Entropy of As

Any change taking place which results in an increase in entropy has a positive entropy change (AS). Most spontaneous thermodynamic processes are accompanied by an increase in entropy. Entropy has units of Joules per degree K per mole. For representative values see table on p. 393. 

If the dependence on temperature as well as on composition is known for a solution, enthalpies and entropies of adsorption may be calculated from the appropriate thermodynamic relationships [82]. Neam and Spaull [147] have, for example, calculated the enthalpies of surface adsorption for a series of straight-chain alcohols. They find an increment in enthalpy of about 1.96 kJ/mol per CH2 group. 

The broken bond approach has been extended by Nason and co-workers (see Ref. 85) to calculate as a function of surface composition for alloys. The surface free energy follows on adding an entropy of mixing term, and the free energy is then minimized. 

Dislocation theory as a portion of the subject of solid-state physics is somewhat beyond the scope of this book, but it is desirable to examine the subject briefly in terms of its implications in surface chemistry. Perhaps the most elementary type of defect is that of an extra or interstitial atom—Frenkel defect [110]—or a missing atom or vacancy—Schottky defect [111]. Such point defects play an important role in the treatment of diffusion and electrical conductivities in solids and the solubility of a salt in the host lattice of another or different valence type [112]. Point defects have a thermodynamic basis for their existence in terms of the energy and entropy of their formation, the situation is similar to the formation of isolated holes and erratic atoms on a surface. Dislocations, on the other hand, may be viewed as an organized concentration of point defects they are lattice defects and play an important role in the mechanism of the plastic deformation of solids. Lattice defects or dislocations are not thermodynamic in the sense of the point defects their formation is intimately connected with the mechanism of nucleation and crystal growth (see Section IX-4), and they constitute an important source of surface imperfection. 

There is no reason why the distortion parameter should not contain an entropy as well as an energy component, and one may therefore write 0 = 0q-sT. The entropy of adsorption, relative to bulk liquid, becomes A5fi = sexp(-ca). A critical temperature is now implied, Tc = 0o/s, at which the contact angle goes to zero [151]. For example, Tc was calculated to be 174°C by fitting adsorption and contact angle data for the -octane-PTFE system. 

Thus the entropy of localized adsorption can range widely, depending on whether the site is viewed as equivalent to a strong adsorption bond of negligible entropy or as a potential box plus a weak bond (see Ref. 12). In addition, estimates of AS ds should include possible surface vibrational contributions in the case of mobile adsorption, and all calculations are faced with possible contributions from a loss in rotational entropy on adsorption as well as from change in the adsorbent structure following adsorption (see Section XVI-4B). These uncertainties make it virtually impossible to affirm what the state of an adsorbed film is from entropy measurements alone for this, additional independent information about surface mobility and vibrational surface states is needed. (However, see Ref. 15 for a somewhat more optimistic conclusion.)... 

Thus from an adsorption isotherm and its temperature variation, one can calculate either the differential or the integral entropy of adsorption as a function of surface coverage. The former probably has the greater direct physical meaning, but the latter is the quantity usually first obtained in a statistical thermodynamic adsorption model. 

Finally, it is perfectly possible to choose a standard state for the surface phase. De Boer [14] makes a plea for taking that value of such that the average distance apart of the molecules is the same as in the gas phase at STP. This is a hypothetical standard state in that for an ideal two-dimensional gas with this molecular separation would be 0.338 dyn/cm at 0°C. The standard molecular area is then 4.08 x 10 T. The main advantage of this choice is that it simplifies the relationship between translational entropies of the two- and the three-dimensional standard states. 

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). 

Some representative plots of entropies of adsorption are shown in Fig. XVII-23, in general, T AS2 is comparable to Ah2, so that the entropy contribution to the free energy of adsorption is important. Notice in Figs. XVII-23 i and b how nearly the entropy plot is a mirror image of the enthalpy plot. As a consequence, the maxima and minima in the separate plots tend to cancel to give a smoothly varying free energy plot, that is, adsorption isotherm. 

As with enthalpies of adsorption, the entropies tend to approach the entropy of condensation as P approaches in further support of the conclusion that the nature of the adsorbate is approaching that of the liquid state. 

Brunauer (see Refs. 136-138) defended these defects as deliberate approximations needed to obtain a practical two-constant equation. The assumption of a constant heat of adsorption in the first layer represents a balance between the effects of surface heterogeneity and of lateral interaction, and the assumption of a constant instead of a decreasing heat of adsorption for the succeeding layers balances the overestimate of the entropy of adsorption. These comments do help to explain why the model works as well as it does. However, since these approximations are inherent in the treatment, one can see why the BET model does not lend itself readily to any detailed insight into the real physical nature of multilayers. In summary, the BET equation will undoubtedly maintain its usefulness in surface area determinations, and it does provide some physical information about the nature of the adsorbed film, but only at the level of approximation inherent in the model. Mainly, the c value provides an estimate of the first layer heat of adsorption, averaged over the region of fit. 

