Calculation at another temperature

Equation (10.4) is important for several reasons. For example, we can determine the heat of vaporization for a liquid by measuring Pvap at several temperatures and then evaluating the slope of a plot of In(Pvap) versus 1/T On the other hand, if we know the values of AWvap and Pvap at one temperature, we can use Equation (10.4) to calculate at another temperature. This can be done by recognizing that the constant C does not... 

Column 2 of Table 26 provides the constants that are used to compute A/f p an alkane at 298 K. To carry out the same calculation at another temperature, for example, the boiling point, one can calculate AfH and AfR by using the appropriate polynomial expansions of equation 50, or use the set of Au,- y p, also tabulated in Table 26, which are the differences in the corresponding a,-g and a, j. (The former are given in Table 7 and the latter in Table 25.)... 

AH, A.S or AG is known at a specified temperature T, say 298 K, its value at another temperature T can be computed using this value and the changes involved m bringing the products and the reactants separately from T to T. If these measurements can be extrapolated to 0 K, the isothennal changes for the reaction at 0 K can be calculated. 

In the preceding sections we discussed methods of obtaining enthalpies of reaction at a fixed temperature (generally 298.15 K). In particular, we pointed out that it is possible to tabulate enthalpies of formation and bond enthalpies and to use these to calculate enthalpies of reaction. Such tables of enthalpies of formation are available for only a few standard temperatures. Frequently, however, it is necessary to know the enthalpy of a reaction at a temperature different from those available in a reference table. Therefore, we consider now the procedures that can be used to calculate the enthalpy of reaction (at constant pressure) at one temperature, from data at another temperature. 

We have observed from Equation (10.24) that we can calculate from the temperature derivative of K, or we can calculate K at one temperature from the value of K at another temperature and the value of The technique of titration calorimetry [5] makes possible the calculation of K and from the same... 

Experimentally, the decay of the reagents at temperature T in the samples, which have initially been kept for some time, ta, at another temperature, T0 < T, is often studied. In this case, the distance of tunneling, Rt, which for lF(i ) of eqn. (22) obeys, by definition, the relationship VE(i t) t = 1, can be calculated for the times t > t0 from the ratio... 

This equation can be used to calculate Ea from rate constants k and k2 at temperatures 7i and T2. By the same token, if we know Ea and the rate constant k at one temperature Tlr we can calculate the rate constant k2 at another temperature T2. Worked Example 12.11 shows how this is done. 

If vapour pressure of a liquid at one temperature is known, that at another temperature can be calculated. 

Looking up a density or specific volume at one temperature and pressure and using it at another temperature and pressure usually works well for a solid or a liquid, but not at all for a gas. An expression is needed for gases that relates specific volume to temperature and pressure, so that if any two of these quantities are known the third can be calculated. 

If Kiq is known at one temperature, the at another temperature can be calculated if AH is known. Or, we can calculate AH from the values at two different temperatures. If Kcq increases with increasing temperature, AH is positive if decreases with increasing temperature, AH is negative. The calculations assume that AH is constant over the temperature range studied. This assumption is reasonably valid for the small temperature ranges usually employed for enzyme-catalyzed reactions (e.g., 20 to 40 C). 

This entry allows the user to calculate VLE at another temperature for the binary mixture under consideration.)... 

Part of the problem with this and other investigations where departures from a slope of 1.0 were noted may be due to the fact that most values of Ho have been determined near 25°, whereas the hydrolyses have been performed at 60-80°. Since there is a small but systematic shift in the relationships between R oRnd of concentration of acid with changing temperature for sulfuric, hydrochloric, and phosphoric acids, values of Ho and relationships between different acids calculated at one temperature will not hold true for another. Therefore, all relationships between hydrolysis rate constants and Ho should be determined at 25° (or the temperature at which Ho was determined). 

If Ag and Ah° are needed at another temperature (besides 298.15°K), then the following sets of relations can be used. Temperature variations of Ag° are calculated from the following Gibbs-Hehnholtz relation ... 

An equation relating the rate constants ki and k2 at temperatures Tj and T2 can be used to calculate the activation energy or to find the rate constant at another temperature if the activation energy is known. To derive such an equation we start with Equation (13.9) ... 

