Equilibrium constants single-temperature

The solution then follows along the same lines as for TCR if the temperature and pressure are known then 7, S and the resulting mole fractions can be determined from the equilibrium constants. The temperature change between inlet and outlet is now likely to be higher than in the TCR reactions, so the determination of the A, s as functions of a single mean temperature for the reaction is more difficult. 

The working capacity of a sorbent depends on fluid concentrations and temperatures. Graphical depiction of soration equilibrium for single component adsorption or binary ion exchange (monovariance) is usually in the form of isotherms [n = /i,(cd or at constant T] or isosteres = pi(T) at constant /ij. Representative forms are shown in Fig. I6-I. An important dimensionless group dependent on adsorption equihbrium is the partition ratio (see Eq. 16-125), which is a measure of the relative affinities of the sorbea and fluid phases for solute. 

These data can be used to obtain the value of the equilibrium constant at any temperature and this in turn can be used to calculate the degree of dissociation through the equation for the conceiiuation dependence of the constant on the two species for a single element, die monomer and the dimer, which coexist. Considering one mole of the diatomic species which dissociates to produce 2x moles of the monatomic gas, leaving (1 — jc) moles of the diatomic gas and producing a resultant total number of moles of (1 +jc) at a total pressure of P atmos, the equation for the equilibrium constant in terms of these conceiiU ations is... 

From such crude data as are to be found in the literature we can calculate approximate values of the equilibrium constants, and hence of the free energies of dissociation for the various hexaarylethanes. From our quantum-mechanical treatment, on the other hand, we obtain only the heats of dissociation, for which, except in the single case of hexaphenylethane, we have no experimental data. Thus, in order that we may compare our results with those of experiment, we must make the plausible assumption that the entropies of dissociation vary only slightly from ethane to ethane. Then at a given temperature the heats of dissociation run parallel to the free energies and can be used instead of the latter in predicting the relative degrees of dissociation of the different molecules. 

Besides the second law method, there is another way of extracting reaction enthalpies from gas-phase equilibrium constants. This alternative involves the determination of a single value of an equilibrium constant at a given temperature and the calculation of the reaction entropy at the same temperature. From equations 2.54 and 2.55, we obtain... 

Single-temperature equilibrium constant values may also yield quantitative information about reaction enthalpies, provided that the entropy term can be estimated. Take, for example, reaction 14.32, which involves hydrogen transfer between two substituted phenols (ArOH and Ar OH see examples in figure 14.4). Note that Kc = Km in this case. 

It should be stressed that the condition ArX 0 is not required to derive a series of relative values of reaction enthalpies from single-temperature equilibrium constants. We only need to ensure that Arremains constant. Consider, for example, the case of reaction 14.33 (see figure 14.5) ... 

A final word of caution regarding the use of single-temperature equilibrium constants. Although this is a rather expeditious method to derive reaction enthalpies, the obtained values may be quite inaccurate. For instance, a small 10 J K I mol-1 error in the estimated Ar,Vy. yields a 3 kJ mol-1 error in Ar at 298.15 K. 

From the value of ATSi at a single temperature, it is possible to calculate the equilibrium constant Kp at that temperature. It is also desirable to be able to calculate Kp as a function of the temperature, so that it is not necessary to have values of AFJ at frequent temperature intervals. All that is required is to differentiate the relationship between AT and In Kp [Equation (10.22)] and to use Equation (7.57) for the derivative of ATA. Then... 

If the three gases in the reaction were at equilibrium and you then increased the carbon monoxide concentration, some Bt2 would combine with added CO to produce COBr2 and thereby minimize the increase in CO. Alternatively, if you decrease the CO concentration, some COBtj would decompose to produce CO and Br2 and thereby minimize any decrease in CO. Notice how the concentrations of all constituents shift to counteract the imposed change in a single substance. Of course, this shift does not affect the value of the equilibrium constant. Only a change in temperature can do that. 

SC (simultaneous correction) method. The MESH equations are reduced to a set of N(2C +1) nonlinear equations in the mass flow rates of liquid components ltJ and vapor components and the temperatures 2J. The enthalpies and equilibrium constants Kg are determined by the primary variables lijt vtj, and Tf. The nonlinear equations are solved by the Newton-Raphson method. A convergence criterion is made up of deviations from material, equilibrium, and enthalpy balances simultaneously, and corrections for the next iterations are made automatically. The method is applicable to distillation, absorption and stripping in single and multiple columns. The calculation flowsketch is in Figure 13.19. A brief description of the method also will be given. The availability of computer programs in the open literature was cited earlier in this section. 

The SOFC consists of cathode, electrolyte and anode collectively referred to as the PEN - positive electrode, electrolyte, negative electrode. A single cell operated with hydrogen and oxygen provides at equilibrium a theoretical reversible (Nernst) or open circuit voltage (OCV) of 1.229 V at standard conditions (STP, T = 273.15 K. i> = 1 atm). With the standard electrode potential E°, universal gas constant R. temperature T. Faraday s constant F, molar concentration x and pressure p, the OCV is given by... 

The values of AtH ° calculated using this table can be used to calculate apparent equilibrium constants at other temperatures not too far from 298.15 K. Note that standard transformed enthalpies of reactants that consist of a single species are not a function of pH (see equation 4.4-12). The standard transformed enthalpies of reactants are functions of pH when there are more than two species because rt depends on pH. As indicated by the pH dependencies in Table 4.3, these differences are often small. 

The apparent rate constant in (2.10), which is obtained by multiplying a true rate constant kc and the square root of an equilibrium constant, Keq, can show a law of dependence on temperature different from the simple Arrhenius law. In some cases, even a negative temperature dependence can be observed. Moreover, if both mechanisms (2.6) and (2.7)-(2.8) are active in parallel, the observed reaction rate is the sum of the single rates, and an effective reaction order variable from 1 /2 to 1 can be observed with respect to reactant A. Variable and fractionary reaction orders can be also encountered in heterogeneous catalytic reactions as a consequence of the adsorption on a solid surface [6],... 

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