the Gibbs-Duhem equation see Section 3.5. 

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... 

Molar entropy of an adsorbed layer perturbed by the solid surface Total enthalpy change for the immersion of an evacuated solid in a solution at a concentration at which monolayer adsorption occurs Heat of dilution of a solute from a solution Enthalpy change for the formation of an interface between an adsorbed mono-layer and solution Integral heat of adsorption of a monolayer of adsorbate vapor onto the solid surface... 

An analytical method for applying Polanyi s theory at temperatures near the critical temperature of the adsorbate is described. The procedure involves the Cohen-Kisarov equation for the characteristic curve as well as extrapolated values from the physical properties of the liquid. This method was adequate for adsorption on various molecular sieves. The range of temperature, where this method is valid, is discussed. The Dubinin-Rad/ush-kevich equation was a limiting case of the Cohen-Kisarov s equation. From the value of the integral molar entropy of adsorption, the adsorbed phase appears to have less freedom than the compressed phase of same density. 

Adsorbed Phase Entropy. Since Equations 7 and 8 can accurately describe the relationship between q, T, and p, we may use them to calculate the integral molar entropy of the adsorbed phase. At temperatures significantly lower than critical for the adsorbate, the entropy of the adsorbed phase is usually compared with the entropy of the liquid at same temperature in order to compare the freedom of each phase. Because our experimental domain was higher, we shall make this comparison with the gaseous phase compressed to the same density p as determined by Equation 8. 

In the range of validity of our method, integral molar entropies were easily computed by the appropriate equations and showed a loss of freedom for the adsorbed phase with respect to the compressed phase of same density. 

Both models give substantially similar integral molar entropies of sorption. 

The integral molar entropy of adsorption is obtained from a well-known thermodynamic relation for a reversible, isothermal process the heat is equal to the change in entropy multiplied by the temperature. This directly leads to... 

Sm is the molar entropy, Vm the molar volume of the gas at a distance x, and Um is the molar internal energy. Integrating the left side from infinite distance to a distance x leads to... 

The integral molar enthalpy and entropy of the Cd-Zn liquid alloy at 432°C are described by the following empirical equations ... 

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