The standard entropy of adsorption AS2 of benzene on a certain surface was found to be -25.2 EU at 323.1 K the standard states being the vapor at 1 atm and the film at an area of 22.5 x T per molecule. Discuss, with appropriate calculations, what the state of the adsorbed film might be, particularly as to whether it is mobile or localized. Take the molecular area of benzene to be 22 A. ... 

Entropies of adsorption are obtainable in the same manner as discussed in Chapter XVII. 

In general, it seems more reasonable to suppose that in chemisorption specific sites are involved and that therefore definite potential barriers to lateral motion should be present. The adsorption should therefore obey the statistical thermodynamics of a localized state. On the other hand, the kinetics of adsorption and of catalytic processes will depend greatly on the frequency and nature of such surface jumps as do occur. A film can be fairly mobile in this kinetic sense and yet not be expected to show any significant deviation from the configurational entropy of a localized state. 

Equation (A2.1.21) includes, as a special case, the statement dS > 0 for adiabatic processes (for which Dq = 0) and, a fortiori, the same statement about processes that may occur in an isolated system (Dq = T)w = 0). If the universe is an isolated system (an assumption that, however plausible, is not yet subject to experimental verification), the first and second laws lead to the famous statement of Clausius The energy of the universe is constant the entropy of the universe tends always toward a maximum. ... 

Figure A2.1.9. Chemically reacting systems, (a) The entropy. S as a fiinction of the degree of advancement of the reaction at constant U and V. (b) The affinity Aas a fiinction of for the same reacting system. Equilibrium is reached at 0.623 where tiis a maxuniim and A= 0.

If the entropy of each element in some crystalline state be taken as zero at the absolute zero of temperature, every substance has a finite positive entropy, but at the absolute zero of temperature the entropy may become zero, and does so become in the case of perfect crystalline substances. 

For those who are familiar with the statistical mechanical interpretation of entropy, which asserts that at 0 K substances are nonnally restricted to a single quantum state, and hence have zero entropy, it should be pointed out that the conventional thennodynamic zero of entropy is not quite that, since most elements and compounds are mixtures of isotopic species that in principle should separate at 0 K, but of course do not. The thennodynamic entropies reported in tables ignore the entropy of isotopic mixing, and m some cases ignore other complications as well, e.g. ortho- and para-hydrogen. 

The principle of tire unattainability of absolute zero in no way limits one s ingenuity in trying to obtain lower and lower thennodynamic temperatures. The third law, in its statistical interpretation, essentially asserts that the ground quantum level of a system is ultimately non-degenerate, that some energy difference As must exist between states, so that at equilibrium at 0 K the system is certainly in that non-degenerate ground state with zero entropy. However, the As may be very small and temperatures of the order of As/Zr (where k is the Boltzmaim constant, the gas constant per molecule) may be obtainable. 

Snch a generalization is consistent with the Second Law of Thennodynamics, since the //theorem and the generalized definition of entropy together lead to the conchision that the entropy of an isolated non-eqnilibrium system increases monotonically, as it approaches equilibrium. 

This completes the heuristic derivation of the Boltzmann transport equation. Now we trim to Boltzmaim s argument that his equation implies the Clausius fonn of the second law of thennodynamics, namely, that the entropy of an isolated system will increase as the result of any irreversible process taking place in the system. This result is referred to as Boltzmann s H-theorem. 

In order to separate the enthalpy and the entropy of activation, the rate is measured as a fiinction of temperature. These data should give a straight line on an Eyrmg plot of log(rate/7) against (1/7) (figure... 

Transient, or time-resolved, techniques measure tire response of a substance after a rapid perturbation. A swift kick can be provided by any means tliat suddenly moves tire system away from equilibrium—a change in reactant concentration, for instance, or tire photodissociation of a chemical bond. Kinetic properties such as rate constants and amplitudes of chemical reactions or transfonnations of physical state taking place in a material are tlien detennined by measuring tire time course of relaxation to some, possibly new, equilibrium state. Detennining how tire kinetic rate constants vary witli temperature can further yield infonnation about tire tliennodynamic properties (activation entlialpies and entropies) of transition states, tire exceedingly ephemeral species tliat he between reactants, intennediates and products in a chemical reaction. 

The Shannon Equation Eq. (1)) [4] enables one to evaluate the information content, I (also known as the Shannon entropy), of the system. 

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