Entropy of electrons

The electrons, if they are separated from the ions, will also contribute to the entropy, and one might naively expect an expression similar to Eqn (7.8). Then the chemical potential for an atom would be the sum of two terms like Eqn (7.10), one from ions and one from electrons, and so the entropy term would be doubled. This is not so, however, in metallic intercalation compounds. In metals, the entropy of electrons is small. Electrons added by intercalation do not have a choice of all the empty states in a band, but only those within kT of the Fermi energy. If the Fermi energy is expressed as a temperature Tp and is measured from the bottom of the band, the change in entropy with the number n of electrons, dS/dn, is of order kTfTj (Kittel, 1971), not of order k like Eqn (7.8) for 

Although the simple mean-field expression (Eqn (7.10)) for a lattice-gas model has been used to understand intercalation systems qualitatively 

The negative value of U in the fit in Fig. 7.11 signifies that the interaction between Li ions is attractive. Under an attractive interaction the ions can cluster, and the compound should separate into two phases at low temperatures. Fig. 7.12 shows one of the Bragg peaks in the X-ray diffraction pattern for Lio.sMoeScg as it cools (Dahn and McKinnon, 

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Boichenko, V.A., Hou, J-M., and Mauzerall, D. (2001) Thermodynamics of electron transfer in oxigen photosynthetic reaction centers volume change, enthalpy and entropy of electron-transfer reactions in intact cells of the cyanobacterium Synechocystis PCC 6803, Biochemistry 40, 7126-7132. 

It is worth noting that the above expression for AS includes a term corresponding to the entropy of electrons. This term was explicitly neglected in Ref 1, but recent work has demonstrated that this term should be included because electrons are involved in the... 

As a first approximation, it is tempting to assume that the measured E1/2 values are not significantly affected by variations of temperature. This is of course not true, even if one assumes that the entropy of electron transfer is small. Although Em values may be virtually temperature independent for many redox systems, it is not altogether evident that this would be the case when dealing with a totally unknown system. 

An example is given in Figure 3 which shows how E1/2 for the first three reductions of 50 varies with temperature in benzonitrile (T = 0 to 80 °C) and dichloromethane (T = -30 to 10 °C) [13]. All three reductions of C q are affected by temperature to a different extent but the shift is largest for the third reduction where a fifty degree increase in temperature results in a decrease of E1/2 by 80 mV. This shift in potential is substantial and is associated with a large negative entropy of electron transfer which simply cannot be ignored. 

AH and AS to various notional subprocesses such as bond dissociation energies, ionization energies, electron affinities, heats and entropies of hydration, etc., which themselves have empirically observed values that are difficult to compute ab initio. 

The reactivities of 4- and 2-halo-l-nitronaphthalenes can usefully be compared with the behavior of azine analogs to aid in delineating any specific effects of the naphthalene 7r-electron system on nucleophilic substitution. With hydroxide ion (75°) as nucleophile (Table XII, lines 1 and 8), the 4-chloro compound reacts four times as fast as the 2-isomer, which has the higher and, with ethoxide ion (65°) (Table XII, lines 2 and 11), it reacts about 10 times as fast. With piperidine (Table XII, lines 5 and 17) the reactivity relation at 80° is reversed, the 2-bromo derivative reacts about 10 times as rapidly as the 4-isomer, presumably due to hydrogen bonding or to electrostatic attraction in the transition state, as postulated for benzene derivatives. 4-Chloro-l-nitronaphthalene reacts 6 times as fast with methanolic methoxide (60°) as does 4-chloroquinoline due to a considerably higher entropy of activation and in spite of a higher Ea (by 2 kcal). ... 

Since the saturated solutions of AgT and AgCl are both very dilute, it is of interest to examine their partial molal entropies, to see whether we can make a comparison between the values of the unitary terms. As mentioned above, the heat of precipitation of silver iodide was found by calorimetric measurement to be 1.16 electron-volts per ion pair, or 26,710 cal/mole. Dividing this by the temperature, we find for the entropy of solution of the crystal in the saturated solution the value... 

Turning next to AgBr, we see from Table 33 that the value of L increases from 0.931 electron-volt at 15° to 0.935 at 35°, a difference of 0.004. Dividing by 20, we find that the average value of dL/dT in the neighborhood of 25° is 2 X 10 1 electron-volt/deg. Multiplying by 23,060, we find this is equivalent to 4.6 cal/mole. It follows that the value of the conventional entropy of solution AS0 in the neighborhood of 25°C is approximately... 

Table III presents integral excess entropies of formation for some solid and liquid solutions obtained by means of equilibrium techniques. Except for the alloys marked by a letter b, the excess entropy can be taken as a measure of the effect of the change of the vibrational spectrum in the formation of the solution. The entropy change associated with the electrons, although a real effect as shown by Rayne s54 measurements of the electronic specific heat of a-brasses, is too small to be of importance in these numbers. Attention is directed to the very appreciable magnitude of the vibrational entropy contribution in many of these alloys, and to the fact that whether the alloy is solid or liquid is not of primary importance. It is difficult to relate even the sign of the excess entropy to the properties of the individual constituents.

Under near-equilibrium conditions the shape of this curve is related to two contributions, the compositional dependence of the configurational entropy of the guest ions, and the contribution to the chemical potential from the electron gas [31]. 

The wide variation in the entropy factors for both the substituted phenyl and heterocyclic compounds and in particular for the methoxyphenyl and furan derivatives was considered to be strong evidence for solvent effects being predominant in determining the activation entropy. Consequently, discussion of the substituent effects in terms of electronic factors alone requires caution in this reaction. Caution is also needed since rates for the substituted phenyl compounds were only determined over a 20 °C range. The significance of entropy factors has also been indicated by the poor correlation of the data of the electrophilic reactivities of the heterocyclic compounds, as derived from protodemercuration, with the data for other electrophilic substitutions and related reactions572. 

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