Van’t Hoff-type plot

In Figure 11 a van t Hoff-type plot is given of the characteristic buildup time as a function of 1/T. The data clearly indicate a linear behavior in this plot with one value of the slope (AG/ ratio) for laminar flow and cinother for turbulent flow. Two lines are shown for the turbulent case, one in which only three of the data points were correlated, with what appears to be an inconsistent point left out, and one in which all four points were correlated. The effective association energies of Eq. (15), as defined by these correlations, are relatively low, ranging between 6.5 and 19 kJ/mol, which is in the range of values corresponding to physical adsorption. These results indicate that the foulant film buildup is controlled by physical rather than chemical phenomena. 

Figure 11. Dependence of characteristic fiim buiid-up time on temperature in van t Hoff-type plot...

Figure 3. Van t Hoff-type plot of intrinsic dissolution rates (log scale) against the reciprocal of the absolute temperature for the anhydrous form 111 and dihydrated forms of carbamazepine in water. Data obtained from Murphy et al. (2002).

For example, Murphy et al. (2002) determined the relative intrinsic dissolution rates of carbamazapine anhydrate (form III) and dihydrate forms in water at 10, 25, and 37 °C. Figure 3 shows the Van t Hoff-type plot of the intrinsic dissolution rates of the forms. A transition temperature of 102 °C was calculated by extrapolation of the intrinsic dissolution rate lines to the point of intersection. The anhydrous form is more stable above the transition temperature whereas the dihydrate is more stable below the transition temperature. 

Coleman et al. uses Fourier transform infrared (ETIR) spectroscopy to obtain the association constants, K, for all of the hydrogen-bonding equilibria occurring in the pure polymers, and in the blends. Using a van t Hoff-type plot, the enthalpies of hydrogen bonding can be calculated. 

Fig. 19. Van t Hoff type plots, describing the temperature dependence of the hydrogen desorption pressures in various materials. (Data are from Huston and Sandrock, 1980 Goudy et al., 1978 Kierstead, 1981a).

These two constants depend only on temperature. If data had been available at a series of temperatures, In k could have been plotted against 1/T to obtain the activation energy E and the frequency factor A, according to the Arrhenius equation. The adsorption equilibrium constant would also be expected to be an exponential function of temperature, according to a van t Hoff type of equation,... 

Henderson-Hasselbalch equation lahn-Teller effect Lee-Yang-Parr method Lineweaver-Burk method Mark-Houwink plot Meerwein-Ponndorf theory Michaelis-Menten kinetics Stem-Volmer plot van t Hoff-Le Bel theory Wolff-Kishner theory Young-Laplace equation Ziegler-Natta-type catalyst... 

Buliginski s and Quincke s linear equation was found to be only approximate by Heydweiller,io who showed that Zlor=100(or—(7o)/c(7o, where c=conc. in equiv. per lit., was not constant. E.g. for KCl c=0 031, Zlor=2 22 c=l. 43, Zlcr=2 44 and for CaCl2, c=0-658, Zlo =2 26 c=4-91, Zlor=2 82. For some salts, A a first increased and then decreased, with increase of c. A plot of Aa against /, van t Hoff s factor, went through a minimum for all the salts. The results could be represented by an equation of the type used by Griineisenii for viscosities ... 

In another study, thermodynamic parameters have been investigated with a tris(4-methyIbenzoate)-derivatized cellulose type CSP (Chiralcel OJ) and a chiral diol compound. It was found that at low temperatures, the enantioselectivity is entropy-driven (region 11), while at higher temperatures the separation is enthalpy-driven (region 1) (see Fig. 9.8). DSC and IR experiments revealed that the transitions between the enthalpic and the entropic regions of the van t Hoff plots are a result of a change in conformation of the stationary phase [76]. 

Henderson—Hasselbalch equation Jahn—Teller effect Lineweaver—Burk method Mark-Houwink plot Meerwein—Ponndorf theory Michaelis—Menten kinetics Stern—Volmer plot van t Hoff—Le Bel theory Wolff—Kishner theory Young—Laplace equation Ziegler—Natta-type catalyst... 

In general, non-linear van t Hoff behavior may be indicative of a change in the mechanism of retention. Basically, any reversible process which alters the enthalpy and entropy of adsorption can, in principle, give rise to non-linear van t Hoff plots. Dissociative processes, such as ionisation, change in conformation, or changes in the extent to which the mobile phase interacts with either the analyte or stationary phase are examples of such reversible processes. In addition, the presence of multiple types of retention mechanisms or multiple types of binding sites may also lead to non-linear van t Hoff plots. 

However, when po is maintained constant, a linear van t Hoff plot can no longer be extracted for the compositional variable. This illustrates the fact that w is a composite variable which, as such, does not represent an actual participant in the equilibrium in eq. (23). Two types of models are used to approach these real components (i) a defect-chemistry model, based on solid-state equilibria among variously charged and clustered defects, and (ii) a lattice-gas model, based on structural features of the oxygen sublattice and statistical thermodynamics. 

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