Molar integral entropy of adsorption

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... 

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. 

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... 

Comparison of Equations (2.61) and (2.67) shows that the second term of the right-hand side of Equation (2.67) is due to the spreading pressure, which is related to the interactions between adsorbed molecules, often referred to as the lateral interactions . It is only at very low coverages, where the spreading pressure is negligible, that the integral molar entropy of adsorption is simply dependent on the adsorbate-adsorbent interaction. 

Finally, when integral molar energies of adsorption are directly measured by gas adsorption calorimetry, it is possible to obtain the corresponding integral molar entropies of adsorption from Equations (2.65) and (2.66). 

The integral molar entropy of adsorption then becomes (as indicated by Jura and Hill, 1952) ... 

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 standard integral molar entropy of adsorption is defined as... 

The computation of the integral molar entropy of adsorption at any coverage requires knowledge of the adsorption isotherm " = "(p) at a given temperature combined with the calorimetric isotherm Q = Q" (p). We must emphasize the fact that Q is measured by definition under reversible conditions. Therefore, when applying Eq. (48) to experimental data, the quasireversibility of the process must be verified. 

The integral molar entropy of adsorption is the difference between the entropy of an adsorbed mole and the entropy of the adsorptive in the ideal gas state, at given p and T. It is a mean integral quantity taken over the whole amount adsorbed and it is characteristic of a given state of equilibrium. This is distinguished by the standard integral molar entropy of adsorption, which is the entropy of one adsorbed mole with respect to the entropy of the adsorptive in the ideal gas state at the same T, but under standard pressure. 

F. Rouquerol, J. Rouquerol, C. Letoquart, Use of isothermal microcalorimetry data for the determination of integral molar entropies of adsorption at the gas-solid interface by a quasi-equilibrium procedure. Thermochim. Acta 39(2), 151-158 (1980). doi 10.1016/0040-... 

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). 

Molar entropy of adsorption is also an important parameter in understanding the state of the adsorbed phase. The molar integral entropy of adsorption AS , can be 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... 

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... 

To integrate the second term of the right-hand side of the Equation (2.65), it must be taken into account that the molar entropy of the gaseous adsorptive depends on the pressure which varies from 0 top when na varies from 0 to na (cf. Equation 2.47). So integration by parts gives ... 

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. 

From a molar point of view, the term entropy of adsorption covers a great number of different functions and it is required to specify whether the function considered is a derivative or an integral, and if it refers to an equilibrium state ip, T) or to a standard state (p°, T) [95, 96],... 

Equation (2.72) is not easy to use in the general case in which the spreading pressure is unknown. But in the particular case of stepwise isotherms where there are two adsorbed phases in equilibrium with the gaseous adsorptive (i.e. in the case of a univariant adsorption system), Larher (1968, 1970) showed that the isosteric method may be used with the transition pressure p" to give integral molar energies un and entropies sn of the quasi-layer ... 

It is assumed that the binding energy of an adsorbed single molecule to the surface approximately equals its partial molar adsorption enthalpy at zero surface coverage. In the adsorbed state at zero surface coverage the individual variations of the entropy are partly but not completely suppressed. Hence, it is expected that this adsorption enthalpy is proportional to the standard sublimation enthalpy, which characterizes the volatility properties of pure solid phases as an integral value, ... 

The principles of phase equilibrium do not apply to excess adsorption variables at high pressure where the excess adsorption passes throu a maximum. Under these conditions, the pressure is no longer a single-valued function of excess adsorption so that n cannot serve as an independent variable for the determination of partial molar quantities such as activity coefficients. Additional complications which arise at high pressure are (1) the selectivity for excess adsorption (S12 = (nf/j/i)/(n2/y2)) approaches infinity as nj — 0 and (2) the differential enthalpy of the ith component has a singularity at the pressure corresponding to maximum nf. For excess variables, the diffierential functions are undefined but the integral functions for enthalpy and entropy are smooth and well-behaved (1). 

The quantities of interest are (i) n, moles of adsorbate (ii) m, mass of adsorbent (iii) V, volume (iv) p, pressure (v) T, absolute temperature (vi) R, molar ideal gas constant (vii) A, surface area of the adsorbent (viii) Q heat (ix) U, internal energy (x) H, enthalpy (xi) 5, entropy and (xii) G, Gibbs free energy. Superscripts refer to differential quantities (d) experimentally measured quantities (exp) integral quantities (int) gas phase (g), adsorbed phase (s) and solid adsorbent (sol) quantities standard state quantities (°). Subscript (a) refers to adsorption phenomena (e.g. AaH) [13, 91]. 

---