Temperature dependence enthalpy

A,B K-value or temperature dependence parameters a,b Activity coefficient composition dependence parameters C,D Vapor enthalpy temperature dependence parameters E,F Liquid enthalpy temperature dependence parameters F Total feed molar flow rate... 

While the P-M model uses the linear extrapolation of the liquid polymer enthalpy-temperature dependence to measure (])(y, the C-F approach is different It considers AH(oo, Ta) to be an adjustable parameter and is measured by curve fitting the AH(ta, Ta) versus logta plots. The function (])(ta) is also redefined as ... 

For many liquid mixtures. Equation (19) can be used to provide a crude estimate of excess enthalpy. A much better estimate is obtained if the UNIQUAC parameters are considered temperature-dependent. For example, suppose Equations (4-9) and (4-10) are modified to = + k /t... 

The Fp correction factor for each enthalpy interval depends both on the assumed value of Xp and the temperatures of the interval on the composite curves. It is possible to modify the simple area target formula to obtain the resulting increased overall area A etwork for a network of 1-2 exchangers ... 

It is not necessary to limit the model to idealized sites Everett [5] has extended the treatment by incorporating surface activity coefficients as corrections to N and N2. The adsorption enthalpy can be calculated from the temperature dependence of the adsorption isotherm [6]. If the solution is taken to be ideal, then... 

The validity of equation (A2.1.70) has sometimes been questioned when enthalpies of reaction detennined from calorimetric experiments fail to agree with those detennined from the temperature dependence of the equilibrium constant. The thennodynamic equation is rigorously correct, so doubters should instead examine the experunental uncertainties and whether the two methods actually relate to exactly the same reaction. 

In order to Introduce thermal effects into the theory, the material balance equations developed in this chapter must be supplemented by a further equation representing the condition of enthalpy balance. This matches the extra dependent variable, namely temperature. Care must also be taken to account properly for the temperature dependence of certain parameters In... 

Figure 1.4. Temperature dependence of the change in Gihhs energy, enthalpy and entropy upon transfer of ethane and butane from the gas phase to water. The data refer to transfer from the vapour phase at 0.101 MPa to a hypothetical solution of unit mole fraction and are taken from ref. 125.

NMR signals of the amino acid ligand that are induced by the ring current of the diamine ligand" ". From the temperature dependence of the stability constants of a number of ternary palladium complexes involving dipeptides and aromatic amines, the arene - arene interaction enthalpies and entropies have been determined" ". It turned out that the interaction is generally enthalpy-driven and counteracted by entropy. Yamauchi et al. hold a charge transfer interaction responsible for this effect. 

Chlorine, a member of the halogen family, is a greenish yellow gas having a pungent odor at ambient temperatures and pressures and a density 2.5 times that of air. In Hquid form it is clear amber SoHd chlorine forms pale yellow crystals. The principal properties of chlorine are presented in Table 15 additional details are available (77—79). The temperature dependence of the density of gaseous (Fig. 31) and Hquid (Fig. 32) chlorine, and vapor pressure (Fig. 33) are illustrated. Enthalpy pressure data can be found in ref. 78. The vapor pressure P can be calculated in the temperature (T) range of 172—417 K from the Martin-Shin-Kapoor equation (80) ... 

Table 2. Temperature Dependence of Vapor Pressure, Density, and Enthalpy of Methyl Chloride...

For experiments conducted at constant pressure, the second term ia equation 36 disappears. The expression for the temperature dependence is then obtained by performing an indefinite integration on the remainder of the equation after assuming that the enthalpy change of volatilization, (/i. — hp ), is constant with respect to temperature. The resulting equation is... 

Tbe enthalpy of vaporization decreases with temperature and is zero at tbe critical point. If tbe value of an enthalpy of vaporization AH is known at temperature T, this temperature dependency can be represented by tbe Watson relation to calculate another enthalpy of vaporization AH at any other temperature To ... 

It follows from this discussion that all of the transport properties can be derived in principle from the simple kinetic dreoty of gases, and their interrelationship tlu ough k and c leads one to expect that they are all characterized by a relatively small temperature coefficient. The simple theory suggests tlrat this should be a dependence on 7 /, but because of intermolecular forces, the experimental results usually indicate a larger temperature dependence even up to for the case of molecular inter-diffusion. The Anhenius equation which would involve an enthalpy of activation does not apply because no activated state is involved in the transport processes. If, however, the temperature dependence of these processes is fitted to such an expression as an algebraic approximation, tlren an activation enthalpy of a few kilojoules is observed. It will thus be found that when tire kinetics of a gas-solid or liquid reaction depends upon the transport properties of the gas phase, the apparent activation entlralpy will be a few kilojoules only (less than 50 kJ). 

