Entropy column

If T —> oo with B fixed, the contribution by the length-B horizontal segment of the outer lined area in figure 4.15 can be effectively ignored. Noting that the entropy is largest when the 2r vertical columns are statistically independent, we find that... 

Table 27 contains data for some uni-univalent solutes for which both the entropy of solution at 25°C and the viscosity //-coefficient in aqueous solution at 18 or 25° are known. In column 3 from the entropy of solution 16.0 e.u. have been subtracted for the cratic term. 

Fig. 37. Dependence of the thermodynamic parameters AH and AS of triple-helix formation on the imino acid content of the peptides (obtained by cleavage of calf skin-type I collagene with cyanogene and subequent isolation by column chromatrography)3) and of the native neutral salt-soluble skin collagene of various animals. The entropy values are denoted by dotted lines...

The idea here was to examine which pair of techniques and individual columns could lead to the best separations in 2DLC. This is achievable by using ID separations and then comparing how the retention of each component varies across the separation space. Another innovation here was the use of IT-derived metrics such as information entropy, informational similarity, and the synentropy. As stated in this paper, The informational similarity of 2D chromatographic systems, H is a measure of global... 

The last column in Table 19 lists the entropy losses due to reduction of freedom of internal rotations around the single bonds upon cyclisation. It is of interest to note, for example, that the conformational entropy lost upon cyclisation of an 8-rotor chain amounts to f of the corresponding quantity related to cyclisation of a 100-rotor chain. 

The procedure followed in the use of the tables of Andersen et al. [1], and Yoneda [4] is illustrated below for the estimation of standard entropies. These tables also include columns of base structure and group contributions for estimating fHm,298.i5K> thc Standard enthalpy of formation of a compound, as well as columns for a, b, and c, the constants in the heat capacity equations that are quadratic in the temperature. Thus it is possible to estimate AfGm gg.isK by appropriate summations of group contributions to Af7/ 298.i5K and to 5m,298.i5K- Then, if information is required at some other temperature, the constants of the heat capacity equations can be inserted into the appropriate equations for AG, as a function of temperature and AGm can be evaluated at any desired temperature (see Equation 7.68 and the relation between AG and In K). 

Table 3.6 Comparison of predictive capacities of various equations in estimating standard molar entropy T = 298.15 K, P = 1 bar). Column I = simple summation of standard molar entropies of constituent oxides. Column II = equation 3.86. Column III = equation 3.86 with procedure of Holland (1989). Column IV = equation 3.85. Values are in J/(mole X K). Lower part of table exchange reactions adopted with equation 3.85 (from Helgeson et al., 1978) and Sj finite differences for structural oxides (Holland, 1989). 

Comparative evaluation of the predictive properties of equations 3.85 and 3.86 is given in table 3.6. Column I lists values obtained by simple summation of the entropies of the constituent oxides. This method (sometimes observed in the literature) should be avoided, because it is a source of significant errors. The lower part of table 3.6 lists the adopted exchange reactions and the Sj terms of Holland s model. 

Barrer (3) makes similar calculations for the entropies of occlusion of substances by zeolites and reaches the conclusion that the adsorbed material is devoid of translational freedom. However, he uses a volume, area or length of unity when considering the partition function for translation of the adsorbed molecules in the cases where they are assumed to be capable of translation in three, two or one dimensions. His entropies are given for the standard state of 6 = 0.5, and the volume, area or length associated with the space available to the adsorbed molecules should be of molecular dimensions, v = 125 X 10-24 cc., a = 25 X 10-16 cm.2 and l = 5 X 10-8 cm. When these values are introduced into his calculations the entropies in column four of Table II of his paper come much closer together, as is shown in Table I. The experimental values for different substances range from zero to —7 cals./deg. mole or entropy units, and so further examination is required in each case to decide... 

The experimental entropies of adsorption were calculated after obtaining the free energies of adsorption at 0 = /% from the gas pressure in equilibrium with half the amount of adsorbate required to form the monolayer. The same principles were used to obtain the figure for the entropy of adsorption of O2 on unreduced steel. The values for carbon tetrachloride were taken directly from Foster s paper (4). The results for adsorption in chabazite were obtained from the work of Barrer and Ibbitson (15) with the slight modification needed to allow for the different standard states in the two phases used by them. The figures in the last column... 

Changing the column temperature can produce a variety of additional effects. Temperature changes the balance between enthalpy and entropy effects on retention mechanisms. Changing the temperature changes the equilibrium constants of both solvent and solutes, and it changes the... 

FIGURE 16.3 Dependences of the polymer retention volume on the logarithm of its molar mass M or hydrodynamic volume log M [T ] (Section 16.2.2). (a) Idealized dependence with a long linear part in absence of enthalpic interactions. Vq is the interstitial volume in the column packed with porous particles, is the total volume of liquid in the column and is the excluded molar mass, (b) log M vs. dependences for the polymer HPLC systems, in which the enthalpic interaction between macromolecules and column packing exceed entropic (exclusion) effects (1-3). Fully retained polymer molar masses are marked with an empty circle. For comparison, the ideal SEC dependence (Figure 16.3a) is shown (4). (c) log M vs. dependences for the polymer HPLC systems, in which the enthalpic interactions are present but the exclusion effects dominate (1), or in which the full (2) or partial (3,4) compensation of enthalpy and entropy appears. For comparison, the ideal SEC dependence (Figure 16.3a) is shown (5). (d) log M vs. dependences for the polymer HPLC systems, in which the enthalpic interactions affect the exclusion based courses. This leads to the enthalpy assisted SEC behavior especially in the vicinity of For comparison, the ideal SEC dependence (Eigure 16.3a) is shown (4). 

The effect of exclusion on the retention volumes of macromolecules was qualitatively explained above. The pioneering work of Casassa [54] has shown that the extent of pore exclusion of macromolecules is controlled by the changes in their (conformational) entropy. The principle is explained in a simplified form in Figure 16.4a through c. A zone of polymer solution with a nonzero concentration travels along a column packed with porous particles. Initially, the concentration of macromolecules within pores is zero (Figure 16.4a). The concentration gradient outside of pore (c > 0) and within pore (c = 0) pulls macromolecules into the pores. 

The principle of the liquid chromatography under critical conditions (LC CC) was elucidated in Section 16.3.3. The mutual compensation of the exclusion—entropy and the interaction—enthalpy-based retention of macromolecules can be attained when applying in the controlled way the interactions that lead to either adsorption or enthalpic partition. The resulting methods are called LC at the critical adsorption point (LC CAP) or LC at the critical partition point (LC CPP), respectively. The term LC at the point of exclusion-adsorption transition (LC PEAT) was also proposed for the procedures employing compensation of exclusion and adsorption [161]. It is anticipated that also other kinds of enthalpic interactions, for example the ion interactions between column packing and macromolecules can be utilized for the exclusion-interaction compensation. 

Entropy changes were estimated with Eq. 4 assuming that V, is equal to the total stationary phase volume existing in the column. Therefore, these values reflect more properly the relative differences in entreaties of transfer instead of the standard molar entropies that would require to use the volume of the active stationary phase. 

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