Separation entropy

Conformational Adjustments The conformations of protein and ligand in the free state may differ from those in the complex. The conformation in the complex may be different from the most stable conformation in solution, and/or a broader range of conformations may be sampled in solution than in the complex. In the former case, the required adjustment raises the energy, in the latter it lowers the entropy in either case this effect favors the dissociated state (although exceptional instances in which the flexibility increases as a result of complex formation seem possible). With current models based on two-body potentials (but not with force fields based on polarizable atoms, currently under development), separate intra-molecular energies of protein and ligand in the complex are, in fact, definable. However, it is impossible to assign separate entropies to the two parts of the complex. 

The formation of the solution may be conceived to occur in two steps disorientation of the polymer molecules and mixing of the disoriented polymer with solvent. The separate entropy changes are readily obtained as follows The first is given by Eq. (8) with ni = 0, i.e. 

The calculations yield S and AS° values of appreciable magnitude, +18 eu (cal moff and +10 eu, respectively, and in contrast to the free energies, calculated S and H quantities depart substantially from the quadratic relationship given by Eq. 112. In the case of small a and also small 7.,s-, one expects, from Eq. 112, the value of S /AS° to be approximately 0.5, whereas the calculations yield a ratio of approximately 2 (the distinction is pronounced even when the sizable estimated statistical uncertainties ( 5 eu) in the calculated entropies is taken account of). For this result to be compatible with Eq. 112, it would require a sizable positive value of /..S, but in fact the simulation results indicated a.s 0. Thus, for reaction 107, as represented by the simulation and model molecular Hamiltonian [36], we infer that near room temperature the separate entropy and enthalpy quantities are not well accounted for by a harmonic model, whereas, due to compensating effects, harmonic behavior is recovered when they are combined in the free-energy quantities. 

Second law entropies were calculated by the review as discussed in Appendix A from those investigations in which the temperature dependence could be measured for equilibria involving Se7(g). Such evaluations were not made in the original papers. The values in [66BER/CHU] were derived in two steps. First the entropy of Sefi(g) was determined from the equilibria Se6(g)/liquid and Se6(g)/solid and then the entropy of Se7(g) was evaluated from the reaction 7Sc6(g) 6Sc7(g). The first step resulted in two different values for Se6(g) and hence also in two separate entropies of Se7(g). The entropy values are summarised in Table V-16. 

Suppose we have a gaseous system consisting of the substances A, B, C, and D m which the reaction A + B = C + D can occur Suppose the total entropy of the system in the equilibrium state is S We have seen already that the total entropy is the sum of the separate entropies of the constituents That is—... 

We may now go a step further and, with Planck, put the separate entropies equal to zero. Then our Heat Theorem states At the absolute zero of temperature the entropy of every chemically homogeneous solid or liquid body has a zero value The entropy of a body at the temperature T is thus... 

Table 11.1 Aggregation temperatures surface entropies relative surface entropies per monomer relative aggregation and fragmentation energies per monomer, and respectively, latent heat per monomer Ar/, and phase-separation entropy per monomer Ar/,/ f, . All quantities for systems consisting of two, three, and four 13-mers with AB sequence FI.

Vibrational energy states are too well separated to contribute much to the entropy or the energy of small molecules at ordinary temperatures, but for higher temperatures this may not be so, and both internal entropy and energy changes may occur due to changes in vibrational levels on adsoiption. From a somewhat different point of view, it is clear that even in physical adsorption, adsorbate molecules should be polarized on the surface (see Section VI-8), and in chemisorption more drastic perturbations should occur. Thus internal bond energies of adsorbed molecules may be affected. 

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. 

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 seen in previous sections, the standard entropy AS of a chemical reaction can be detemiined from the equilibrium constant K and its temperature derivative, or equivalently from the temperature derivative of the standard emf of a reversible electrochemical cell. As in the previous case, calorimetric measurements on the separate reactants and products, plus the usual extrapolation, will... 

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. 

If the entropy and the enthalpy for the separate mixing in each of the half-mole superlattices are calculated and then combined, the following equation is obtained ... 

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... 

There are two ways in which the volume occupied by a sample can influence the Gibbs free energy of the system. One of these involves the average distance of separation between the molecules and therefore influences G through the energetics of molecular interactions. The second volume effect on G arises from the contribution of free-volume considerations. In Chap. 2 we described the molecular texture of the liquid state in terms of a model which allowed for vacancies or holes. The number and size of the holes influence G through entropy considerations. Each of these volume effects varies differently with changing temperature and each behaves differently on opposite sides of Tg. We shall call free volume that volume which makes the second type of contribution to G. 

In Chap. 8 we discuss the thermodynamics of polymer solutions, specifically with respect to phase separation and osmotic pressure. We shall devote considerable attention to statistical models to describe both the entropy and the enthalpy of mixtures. Of particular interest is the idea that the thermodynamic... 

Remember that the hump which causes the instability with respect to phase separation arises from an unfavorable AH considerations of configurational entropy alone favor mixing. Since AS is multiplied by T in the evaluation of AGj, we anticipate that as the temperature increases, curves like that shown in Fig. 8.2b will gradually smooth out and eventually pass over to the form shown in Fig. 8.2a. The temperature at which the wiggles in the curve finally vanish will be a critical temperature for this particular phase separation. We shall presently turn to the Flory-Huggins theory for some mathematical descriptions of this critical point. The following example reminds us of a similar problem encountered elsewhere in physical chemistry. 

The entropy value of gaseous HCl is a sum of contributions from the various transitions summarized in Table 4. Independent calculations based on the spectroscopic data of H Cl and H Cl separately, show the entropy of HCl at 298 K to be 186.686 and 187.372 J/(mol K) (44.619 and 44.783 cal/(mol K), respectively. The low temperature (rhombic) phase is ferroelectric (6). SoHd hydrogen chloride consists of hydrogen-bonded molecular crystals consisting of zigzag chains having an angle of 93.5° (6). Proton nmr studies at low temperatures have also shown the existence of a dimer (HC1)2 (7). 

The separation of Hquid crystals as the concentration of ceUulose increases above a critical value (30%) is mosdy because of the higher combinatorial entropy of mixing of the conformationaHy extended ceUulosic chains in the ordered phase. The critical concentration depends on solvent and temperature, and has been estimated from the polymer chain conformation using lattice and virial theories of nematic ordering (102—107). The side-chain substituents govern solubiHty, and if sufficiently bulky and flexible can yield a thermotropic mesophase in an accessible temperature range. AcetoxypropylceUulose [96420-45-8], prepared by acetylating HPC, was the first reported thermotropic ceUulosic (108), and numerous other heavily substituted esters and ethers of hydroxyalkyl ceUuloses also form equUibrium chiral nematic phases, even at ambient temperatures. 

The enthalpy and entropy are simple sums of the ideal gas and residual properties, which are evaluated separately. 

A turboexpander generates the deep, low-temperature refrigeration industrially used for gas separation and liquefaetion, and a number of related purposes. It does so by the meehanism of eonstant entropy expansion, together with the produetion of power (a byproduet). The power is generated from the deerease in enthalpy of the stream itself. A turboexpander is a high effieieney turbine with numerous speeial features. These features make it eonveniently usable and reliable for small volumetrie flows at the low temperatures (and often rather high pressures) usually found in these applieations. 

---