Standard-state corrections

To our knowledge, the question of the standard state corrections in DSC experiments has never been addressed. These corrections may in general be negligible, because most studies only involve condensed phases and are performed at pressures not too far from atmospheric. This may not be the case if, for example, a decomposition reaction of a solid compound that generates a gas is studied in a hermetically closed crucible, or high pressures are applied to the sample and reference cells. The strategies for the calculation of standard state corrections in calorimetric experiments have been illustrated in chapter 7 for combustion calorimetry. 

Equation 7.31 was derived from a least squares fit to the data given in W.N. Hubbard, D. W. Scott, G. Waddington. Standard States Corrections for Combustions inaBomb at Constant Volume. In Experimental Thermochemistry, vol. 1 F. D. Rossini, Ed. Interscience New York, 1956 p. 93. 

Table 1 Temperature dependence of effective Henry s Law constants for several species in H2SO4. standard state correction was used to convert AS = -30 cal/mol-K to an intercept. Assuming a standard state of 1 atm, the intercept is equ to (AS/4.58 + log [solvent]), where [solvent] is the molarity of the sulfuric acid. ( )A standard state correction was used to convert AS = -21 cal/mol-K to an intercept. Same as in (a). )A standard state correction was used to convert AS = -27 cal/mol-K to an intercept. Same as in (a).

Enthalpies are referred to the ideal vapor. The enthalpy of the real vapor is found from zero-pressure heat capacities and from the virial equation of state for non-associated species or, for vapors containing highly dimerized vapors (e.g. organic acids), from the chemical theory of vapor imperfections, as discussed in Chapter 3. For pure components, liquid-phase enthalpies (relative to the ideal vapor) are found from differentiation of the zero-pressure standard-state fugacities these, in turn, are determined from vapor-pressure data, from vapor-phase corrections and liquid-phase densities. If good experimental data are used to determine the standard-state fugacity, the derivative gives enthalpies of liquids to nearly the same precision as that obtained with calorimetric data, and provides reliable heats of vaporization. 

Standard-state fugacities at zero pressure are evaluated using the Equation (A-2) for both condensable and noncondensable components. The Rackett Equation (B-2) is evaluated to determine the liquid molar volumes as a function of temperature. Standard-state fugacities at system temperature and pressure are given by the product of the standard-state fugacity at zero pressure and the Poynting correction shown in Equation (4-1). Double precision is advisable. 

Only those components which are gases contribute to powers of RT. More fundamentally, the equiUbrium constant should be defined only after standard states are specified, the factors in the equiUbrium constant should be ratios of concentrations or pressures to those of the standard states, the equiUbrium constant should be dimensionless, and all references to pressures or concentrations should really be references to fugacities or activities. Eor reactions involving moderately concentrated ionic species (>1 mM) or moderately large molecules at high pressures (- 1—10 MPa), the activity and fugacity corrections become important in those instances, kineticists do use the proper relations. In some other situations, eg, reactions on a surface, measures of chemical activity must be introduced. Such cases may often be treated by straightforward modifications of the basic approach covered herein. 

From this equation, the temperature dependence of is known, and vice versa (21). The ideal-gas state at a pressure of 101.3 kPa (1 atm) is often regarded as a standard state, for which the heat capacities are denoted by CP and Real gases rarely depart significantly from ideaHty at near-ambient pressures (3) therefore, and usually represent good estimates of the heat capacities of real gases at low to moderate, eg, up to several hundred kPa, pressures. Otherwise thermodynamic excess functions are used to correct for deviations from ideal behavior when such situations occur (3). 

Fan Performance The performance of a centrifugal fan varies with changes in conditions such as temperature, speed, and density of the gas being handled. It is important to keep this in mind in using the catalog data of various fan manufacturers, since such data are usually based on stated standard conditions. Corrections must be made for variations from these standards. The usual variations are as follows ... 

The ratio f/f° is called activity, a. Note This is not the activity coefficient. The activity is an indication of how active a substance is relative to its standard state (not necessarily zero pressure), f°. The standard state is the reference condition, which may be anything however, most references are to constant temperature, with composition and pressure varying as required. Fugacity becomes a corrected pressure, representing a specific component s deviation from ideal. The fugacity coefficient is ... 

The standard rates the offending noise according to its nature 5dB(A) is added where the noise has a definite continuous note and a further 5 dB(A) added for noise of an intermittent nature. The number of occasions that happen in an 8-hour period is then plotted on a graph and the correction for intermittency is derived. When these calculations have been performed, the noise level is compared to the background level. The standard states that where the noise exceeds the background by 5 dB or more, the nuisance is to be classed as marginal, and where the background is exceeded by lOdB(A) or more, complaints are to be expected. 