The enthalpy of the adsorption of chlorobenzene was determined from the temperature-dependence of the SERS intensities [4.301]. 

The temperature dependence of reaction rates permits evaluation of the enthalpy and entropy components of the free energy of activation. The terms in Eq. (4.4) corresponding to can be expressed as... 

Enthalpy changes for biochemical processes can be determined experimentally by measuring the heat absorbed (or given off) by the process in a calorimeter (Figure 3.2). Alternatively, for any process B at equilibrium, the standard-state enthalpy change for the process can be determined from the temperature dependence of the equilibrium constant ... 

A simpler phenomenological form of Eq. 13 or 12 is useful. This may be approached by using Eq. 4 or its equivalent, Eq. 9, with the rate constants determined for Na+ transport. Solving for the AG using Eqn. (3) and taking AG to equal AHf, that is the AS = 0, the temperature dependence of ix can be calculated as shown in Fig. 16A. In spite of the complex series of barriers and states of the channel, a plot of log ix vs the inverse temperature (°K) is linear. Accordingly, the series of barriers can be expressed as a simple rate process with a mean enthalpy of activation AH even though the transport requires ten rate constants to describe it mechanistically. This... 

The free-energy change AG is made up of two terms, an enthalpy term, AH, and a temperature-dependent entropy term, TAS. Of the two terms, the enthalpy term is often larger and more dominant. 

Apparently, the 1H NMR spectra of 1 //-azepines are invariant over substantial temperature ranges.61 However, temperature dependence has been noted69 in the 13CNMR spectra of some 1 -acyl-1 //-azepines, and is attributed to hindered rotation about the N-CO bond rather than to ring-inversion phenomena AG free enthalpies of activation for hindered rotation of 62-66 kJ moP1 have been calculated. E/Z-rotamcr ratios for l-aroyl-l//-azepines have been assessed and show a slight preference for the -rotamer 22 however, an X-ray structural analysis of l-(4-bromobenzoyl)-2-methyl-3.5,7-triphenyl-l//-azepine demonstrates that in the crystal state it is exclusively in the E configuration.22... 

The partial molar entropy of a component may be measured from the temperature dependence of the activity at constant composition the partial molar enthalpy is then determined as a difference between the partial molar Gibbs free energy and the product of temperature and partial molar entropy. As a consequence, entropy and enthalpy data derived from equilibrium measurements generally have much larger errors than do the data for the free energy. Calorimetric techniques should be used whenever possible to measure the enthalpy of solution. Such techniques are relatively easy for liquid metallic solutions, but decidedly difficult for solid solutions. The most accurate data on solid metallic solutions have been obtained by the indirect method of measuring the heats of dissolution of both the alloy and the mechanical mixture of the components into a liquid metal solvent.05... 

An implicit assumption of the foregoing treatment is that A// remains independent of temperature over the range investigated. This is very nearly correct in general, and is particularly the case given that studies of reactions in solution are usually conducted over a temperature interval of only some 30-50°. In certain circumstances the temperature profiles show curvature outside the experimental error. Such cases have, or appear to have, temperature-dependent activation enthalpies. Here we explore one of the reasons for that another is given in Section 7.3. 

Clearly, the temperature profile is linear. The activation parameters are the sums shown in general, a sum of entropies and enthalpies is the result when constants are multiplied. If values of AS% and Aare known independently, from the temperature dependence of Ka for example, one can then calculate AS and AH by difference. 

The Arrhenius activation energy,3 obtained from the temperature dependence of the three-halves-order rate constant, is Ea = 201 kJ mol-1. This is considerably less than the standard enthalpy change for the homolysis of acetaldehyde, determined by the usual thermodynamic methods. That is, reaction (8-5) has AH = 345 kJ mol-1. At first glance, this disparity makes it seem as if dissociation of acetaldehyde could not be a predecessor step. Actually, however, the agreement is excellent when properly interpreted. 

For the temperature dependency of Pu(g), Battles et al. (15) combined the partial pressure of Pu(g) determined at 2219 K and an assumed partial enthalpy of vaporization value of 118 kcal/mol, leading to the expression... 

