Equilibrium second-order

Consider a simple equilibrium, second-order in the forward direction and first-order in the reverse ... 

Gibbs and Di Marzio [23,24] proposed that the dilatometric Tg is a manifestation of a true equilibrium second-order transition at the temperature T2. In a further development, Adams and Gibbs [21] have shown how the WLF equation can then be derived. On their theory, the frequency of molecular jumps is given by... 

The subroutine is well suited to the typical problems of liquid-liquid separation calculations wehre good estimates of equilibrium phase compositions are not available. However, if very good initial estimates of conjugate-phase compositions are available h. priori, more effective procedures, with second-order convergence, can probably be developed for special applications such as tracing the entire boundary of a two-phase region. 

This equilibrium has been extensively studied by Bodenstein. Unlike the other halogen-hydrogen reactions, it is not a chain reaction but a second order, bimolecular, combination. 

Equilibrium constants and second-order rate constants for Co, Ni, Cu and Zn catalysed of the... 

Table 2.6. Equilibrium constants from complexation of 2.4a, 2.4b, and 2.4d to different metal ions (Kj) and second-order rate constants for the Diels-Alder reaction of these complexes with 2 (%cd) in water at 2.00 M ionic strength and 25°C. ...

Table 3.1 summarises the influence of the diamine ligands on the equilibrium constant for binding of 3.8c to the ligand-metal ion complex (K ) and the second-order rate constant for reaction of the ternary complex (ICjat) (Scheme 3.5) with diene 3.9. 

All these facts—the observation of second order kinetics nucleophilic attack at the carbonyl group and the involvement of a tetrahedral intermediate—are accommodated by the reaction mechanism shown m Figure 20 5 Like the acid catalyzed mechanism it has two distinct stages namely formation of the tetrahedral intermediate and its subsequent dissociation All the steps are reversible except the last one The equilibrium constant for proton abstraction from the carboxylic acid by hydroxide is so large that step 4 is for all intents and purposes irreversible and this makes the overall reaction irreversible... 

Estr is the energy function for stretching a bond between two atom types A and B. In its simplest form, it is written as a Taylor expansion around a namral , or equilibrium , bond length Rq- Tenninating the expansion at second order gives the expression... 

There is a rather wide disparity between experiment and the second-order cumulate prediction for small T, equation 7.123 correctly predicts both the values of equilibrium density for large T and the fact that the system undergoes a sharp... 

Note that while the power-law distribution is reminiscent of that observed in equilibrium thermodynamic systems near a second-order phase transition, the mechanism behind it is quite different. Here the critical state is effectively an attractor of the system, and no external fields are involved. 

A consideration of the same example also illustrates the result established in treatises on dynamics that the condition for stable, unstable, or neutral equilibrium of a mechanical system is that, for any small displacement which does not violate the constraints, the change of potential energy shall vanish to the first order, and be positive, negative, or zero respectively to the second order. When the system is in stable, unstable, or neutral equilibrium, the potential energy is a minimum, a maximum, or stationary respectively (Theorem of Dirichlet). Thus the work done by the system in any infinitesimal displacement is zero to the first order, and negative, positive, or zero to the second order, for the three cases. All these conditions refer only to a par-... 

Note that there is no net change in the number of moles of gas in this equilibrium. Therefore, by Le Chatelier s principle, this reaction will be independent of external pressure (ignoring second-order effects due to gas imperfections). Under these conditions the N of the expl will... 

The rate is no longer second-order with respect to nitrous acid, but first-order. Therefore, N203 cannot be the nitrosating reagent. The marked acid catalysis, as seen in the term h0, indicates that the new nitrosating agent is some species whose equilibrium concentration increases rapidly with increasing acidity. As shown in Scheme 3-8, this may be the nitrosoacidium ion (H20 —NO), but could also be the nitrosyl ion (NO+). 

Bradfield et al.21g first studied the kinetics of molecular bromination using aromatic ethers in 50% aqueous acetic acid at 18 °C. They showed that the kinetics are complicated by the hydrogen bromide produced in the reaction which reacts with free bromine to give the tribromide in BrJ, a very unreactive electrophile. To avoid this complication, reactions were carried out in the presence of 5-10 molar excess of hydrogen bromide, and under these conditions second-order rate coefficients (believed to be I02k2 by comparison with later data) were obtained as follows after making allowance for the equilibrium Br2 + Br7 Bn, for which K = 50 at 18 °C 4-chloroanisole (1.12), 4-bromoanisole (1.20), 4-... 

In 75 % aqueous acetic acid, the bromination of fluorene at 25 °C obeys second-order kinetics in the presence of bromide ion and higher orders in its absence287, with Ea (17.85-44.85 °C) = 17.4, log A = 10.5 and AS = —12.4 however, these values were not corrected for the bromine-tribromide ion equilibrium, the constant for which is not known in this medium, and so they are not directly comparable with the proceeding values. In the absence of bromide ion the order with respect to bromine was 2.7-2.0, being lowest when [Br2]initial was least. Second- and third-order rate coefficients were determined for reaction in 90 and 75 wt. % aqueous acetic acid as 0.0026 and 1.61 (k3/k2 = 619), 0.115 and 12.2 (k3/k2 = 106) respectively, confirming the earlier observation that the second-order reaction becomes more important as the water content is increased. A value of 7.25 x 106 was determined for f3 6 (i.e. the 2 position of fluorene). 

Subsequently, rate coefficients were determined for the zinc chloride-catalysed bromination of benzene, toluene, i-propyl-benzene, r-butylbenzene, xylenes, p-di-f-butylbenzene, mesitylene, 1,2,4-trimethyl-, sym-triethyl-, sym-tri-f-butyl-, 1,2,3,5-and 1,2,4,5-tetramethyl- and pentamethylbenzenes, all at 25.4 °C and in acetic acid, and it was shown that the reaction was inhibited by HBr.ZnCl2 which accumulates during the bromination and was considered to cause the first step of the reaction (formation of ArHBr2) to reverse320. The second-order coefficients for bromination of o-xylene at 25.0 °C were shown to be inversely dependent upon the hydrogen bromide concentration and the reversal of equilibrium (155)... 

Iodine acetate would seem to be unambiguously present in the iodination of pentamethylbenzene in acetic acid by iodine and mercuric acetate, since the latter components form an equilibrium mixture of iodine acetate and acetoxy-mercuric iodide and mercuric acetate speeds up the iodination332. Second-order rate coefficients of 0.078 (25 °C) and 0.299 (45 °C) were obtained, and these values are intermediate between those obtained for the reaction of bromine acetate with benzene (2.5 xlO-3) and toluene (1.2) at 25 °C, indicating that bromine acetate is the stronger electrophile. 

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