Activity can be thought of as the quantity that corrects the chemical potential at some pressure and/or composition condition" to a standard or reference state. The concept of a standard state is an important one in thermodynamics. The choice of the pressure and composition conditions for the standard state are completely arbitrary, and unusual choices are sometimes made. The common choices are those of convenience. In the next section, we will describe and summarize the usual choices of standard states. But, first, we want to describe the effect of pressure and temperature on a,. 

Later, we will make equilibrium calculations that involve activities, and we will see why it is convenient to choose the ideal gas as a part of the standard state condition, even though it is a hypothetical state/ With this choice of standard state, equations (6.94) and (6.95) allow us to use pressures, corrected for non-ideality, for activities as we make equilibrium calculations for real gases.s... 

The first ACH° is AfH for C02 at 298.15 K, since elements in their naturally occurring state are combining to give C02(g). This combustion reaction is the standard state enthalpy of formation if we carry it out at p = 1 bar and make small corrections to change the C02(g) to the ideal gas condition. 

These four equations are perfectly adequate for equilibrium calculations although they are nonsense with respect to mechanism. Table 7.2 has the data needed to calculate the four equilibrium constants at the standard state of 298.15 K and 1 bar. Table 7.1 has the necessary data to correct for temperature. The composition at equilibrium can be found using the reaction coordinate method or the method of false transients. The four chemical equations are not unique since various members of the set can be combined algebraically without reducing the dimensionality, M=4. Various equivalent sets can be derived, but none can even approximate a plausible mechanism since one of the starting materials, oxygen, has been assumed to be absent at equilibrium. Thermodynamics provides the destination but not the route. 

The fundamental fact on which the analysis of heterogeneous reactions is based is that when a component is present as a pure liquid or as a pure solid, its activity may be taken as unity, provided the pressure on the system does not differ very much from the chosen standard state pressure. At very high pressures, the effect of pressure on solid or liquid activity may be determined using the Poynting correction factor. 

Thermochemical data from the compilation of Stull et at., 1969. Entropy values are based on a 1 M standard state. The asterisk denotes symmetry-corrected quantities. Symmetry numbers were chosen as follows 18 for the n-alkanes, cis-3-hexene, dibuthyl sulphide, diethyl ether, and diethyl amine 2n for the cycloalkanes and 2 for all of the remaining ring compounds 3 for the alkanols, alkanethiols and alkyl amines 9 for the methyl alkyl sulphides... 

Information on partial molar heat capacities [1,18] is indeed very scarce, hindering the calculation of the temperature correction terms for reactions in solution. In most practical situations, we can only hope that these temperature corrections are similar to those derived for the standard state reactions. Fortunately, due to the upper limits set by the normal boiling temperatures of the solvents, the temperatures of reactions in solution are not substantially different from 298.15 K, so large ArCp(T - 298.15) corrections are uncommon. 

As mentioned, AvapH refers to 326 K and to 7848 Pa, that is, the calculated value is not the standard enthalpy of vaporization. The correction to the standard states (at 326 K) could be estimated with equation 2.16, but there are more... 

In both cases, the variation of the vaporization enthalpy with temperature is implicit. AVap// is now calculated, at any temperature, from equation 2.40, and the correction to the standard states, if required, is made as already described. 

There is an additional advantage in using relative solution phase bond dissociation enthalpies. In most cases, solution phase bond dissociation enthalpies do not refer to standard states (see section 2.3), and the required corrections are hard to predict. When we consider relative bond dissociation enthalpies in a series of similar molecules in solution it is likely that the unknown corrections to standard states are nearly constant. 

The Washburn corrections for the initial state, At/wi (figure 7.5) correspond to the energy changes for bringing the bomb contents from their standard state to the initial bomb conditions. The traces of N2 inevitably present as an impurity in the O2 are ignored in the computation. 

Step 26 completes the Washburn corrections that bring the products from the final bomb conditions to their standard states. For our example,... 

The experimental data and the calculations involved in the determination of a reaction enthalpy by isoperibol flame combustion calorimetry are in many aspects similar to those described for bomb combustion calorimetry (see section 7.1) It is necessary to obtain the adiabatic temperature rise, A Tad, from a temperaturetime curve such as that in figure 7.2, to determine the energy equivalent of the calorimeter in an separate experiment and to compute the enthalpy of the isothermal calorimetric process, AI/icp, by an analogous scheme to that used in the case of equations 7.17-7.19 and A /ibp. The corrections to the standard state are, however, much less important because the pressure inside the burner vessel is very close to 0.1 MPa. 

The obtained A 7 a() value and the energy equivalent of the calorimeter, e, are then used to calculate the energy change associated with the isothermal bomb process, AE/mp. Conversion of AE/ibp to the standard state, and subtraction from A f/jgp of the thermal corrections due to secondary reactions, finally yield Ac f/°(298.15 K). The energy equivalent of the calorimeter, e, is obtained by electrical calibration or, most commonly, by combustion of benzoic acid in oxygen [110,111,113]. The reduction of fluorine bomb calorimetric data to the standard state was discussed by Hubbard and co-workers [110,111]. 

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