In this chapter we shall first look in more detail at the various experimental methods of measurement. These include determinations not only of heats of reaction, as AH or A U, but also of the heat capacities of individual materials. Armed with such information, we shall see that if the value of AH for a reaction is known at one temperature, it is possible to calculate it at another temperature. Finally, by using the concept of bond energies, we shall learn how to estimate values of Aft for compounds which are rare or uncharacterized. 

Empirical potentials are only applicable with certainty over the range of interatomic distances used in the fitting procedure, which can lead to problems if the potential is used in a calculation that accesses distances outside this range. This can happen in defect calculations, molecular dynamics simulations or lattice dynamics calculations at high temperature and/or pressure. In addition experimental data is required and thus direct calculation is the only method available when there is no relevant experimental data. It may, of course, be possible to take potentials derived for one system and transfer them to another. This method has been successful with potentials derived for binary oxides (Lewis and Catlow, 1985 Bush et al., 1994) being transferred to ternary systems (Lewis and Catlow, 1985 Price et al., 1987 Cormack et al., 1988 Purton and Catlow, 1990 Bush et al., 1994). 

The van t Hoff equation allows the calculation of an equilibrium constant at a different temperature if the value of the equilibrium constant at another temperature and A/f° for the reaction are known. 

Provided that we know the Arrhenius activation eneii for the reaction, this equation may be used to calculate the rate constant at one temperature, knowing the rate constant at another temperature. For example, if for a particular chemical reaction,... 

The enthalpy of vaporization of a liquid at some temperature T can be calculated from its value at another temperature Tj by means of the Watson equation ... 

A common problem is to calculate the composition of a reacting mixture at equilibrium at a specified temperature. To do this, it is always easier if we start with the stoichiometric table of the reaction. The first step is to express all the concentrations in terms of the extent of reaction, . We then calculate the activity of each species and finally, we equate the product of activities to the equilibrium constant. This produces an equation where the only unknown is Once the extent of reaction is known, all the mole fractions can be computed from the stoichiometric table. If the temperature of the calculation is at 25 C, the equilibrium constant is obtained directly from tabulated values of the standard Gibbs free energy of formation. To calculate the equilibrium constant at another temperature, an additional step is needed to obtain the heat of reaction and the Gibbs energy at the desired temperature. This procedure is demonstrated with examples below. 

The effects of temperature on chemical reactions, including respiratory rate, traditionatly quantified by Qio, which is a coefficient by which it is pessible to calculate how many times increases the rate of a reaction for each increase in temperature of 10 °C. The effect of temperature can also be quantified by the Arrhenius model, where the effect of temperature increase is given by the activation energy (Ea) (Cameron et al., 1995). The temperature quotient is useful because it allows us to calculate the respiration rates at one temperature from a known rate at another temperature. However, the resparation rate does not follow ideal behavior, and the Qio can vary considerably with temperature. At higher temperatures, the Qio is usually smaller than at lower temperatures. 

This means that the standard enthalpy of reaction at another temperature can be calculated when the heat capacities Cp. in the standard state (liquid, solid, and hypothetical ideal gas) of the compounds involved are known. The temperature dependence of the heat capacities can be described, for example, by a polynomial of the following form ... 

Therefore, if the standard emf for a cell at a certain temperature is given, its value at another temperature can be determined using Equation 13.6 if the enthalpy change for the cell reaction is known. On the other hand, if the emf of a cell can be determined at two or more different temperatures. Equation 13.6 can be used to estimate the enthalpy change AH° for a cell reaction. Also, because AG° can be determined from one of the iE° measurements, the entropy change A.S° can be determined using AG° = AH° — TAS° (Equation 8.35). Thus, the standard enthalpy and entropy of a reaction can be determined directly from the emf as a function of temperature, which gives us a noncalorimetric method for the calculation of these important thermodynamic quantities. 

Each of a number of polymerization reactions may be carried out in many different ways. Each reaction carried out in a particular way is accompanied by a particular heat effect. Tabulation of all possible heat effects for all possible polymerization reactions is a herculean task. The heat effects for a desired polymerization reaction can be calcnlated from data for reactions carried out in a standard way. When heat of polymerization is available at one temperature and pressure, it can be calculated for another temperature and pressure by extrapolation. The Clapeyron equation can be derived as follows to aid this process. 

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