Temperature dependence of equilibrium

The Temperature Dependence of Equilibrium Is Described by the van t Hoff Equation 

If you measure an equilibrium constant K T) at different temperatures, you can learn the enthalpy and entropy of the reaction, which are often useful for constructing or testing microscopic models. To illustrate, we return to the two-state equilibrium, 

At constant T and p, the condition for equilibrium is pa = ps- The pressure-based equilibrium constant is Kp = Pb/Pa- Using Equation (13.30) you have, at equilibrium  

Ap° depends only on temperature (see Equation (13.30), not on pressure, and can be expressed in terms of components (see Equation (9.25)). 

When Ah° is independent of temperature, integration of Equation (13.38) gives 

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The temperature dependency of equilibrium constants and of Henry s constants are compiled in tables A II.I and A II.II. 

The temperature dependency of equilibrium as well as of Henry s constants is given in tables A II.I and A II.II (cf. Appendix II). In comparison with the original publication for the equilibrium constants of the second dissociations of hydrogen sulfide and sulfur dioxide (reactions 6 and 7) numbers derived from Cobble (18) and Arkhipova et al. (J 9) were used. 

Also, AH values are required to calculate the temperature dependence of equilibrium constants. For aU these reasons, it is desirable to have tables of AH values available, so that the enthalpies of various transformations can be calculated readily. In many of these calculations, we make use of Hess s law, which is now firmly established on the basis of the first law of thermodynamics. We can then calculate AH for reactions for which the heat effect is difficult to measure but that can be expressed as sums of reactions with known values of AH. 

Based on the solution property of poly (DMAEMA-co-AAm) in response to temperature, the temperature dependence of equilibrium swelling of poly (DMAEMA-c6>-AAm) gel as a function of chemical composition was observed as shown in Figure 6. The transition temperature of copolymer gel between the shrunken and swollen state was shifted to the lower temperature with increases in AAm content in the gel network. This is attributed to the hydrogen bond in the copolymer gel network and its hydrophobic contribution to the LCST Copolymer II gel was selected as a model polymer network for permeation study because it showed the sharp swelling transition around 34°C. 

Table 3.5 gives the average change in Kin per 10°C increase/decrease in temperature for various A12H, values. A much more comprehensive table which is extremely useful for assessing the temperature dependence of equilibrium constants as well as of reaction rate constants is Table D1 in Appendix D. 

Which thermodynamic function needs to be known for assessing the temperature dependence of equilibrium partitioning How can this function be derived from experimental data What caution is advised when extrapolating partition constants from one temperature to another temperature ... 

In analogy to the temperature dependence of equilibrium partition constants (Eqs. 3-47 to 3-54, Section 3.4), the effect of temperature on Xja over a small temperature range can be described by ... 

Phyiscal Constants and Units Physicochemical Properties of Organic Compounds Temperature Dependence of Equilibrium and Kinetic Constants Properties of the Earth... 

Temperature Dependence of Equilibrium Constants and Rate Constants... 

Table D.l Temperature Dependence of Equilibrium Constants Km, K ) or Rate Constants (k) as a Function of the Corresponding Enthalpy Changes [AI277, (Eq. 3-51), Ar//° (Eq. 8-20)] or Activation Energies [ a (Eq. 12-30)], Respectively. Values Given as Percent of the Value at 25°C (T = 298 K)...

Fig. 5. Temperature dependence of equilibrium swelling ratio of poly(APy)/PEO IPNs in water PEO repeating unit 20.6% mole fraction (A) 36.3% mole fraction (V) 49.2% mole fraction ( ) 58.7% mole fraction ( ) crosslinked polyfAPy) ( ) crosslinked PEO (O)...

So there is an underlying basis related to the standard enthalpy of reaction (or heat capacities, Eq. 2.26) for the equation form used in this work to characterize the temperature dependence of equilibrium constants and Pitzer parameters (cf. Eqs. 2.70 and 2.73). 

