Entropy inverse temperature

Figure 6 Entropy per monomer in the bond-fluctuation model as a function of inverse temperature. The results from the simulation (filled circles) are compared with the theoretical predictions discussed in the text.

The diastereofacial selectivity of the addition of lithioacetonitrile to 2-phenylpro-panol has been studied over a wide range of temperatures, solvents, and bases.256 Eyring plots [In(dr) vs 1 IT], activation parameters, and inversion temperatures have been characterized. In some cases, the differential entropy of activation, AAS, plays an exclusive role in determining anti -selectivity. 

The effect of the addition of water and the presence of Li on the disproportionation equilibria was also examined (Ammar and Saveant, 1973). The effect of water was observed to be a decrease in AS with little change in the entropy term. On the other hand, Li also brought about a decrease in A but the entropy increased and became positive. The latter brought about an inverse temperature effect instead of AE increasing with temperature a decrease was observed. 

In addition, since it is a lattice model, the bond fluctuation model can be used to assess the validity of the Gibbs-DiMarzio theory described in Section 4.4.1. Baschnagel et al. (1997) show that the curve of entropy versus inverse temperatures (Sc versus l/T) does not cross zero when the temperature is lowered, but instead levels out at large 1/7, in disagreement with the Gibbs-DiMarzio theory but qualitatively similar to the prediction of Miller s theory in Fig. 4-13. 

FIGURE 12 Sketch of the expected probability density = P/3,expectation values of the inversion coordinate operator q with respect to the maximum entropy decomposition of the molecular thermal state of an ammonia-type molecule. This figure refers to some fixed inverse temperature. The expectation values of q concentrate around two possible values for positive and negative inversion coordinate, respectively. For increasing nuclear masses Mj, j= 1,2,..., the density is expected to become more sharply... 

Figure 2.5 shows also the concentration dependence of the inverse Kauzmann temperature T (entropy catastrophe temperature). For the pure metal, T is much higher than the temperature T0 as discussed. The 77-line should also decrease with increasing concentration and end in the triple point(C, 7 )[2.21] as follows from its definition (AS = 0). It is interesting to note that at this point the real Kauzmann temperature and the inverse Kauzmann temperature meet. But in real systems, the amorphous phase has an excess entropy (small fraction of the entropy of fusion) when compared to the corresponding crystal, the exact amount determined from the kinetics and timescale of the glass transformation. Therefore, another glass transition temperature line with finite excess entropy must be considered, which will be parallel to the Tg-line (above it) and cross the T0- and 77-lines not exactly in the triple point. 

To identify the inversion temperatures of nitrogen we can use Fig. 2.4-2b, a temperature-entropy diagram. From part a of this problem we note that at J mv... 

Polyethylene oxide) and poly(N-alkylacrylamide)s are known to undergo a temperature-dependent phase change whereupon they separate from an aqueous phase at increased temperatures [14]. This inverse temperature dependence, i.e., the occurrence of a lower critical solution temperature, can be related to an entropi-cally favorable decrease in hydrogen bonding between water and the polymer with increasing temperature. In order to exploit this physical property for catalyst recovery, Bergbreiter et al. attached phosphines covalently to commercially available PEO or PEO-b-PPO-b-PEO block copolymers [Schemes 1 and 2 PPO = polypropylene oxide)] [9a],... 

We find two linear regions in the Eyring diagram for almost all the systems studied. The characteristic point where a change is observed in the slopes is called the inversion point, and the corresponding temperature is the inversion temperature (7] ,). From the two linear correlations determined for each aminoalcohol, two different sets of discrimination parameters can be observed AAHf and AASf) and defined for T> 7] v when AAH and AAS are obtained for T < ] . The two regions correspond to the dominance of enthalpy or entropy factors in the selection [46] (see Table 4). 

Thermal precipitation by cooling is the scheme chemists normally use in recrystallizations and is the normal behavior of small molecules. Macromolecules are different in that they can often be phase separated from solution by heating [ 119,120]. Thermal precipitation by heating is a process that produces a solid polymer without addition of anything other than heat. It is the inverse of the process used with the polyethylene oligomers discussed above. This inverse temperature-dependent solubility of macromolecules is a phenomenon that is most simply ascribed to the unfavorable entropy of solvation of a macro-... 

The Joule-Thomson inversion curve as calculated from (1) is in some doubt. This is apparent at high temperatures where (1) predicts a maximum inversion temperature of 293 °K, whereas a more probable value would be 225 °K. Although the effect of such an uncertainty on the temperature-entropy diagram is expected to be small, further efforts are being made to reconcile this problem. 

T[ and Tj, are the on-set temperature for the hydrophobic folding and assembly transition, that is, inverse temperature transition, in pbs (0.15 N NaCl, 0.01 M phosphate) as determined by light scattering and in water as determined by DSC, respectively. Both values are linearly extrapolated tofx = 1 and rounded to a number divisible by 5. AH and AS are the values at/x = 0.2 on the curve for a linear fit of the DSC derived endothermic heats and entropies of the transitions for die polymers in water. 

To obtain these equations we integrated Cp.e=TdSe/3T, Se=-3Gc/3T and ignored any pressure dependence. Below the traitsition temperature T2 the configurational entropy is zero according to the simple version of the GD theory so that we have only energy of activation while above Tj the free energy is consistent with the inverse temperature dependence of the specific heat. 

Figure 2.8. Ordered water molecules surround oillike groups, as first shown by Stackelberg and Muller. As oil-like groups associate, this structured water becomes less ordered liquid water this is the large positive change in entropy responsible for the inverse temperature transition, the phase transition that is fundamental to protein function. As shown in the upper left, hydration of a proximal carboxylate...

Inverse Temperature Transitions Provide Negative Entropy to Protein... 

The inverse temperature transition is a specific mechanism whereby thermal energy (heat) provides an increase in order of the protein part of the system. A decrease in entropy of this sort has been termed negative entropy by Schrodinger. ° While the total entropy (disorder) for the complete system of protein and water increases as the temperature is raised, the structural protein component, critical to the conversion of thermal energy to mechanical work, increases in negative entropy. The protein component increases in order by the folding that shortens length and by the assembly of oillike domains that builds structures. 

Inverse Temperature Transitions Extract Order (Negative Entropy) from Energy Sources ... 

Despite the absorption of heat for the transition and the overall increase in entropy of -(-4.0 EU for the water plus protein, the protein component actually increases in order on raising the temperature. As unambiguously demonstrated by crystallization of a cyclic analog (see Figure 2.7), in this case the protein component of the water plus protein system becomes more ordered as the temperature is raised. For this and additional reasons, noted below in section 5.1.3, we call this transition exhibited by our model protein, poly (GVGVP), an inverse temperature transition. 

---