The Entropy Contribution

Now let s concentrate on the entropy contribution Heff (a) to the free energy. In the case of a swollen coil, this contribution was described by Equation (8.10) resulting from (7.5). Would it be valid for a 1 as well Let s think. Equation (7.5) gives the free energy of an ideal coil whose end-to-end distance is of order R. This is the only condition on the coil s 

So how can we find a reasonable estimate for Ues a) = —TS a)7 Let s look at the Boltzmann equation (7.2). It suggests that the entropy (as well as the entropy loss) should not depend on the actual cause of the 

Take a certain monomer rniit of the chain. Suppose it is currently far away from the cavity walls. What can we say about a strand of the chain near the chosen monomer It does not seem to know anjdhing about the surroundings. It can sense neither the walls of the cavity (since they are far away), nor the presence of other bits of the chain (since the chain is ideal). Therefore, this strand merely acts as a Gaussian cod. If its length is g, then its size will be about Of com e, this would only be right 

Now let s look at a -strand. The monomers deep inside it can arrange themselves in any sort of shape. Therefore, they do not contribute to the entropy (since they do not reduce the choice of possible conformations of the chain). On the other hand, the ends of the -strand must be near the walls, even though they cannot leave the cavity. This restricts the munber of possible conformations, so that each end loses a bit of entropy, of order (To see this suppose the number of conformations fi of a monomer segment in (7.2) drops by half. Then the entropy decreases by ks In 2 0.76 ks)- 

In a chain of N monomers, there are W/g -strands altogether. So the loss of entropy is 

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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. 

Subtracting the entropy contributions of the pure components from gives the entropy of mixing according to the present model ... 

The total heat released is the sum of the entropy contribution plus the irreversible contribution. This heat is released inside the battery at the reaction site. Heat release is not a problem for low rate appHcations however, high rate batteries must make provisions for heat dissipation. Failure to accommodate heat can lead to thermal mnaway and other catastrophic situations. 

Although AGrxn depends on both enthalpy and entropy, there are many reactions for which the entropy contribution is small, and can be neglected. Thus, if AHjxn = AErxn, wc cuu estimate equilibrium constants for such reactions by the following equation ... 

Extrapolations are always subject to error, but fortunately the contribution to the entropy resulting from the extrapolation is a small part of the total. In glucose, for example, S g = 219.2 0.4 J-K -moF1, but the entropy contribution at 10 K obtained from the Debye extrapolation is only 0.28 J-K 1-mol 1. Well-designed cryogenic calorimeters are able to produce Cp measurements of high accuracy hence, the Third Law entropy obtained from the Cp measurements can also be of high accuracy. 

We again assume that the pre-exponential factor and the entropy contributions do not depend on temperature. This assumption is not strictly correct but, as we shall see in Chapter 3, the latter dependence is much weaker than that of the energy in the exponential terms. The normalized activation energy is also shown in Fig. 2.11 as a function of mole fraction. Notice that the activation energy is not just that of the rate-limiting step. It also depends on the adsorption enthalpies of the steps prior to the rate-limiting step and the coverages. 

An important property of chain molecules is that a major contribution to the standard entropy is conformational in nature, i.e. is due to hindered internal rotations around single bonds. This property is most relevant to cyclisation phenomena, since a significant change of conformational entropy is expected to take place upon cyclisation. Pitzer (1940) has estimated that the entropy contribution on one C—C internal rotor amounts to 4.43 e.u, A slightly different estimate, namely, 4.52 e.u. has been reported by Person and Pimentel (1953). Thus, it appears that nearly one-half of the constant CH2 increment of 9.3 e.u. arises from the conformational contribution of the additional C—C internal rotor. 

The tight and loose transition-state hypothesis is in contrast with the assumption that there is extensive cancellation of contributions due to chemical change in the entropic component of the EM (p. 81). Indeed, the uniform behaviours displayed by 0AS-data for reactions widely differing in nature (Figs 5, 23, and 24) clearly shows that no matter how loose a transition state or product is, the entropy contribution from such looseness will be cancelled out extensively by virtue of the operator 0. 

It will also be noted from the results in Table 4 that, unlike the saturated cyclopropanes, the vinylcyclopropanes isomerize with normal preexponential factors. Consideration of the postulated transition complex and the reactant molecule makes it clear why this is so. In the reactant the vinyl group can undergo essentially free rotation. In the transition complex the allylic part of the biradical is rigid and cannot rotate. Thus the entropy contribution of this free rotation in the reactant is lost on forming the transition complex. As a result of ring rupture one new centre of free rotation is produced which is not present in the reactant. The result of these effects is that on passing from the reactant to the... 

If the assumption that the entropy of the reaction according to equation (5) is independent of the aromatic substance applies, the entropy contribution TAS must assume a constant value. Mackor et al. (1958b) were able to demonstrate the correctness of this assumption by determining the thermodynamic data for some methylbenzenes and condensed aromatic hydrocarbons. Whereas AH and AO change considerably, the entropy term TAS remains largely unaltered (Table 21). 

The calculated energy difference is 7.5 kcal/mol in favor of cyclohexyl radical according to the 6-31G calculations. Including the entropy contribution lowers this number to around 5 kcal/mol. Were the reaction under thermodynamic control, only cyclohexane would be observed, and interpretations (b) and (c) cannot be correct. 

The resulting equilibrium concentrations of these point defects (vacancies and interstitials) are the consequence of a compromise between the ordering interaction energy and the entropy contribution of disorder (point defects, in this case). To be sure, the importance of Frenkel s basic work for the further development of solid state kinetics can hardly be overstated. From here on one knew that, in a crystal, the concentration of irregular structure elements (in thermal equilibrium) is a function of state. Therefore the conductivity of an ionic crystal, for example, which is caused by mobile, point defects, is a well defined physical property. However, contributions to the conductivity due to dislocations, grain boundaries, and other non-equilibrium defects can sometimes be quite significant. 

At the critical value of interaction energy, the enthalpy term just compensates for the entropy contribution, and samples of any molar mass are equally retained. The balance is limited to a certain temperature. Increasing temperature yields prolonged retention and vice versa (see Fig. 2). This might seem unusual at a first glance, but can be under-... 

ASf° = -68.78 J-mol 1-K 1 (b) Because AGf°(03, g) is positive at all temperatures, the reaction is not spontaneous at any temperature. It is less favored at high temperatures, (c) Because the reaction entropy is negative, the —TAS° term is always positive, so the entropy contribution to AGf° is always positive. The entropy does not favor the spontaneous formation of ozone. 15.29 (a) -219.27 kj-mol1 (b) 77.4°C 15.31 Fluorine comes from the minerals fluorspar (CaF2), cryolite (Na3AlF6), and the fluorapatites (Ca5F(P04)3). The free element is prepared from HF and KF by electrolysis, but the HF and the KF needed are prepared in the laboratory. 

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