Van t Hoff plots

A connnon approach has been to measure the equilibrium constant, K, for these reactions as a fiinction of temperature with the use of a variable temperature high pressure ion source (see section (Bl.7.2)1. The ion concentrations are approximated by their abundance in the mass spectrum, while the neutral concentrations are known from the sample mlet pressure. A van t Hoff plot of In K versus /T should yield a straight Ime with slope equal to the reaction enthalpy (figure B1.7.11). Combining the PA with a value for basicityG at one temperature yields a value for A.S for the half-reaction involving addition of a proton to a species. While quadnipoles have been tire instruments of choice for many of these studies, other mass spectrometers can act as suitable detectors [19, 20]. 

Figure Bl.7.11. Van t HofF plot for equilibrium data obtained for die reaetion of isobutene with anuuonia in a high pressure ion soiiree (reprodueed from data in [19]).

Figure 2. The enthalpy, AH, of the phase transformation can be calculated from the variation of In P laleai] with reciprocical temperature in a van t Hoff plot.

Stability constants, enthalpies, and entropies determined in the Pu(IV)-HS0I system are presented in Table IV. The enthalpy and entropy for each of the complexation steps were determined by a van t Hoff plot of the stability constant data. These stability constants include a correction for nitrate complexing of Pu(IV) [(Kj = 2.88 (13)) [N0j] = 0.074 M]. Error limits of the... 

Many chromatographic systems show linear relationships between the logarithm of the capacity factor and the reciprocal of the column temperature (van t Hoff plots) [255,258-261]. In thermodynamic terms the interaction of the solute with the stationary phase can be described by... 

The linearity of van t Hoff plots, such as Figure 3.14, depends on the degree to which the isobaric heat capacity of the system (Cp) remains constant between the... 

Figure 3.14 Idealized van t Hoff plot of the temperature dependence of the affinity of a peptide inhibitor for die enzyme hdm2.

Figure 11 Van t Hoff plot of log solubility of methylprednisolone in water as a function of reciprocal water temperature. (Reprinted with permission from Ref. 47.)...

Also, hydrates are more soluble in water-miscible solvents than are the corresponding anhydrous forms. For example, the solubility of caffeine hydrate is lower than that of anhydrous caffeine in water but higher in ethanol. The maximum concentration seen may be due to the solubility of the anhydrous crystalline phase or due to a temporary steady state in which the rate of dissolution of the metastable anhydrous form and the rate of crystallization of the stable hydrate are equal. The decreasing concentration represents crystallization of the stable hydrate from a solution supersaturated with respect to it. If the maximum concentration of the solute in the dissolution experiment corresponds to the solubility, then the initial increase in concentration follows the Noyes-Whitney equation [15]. Van t Hoff plots of log solubility versus the reciprocal of temperature give linear relationships (Fig. 16). 

Figure 16 Van t Hoff plot of molal solubility versus reciprocal absolute temperature for theophylline in water. (Reproduced with permission from Ref. 50.)...

Figure 8. van t Hoff plots of equilibrium constant K, standard state 1 atm., for 0,1 equilibrium H3N(CH2)pNH2++ H20H3N(CH2)oNH3(H20)2+. Values of chain length p given beside each plot in the figure. Equilibrium constant K decreases as p is increased. From Klassen, J. A. Blades, A. T. Kebarle, P. ) Am. Chem. Soc. 1996, with permission. 

First, we examine the sequential hydration of a doubly protonated diamine, H3N(CH2)12NH2+. The sequential enthalpies, Atf n, free energies AG , and entropies obtained from van t Hoff plots of the hydration equilibria,62 are... 

Van t Hoff plots for different metal hydrides. (Reproduced with permission from Schuth, F., Bogdanovic, B., and Felderhoff, M., Chem. Commun., 20,2249-2258, 2004.)... 

Methods for the determination of solubility have been thoroughly reviewed [21,22], Solubility is normally highly dependent on temperature, and so the temperature must be recorded for each solubility measurement. Plots of solubility against temperature, as exemplified by Fig. 4 [23,24], are commonly used for characterizing pharmaceutical solids and have been extensively discussed [1,24]. Frequently (especially over a relatively narrow temperature range), a linear relationship may be given either by a van t Hoff plot according to [23]... 

Fig. 4 Van t Hoff plots (left) and Hildebrand plots (right) representing the influence of temperature (absolute) on the aqueous mole fraction solubility of adipic acid (AD), acetaminophen (AM), methylp-hydroxybenzoate (MP), ethylp-hydroxybenzoate (EP), propyl p-hydroxybenzoate (PP), and butyl p-hydroxybenzoate (BP). The data represented by O were taken from Ref. 24, while the data represented by x were obtained from Ref. 23. (Reproduced with permission of the copyright owner, Elsevier Science Publishing Co.)...

When a simple van t Hoff plot is applied (i.e., when InKm is plotted against 1 /T), the reaction enthalpy and entropy at the mean temperature T of the experimental temperature interval are calculated from the slope and the intercept of equation 14.4 ... 

One set of Kc values at several temperatures, obtained by Sousa Lopes and Thompson, is shown in table 14.1 and plotted in figure 14.1. This van t Hoff plot leads to equation 14.9. 

Figure 14.1 A van t Hoff plot for reaction 14.6. Data from [313],...

What main conclusions can we draw from the three examples discussed here First, although van t Hoff plots should involve Km rather than Kc data, the use of the latter may afford sensible and possibly accurate thermochemical values (always under the assumption of ideal solutions ), particularly if the density term of equation 14.5 is considered in the calculation of the reaction entropy. Second, due to the lack of gas solubility data, the second law method is much... 

The equilibrium concentration of the ions A- and B- participating in the equlibrium can be directly observed by mass spectrometry. Thus, the free-energy change can be derived from the equilibrium constant, since the concentrations of the neutral species are known in advance. Similarly, by measuring the temperature dependence of the equilibrium constants, the associated enthalpy and entropy can be obtained from van t Hoff plots. By measuring a series of interconnecting equlibria, an appropriate scale can be established. The primary standard in such work has frequently been SO2 whose electron affinity is well established by electron photodetachment36. 

A van t Hoff plot of In k against l/T yields a straight line if the stationary phase is not deformed. The relation Ink and —AH/RT is used to study the temperature-dependent selectivity of the stationary phase.33-35... 

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