A recent paper by Leffek and Matheson (1971) nicely complements this work, as it describes the results of a careful investigation of the temperature dependence of the kinetic isotope effect in the reaction studied by Kaplan and Thornton (1967). It is found that AAH = 134 + 30 cal mol and dd/S = 0-15 + 0-09 cal mol deg , demonstrating that the isotope effect is primarily due to an enthalpy difference, and providing support for the steric interpretation suggested by Kaplan and Thornton (1967). 

Reliable information on the thermodynamic stability of group 13/15 adducts is usually obtained by gas phase measurements. However, due to the lability of stibine and bismuthine adducts in the gas phase toward dissociation, temperature-dependent H-NMR studies are also useful for the determination of their dissociation enthalpies in solution [41b], We focussed on analogously substituted adducts t-BusAl—E(f-Pr)3 (E = P 9, As 10, Sb 11, Bi 12) since they have been fully characterized by single crystal X-ray diffraction, allowing comparisons of their thermodynamic stability in solution with structural trends as found in their solid state structures. 

Fig. 3. Dissociation enthalpies of adducts f-Bu3Al-E( -Pr)3 as obtained from temperature-dependent NMR Studies...

In addition to chemical reactions, the isokinetic relationship can be applied to various physical processes accompanied by enthalpy change. Correlations of this kind were found between enthalpies and entropies of solution (20, 83-92), vaporization (86, 91), sublimation (93, 94), desorption (95), and diffusion (96, 97) and between the two parameters characterizing the temperature dependence of thermochromic transitions (98). A kind of isokinetic relationship was claimed even for enthalpy and entropy of pure substances when relative values referred to those at 298° K are used (99). Enthalpies and entropies of intermolecular interaction were correlated for solutions, pure liquids, and crystals (6). Quite generally, for any temperature-dependent physical quantity, the activation parameters can be computed in a formal way, and correlations between them have been observed for dielectric absorption (100) and resistance of semiconductors (101-105) or fluidity (40, 106). On the other hand, the isokinetic relationship seems to hold in reactions of widely different kinds, starting from elementary processes in the gas phase (107) and including recombination reactions in the solid phase (108), polymerization reactions (109), and inorganic complex formation (110-112), up to such biochemical reactions as denaturation of proteins (113) and even such biological processes as hemolysis of erythrocytes (114). 

Theoretically, the problem has been attacked by various approaches and on different levels. Simple derivations are connected with the theory of extrathermodynamic relationships and consider a single and simple mechanism of interaction to be a sufficient condition (2, 120). Alternative simple derivations depend on a plurality of mechanisms (4, 121, 122) or a complex mechanism of so called cooperative processes (113), or a particular form of temperature dependence (123). Fundamental studies in the framework of statistical mechanics have been done by Riietschi (96), Ritchie and Sager (124), and Thorn (125). Theories of more limited range of application have been advanced for heterogeneous catalysis (4, 5, 46-48, 122) and for solution enthalpies and entropies (126). However, most theories are concerned with reactions in the condensed phase (6, 127) and assume the controlling factors to be solvent effects (13, 21, 56, 109, 116, 128-130), hydrogen bonding (131), steric (13, 116, 132) and electrostatic (37, 133) effects, and the tunnel effect (4,... 

As to the computation of reaction enthalpies and entropies, AH and AS , the same arguments apply if they have been obtained from the temperature dependence of the equilibrium constant. A different situation arises vdien AH is determined directly from calorimetry, say with a constant relative error 6. The standard entropy AS then has the standard error... 

Many workers have offered the opinion that the isokinetic relationship is confined to reactions in condensed phase (6, 122) or, more specially, may be attributed to solvation effects (13, 21, 37, 43, 56, 112, 116, 124, 126-130) which affect both enthalpy and entropy in the same direction. The most developed theories are based on a model of the half-specific quasi-crystalline solvation (129, 130), or of the nonideal conformal solutions (126). Other explanations have been given in terms of vibrational frequencies involving solute and solvent (13, 124), temperature dependence of solvent fluidity in the quasi-crystalline model (40), or changes of enthalpy and entropy to produce a hole in the solvent (87). 

Note that the apparent activation energy is the activation energy of the activated process modified by the equilibrium enthalpies. Thus the apparent activation energy depends on both the pressure and temperature in this case. Note also that we have neglected any non-exponential temperature dependence. As we shall see in Chapter 3, V, AH, and AS are to some degree functions of temperature. 

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