Van’t Hoff coefficient

Panagiotopoulos et al. [16] studied only a few ideal LJ mixtures, since their main objective was only to demonstrate the accuracy of the method. Murad et al. [17] have recently studied a wide range of ideal and nonideal LJ mixtures, and compared results obtained for osmotic pressure with the van t Hoff [17a] and other equations. Results for a wide range of other properties such as solvent exchange, chemical potentials and activity coefficients [18] were compared with the van der Waals 1 (vdWl) fluid approximation [19]. The vdWl theory replaces the mixture by one fictitious pure liquid with judiciously chosen potential parameters. It is defined for potentials with only two parameters, see Ref. 19. A summary of their most important conclusions include ... 

T V — nRT. Van t Hoff noted the parallel between this law and the ideal gas equation, and he proposed that solute molecules in solution act independently of one another. Van t Hoffs law worked for solutions of nonelectrolytes and weak electrolytes, but for strong electrolytes, van t Hoff had to multiply n by a coefficient, i. For HCl and NaCl the value of i was close to 2, and for CaCl2, i was close to 3. For this reason, strong electrolytes were considered to be exceptions to van t Hoffs law. 

A theory close to modem concepts was developed by a Swede, Svante Arrhenins. The hrst version of the theory was outlined in his doctoral dissertation of 1883, the hnal version in a classical paper published at the end of 1887. This theory took up van t Hoff s suggeshons, published some years earlier, that ideal gas laws could be used for the osmotic pressure in soluhons. It had been fonnd that anomalously high values of osmotic pressure which cannot be ascribed to nonideality sometimes occur even in highly dilute solutions. To explain the anomaly, van t Hoff had introduced an empirical correchon factor i larger than nnity, called the isotonic coefficient or van t Hoff factor,... 

Similar procedures can be used to estimate coefficients in the van t Hoff equation. The least squares method for estimation of pre-exponential factor and activation energy is illustrated in Example 5.4.4.2b. 

Van t Hoff introduced the correction factor i for electrolyte solutions the measured quantity (e.g. the osmotic pressure, Jt) must be divided by this factor to obtain agreement with the theory of dilute solutions of nonelectrolytes (jt/i = RTc). For the dilute solutions of some electrolytes (now called strong), this factor approaches small integers. Thus, for a dilute sodium chloride solution with concentration c, an osmotic pressure of 2RTc was always measured, which could readily be explained by the fact that the solution, in fact, actually contains twice the number of species corresponding to concentration c calculated in the usual manner from the weighed amount of substance dissolved in the solution. Small deviations from integral numbers were attributed to experimental errors (they are now attributed to the effect of the activity coefficient). 

All partitioning properties change with temperature. The partition coefficients, vapor pressure, KAW and KqA, are more sensitive to temperature variation because of the large enthalpy change associated with transfer to the vapor phase. The simplest general expression theoretically based temperature dependence correlation is derived from the integrated Clausius-Clapeyron equation, or van t Hoff form expressing the effect of temperature on an equilibrium constant Kp,... 

Fig. 2.11 (a) PCX desorption curves at various temperatures for the activated commercial MgH Tego Magnan powder numbers indicate the average mid-plateau pressure, (b) The Van t Hoff plot for finding the enthalpy and entropy of decomposition, which is equal to -71 kJ/mol and-134 J/ mol K, respectively. Note excellent coefficient of fit = 0.991 (p - pressure)... 

As shown in Fig. 2.43b, the enthalpy of absorption and desorption calculated from the Van t Hoff plots using the mid-plateau pressures of PCT curves in Fig. 2.43a, which are listed in Table 2.18, is equal to -72 and 83 kJ/mol, respectively. The value of entropy is 138 and 151 J/mol K for absorption and desorption, respectively. The enthalpy value for absorption is very close to the values found in the literature for MgHj as discussed in Sect. 2.1.2 and 2.1.3. Surprisingly, however, the enthalpy of desorption at 83 kJ/mol is much greater than the former and also greater than the enthalpy of desorption of the as-received and activated MgH as shown in Fig. 2.11. The coefficients of fit are excellent and give good credibility to the obtained values. 

Binding of heme by isolated N-domain causes a change in sedimentation coefficient consistent with a more compact conformation and leads to the more avid association with the C-domain (125). Sedimentation equilibrium analysis showed that the Kd decreases from 55 pM to 0.8 pM (Fig. 5) (106). In addition, the calorimetric AH (-1-11 kcal/mol) and AS (-1-65 kcal/mol K) for the heme-N-domain-C-domain interaction and the AH (-3.6 kcal/mol) and AS (-1-8.1 kcal/mol K) derived from van t Hoff analysis of ultracentrifuge data for the interaction in the absence of heme indicate that hydrophobic interactions predominate in the presence of heme and a mix (e.g., hydrophobic and van der Waals forces) drives the interaction in the absence of heme. However, FTIR spectra (Fig. 6) indicate that little change in the secondary structure of domains or intact hemopexin occurs upon heme binding (104). 

Partition coefficients at different temperatures should follow the van t Hoff equation ... 

The coefficients A, B, . . . must all equal unity in ideal solutions in order to recover Equation (22). Since the van t Hoff equation is a limiting law, the coefficient A must equal unity in all solutions. Therefore Equation (29) becomes... 

Note that the value of the intercept, the value of r/RTc at infinite dilution, obeys the van t Hoff equation, Equation (25). At infinite dilution even nonideal solutions reduce to this limit. The value of the slope is called the second virial coefficient by analogy with Equation (27). Note that the second virial coefficient is the composite of two factors, B and (1/2) Vx/M. The factor B describes the first deviation from ideality in a solution it equals unity in an ideal solution. The second cluster of constants in B arises from the conversion of practical concentration units to mole fractions. Although it is the nonideality correction in which we are primarily interested, we discuss it in terms of B rather than B since the former is the quantity that is measured directly. We return to an interpretation of the second virial coefficient in Section 3.4. 

Musatov, A.P. (1993). Van t Hoff temperature coefficient for energy metabolism in lower vertebrates (In Russian). Gydnobiologicheskii Zhumal 29,77-80. 

Weiss (1970a) reviewed and evaluated the literature data then available for N2, 02, and Ar and fitted them to a smooth functional dependence. His results are reported in terms of the Bunsen coefficient j5, defined as the quantity of gas (in cm3 STP) dissolved in unit volume (1 cm3) under unit partial pressure (1 atm). Pure water data were fitted to an integrated van t Hoff equation ... 

For many reactions taking place at ordinary temperatures, the energy of activation is of the order of 20,000 cal/mole and under such circumstances the temperature coefficient (tc) is found to satisfy van t Hoff rule, e.g., at 300K, the value of tc is given by... 

Very little work has been reported on the gas-phase pyrolysis of acid halides and two excellent reviews are available4,185. Acetyl chloride was decomposed in a static system at 242-491 °C186. The reversible reaction (equation 97) occurs at 242-350 °C, where the equilibrium lies to the left. The equilibrium constant, Kp, was found to be invariable with initial pressure, and temperature-dependent according to the van t Hoff equation 8.314 In Kp (-100.3 2.0) 103/T + (132.9 3.2). Addition of HC1 reduced the extent of the reaction but did not alter the value of Kp. However, at 270-329 °C the reaction is found to be homogeneous, molecular, and to obey a first-order rate law. The rate coefficients were given by... 

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