Temperature Dependence of Reaction Enthalpies

Do all reactions occur at 25°C Of course not Thus, we need a way to use Hf values for reactions at 

Since we may want to use a Cp value not at one of the tabulated temperatures we can fit a polynomial to discrete values of Cp in terms of T. Here e is simply the fifth numerical coefficient. 

FIGURE 4.6 A fourth-order polynomial fit to heat capacity data for H2 data points from the CRC Handbook in J/°K mol. Here x is the Kelvin temperature and we can see that even with a T term the fit is not perfect near 400°K and that is very good hut less than 1.000. However, the fit near 1200°K is quite good. 

From equations such as (3.106) one can readily determine AHrxn(Ti) at the standard temperature 7) of thermochemical compilations. How can we find the value AHrxn(T2) at other temperatures 72 of interest This is the subject of Kirchhoff s equation, which we shall derive as a simple consequence of the first law. 

To evaluate the formal temperature derivative of AH = AHrxn (under isobaric conditions, assumed throughout this section), we note that (3.106) is of the form [cf. (3.103)] 

Differentiating (3.107) with respect to 7 (at constant P), we obtain Kirchhoff s equation in the form 

Here CPj is the molar heat capacity of species At at constant pressure and ACP is the overall change in heat capacity (products minus reactants) 

We can integrate Kirchhoff s equation (3.109) from Tx to T2 to obtain the desired AH(T2) as 

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The temperature dependence of reaction enthalpies can be determined from the heat capacity of the reactants and products. When a substance is heated from T to T2 at a particular pressurep, assuming no phase transition is taking place, its molar enthalpy change from AHm (T]) to AHm (T2) is... 

The Temperature Dependence of Reaction Enthalpies Can Be Determined from Heat Capacity Data... 

The temperature dependence of reaction enthalpies can be determined from heat capacity data using Kirchhoff s law. 

The Variation of Reaction Enthalpy with Temperature Temperature dependence of reaction enthalpy... 

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... 

We introduced the enthalpy function particularly because of its usefulness as a measure of the heat that accompanies chemical reactions at constant pressure. We will find it convenient also to have a function to describe the temperature dependence of the enthalpy at constant pressure and the temperature dependence of the energy at constant volume. Eor this purpose, we will consider a new quantity, the heat capacity. (Historically, heat capacity was defined and measured much earlier than were enthalpy and energy.)... 

Note that Eqs. 3-49 and 3-50 are very general equations which also apply, for example, to describing temperature dependencies of reaction equilibrium constants, as will be discussed in Chapters 8 and 12 (of course, with the appropriate reaction free energy and enthalpy terms). 

In the high-temperature region, the main method of measurement is the drop calorimetry, where the sample is heated to the chosen temperature outside the calorimeter in a furnace and the heat capacity is calculated from the temperature dependence of the enthalpy changes measured after dropping the sample into the calorimeter. The application of this technique affects, however, the behavior of the sample heated in the furnace (decomposition, reaction with the crucible, etc. should be avoided) as well as at the cooling from the furnace temperature to that of the calorimeter. Sometimes the sample does not complete its phase transition at cooling (e.g. at the temperature of fusion, a part of the sample crystallizes while the other part becomes glassy). In such a case, the drop calorimeter must be supplemented by a solution calorimeter in order to get the enthalpy differences of all the samples to a defined reference state. 

The temperature dependence of the enthalpy increment of reaction (3-42) is easily determined. Differentiation of Eq. (3-50) with respect to temperature at constant pressure gives... 

The enthalpy was calculated by the scheme taking into account the condition of congruent vaporization (Sect. 16.6). The thermodynamic functions for kaolinite, muscovite, and talc and of their decomposition products at 298 K are given in Table 16.28. However, published data on the temperature dependences of the enthalpies and entropies of the reactants are lacking. Therefore, to estimate the molar entropies of the reactions the approximation = 160 9 J moP K, valid for reactants decomposing to... 

Equation 10.16 is valid as long as the temperature dependence of the enthalpy of reaction AH° is small. Combining Equation 13.4 [Eceii = (RTInF)ln K] with Equation 10.16 gives... 

Figure 11.7 Dependence of reaction enthalpy on temperature at constant pressure.

A fiirther thermodynamic parameter that provides important information to support the analysis of a bimolecular reaction between a ligand and a protein is the change in heat capacity at constant pressure ACp upon complex formation. ACp is of particular interest for the study of interactions that are largely driven by protein, ligand, and solvent hydrogen bond dissociation and formation and describes the temperature dependence of the enthalpy of binding [32]. 

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. 

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 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. 

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). 

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). 

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). 

Standard heat data are usually compiled at 298 K, and to calculate the heat of reaction at an arbitrary temperature, the temperature dependency of enthalpies of reaction species have to be considered. These are generally dependent on temperature as follows... 

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