Spiro orthoesters (92, R = Me, Ph, and H) show typical equilibrium polymerization behavior at or below ambient temperature. [92] The poly(cyclic orthoester)s derived from 92 depolymerize to the monomers, although they have sufficient strains to be able to undergo ring-opening polymerization. The polymerization enthalpies and entropies for these three monomers were evaluated from the temperature dependence of equilibrium monomer concentrations (Table 5). The enthalpy became less negative as the size of the substituent at the 2-position in 92 was increased H < Me < Ph. This behavior can be explained in terms of the polymer state being made less stable by steric repulsion between the bulky substituents and/or between the substituent and the polymer main chain. The entropy also changed in a similar manner with the size of the substituents. 

Figure 2a. Temperature dependences of equilibrium order parameters, r i(T), r 2(T).

The equation AG° - AIT - TA5° tells us that how ACT1 varies with temperature depends mainly on the entropy change for the reaction (AS°). We need these terms to explain the temperature dependence of equilibrium constants and to explain why some reactions may absorb heat (endothermic) while others give out heat (exothermic). 

While the temperature dependence of equilibrium capacity for adsorption is defined by the parameter AH, the dependence of rate of adsorption is usually expressed in terms of activation energy, E. Rate of adsorption is related to the activation energy by the equation,... 

Two different types of dynamic mechanical experiments were performed. First, the temperature dependence of "equilibrium" dynamic mechanical properties for all epoxy samples were obtained... 

In contrast to the formally analogous van t Hoff equation [10] for the temperature dependence of equilibrium constants, the Arrhenius equation 1.3 is empirical and not exact The pre-exponential factor A is not entirely independent of temperature. Slight deviations from straight-line behavior must therefore be expected. In terms of collision theory, the exponential factor stems from Boltzmann s law and reflects the fact that a collision will only be successful if the energy of the molecules exceeds a critical value. In addition, however, the frequency of collisions, reflected by the pre-exponential factor A, increases in proportion to the square root of temperature (at least in gases). This relatively small contribution to the temperature dependence is not correctly accounted for in eqns 2.2 and 2.3. [For more detail, see general references at end of chapter.]... 

Direct calorimetric methods or temperature dependence of equilibrium constants can be used to measure enthalpies and entropies of acid-base reactions. The following section gives more details on use of data from these measurements. 

The temperature dependence of equilibrium isotope exchange in the calcite-water system has been intensively studied since Urey (1947) first suggested that the paleotemperature of the ancient oceans could be estimated by the 0-isotope distribution between seawater and the calcium carbonate precipitated from it. Urey et al. (1951) argued that O-isotope equilibrium between seawater and CaC03 was likely and support for this idea has come from the close agreement between the CaC03-H20 isotopic fractionation observed in natural systems and those derived from both thermodynamic calculations and laboratory experiments (e.g. Epstein et al., 1951, 1953 Emiliani, 1955 O Neil et al., 1969 O Neil et al., 1975). 

Yanson et al. [41] using field-ionization mass spectrometry studied the formation of gas-phase GC, CC, AT and TT pairs. From measurements of temperature dependence of equilibrium constants, an interaction enthalpy for the base pair formation was derived. This technique was sometimes questioned because the determination of enthalpy from the slopes of appropriate van t Hoff curves might not be unambiguous. From Table 6 it is evident that the agreement with the present theoretical values is good, and concerns not only the relative interaction enthalpies but even the absolute values the average absolute error is less than 1.5 kcal/mol. 

Figure 17. Molecular free energies (a) and internal energies (b) vs surface roughness parameter d at different temperatures for once-folded alkane crystal, using the model in Figure 16. Circles mark the free energy minima. The dashed line shows the temperature dependence of equilibrium energy (after ref 40).

This system shows additional anomalies, the total uptake is high and also some temperature dependence of equilibrium sorption occurs, contrary to the behavior of other systems (6,7, 8) with the same epoxide but different curing agent. 

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