Measurement of time, isothermal

The yield of a gas-solid heterogeneous reaction depends not on the total time that molecules spend in the reactor but on the time that they spend on the catalyst surface. The contact time distribution provides a standardized measure of times spent in the absorbed state. A functional definition is provided by the following equation applicable to a first-order, heterogeneous reaction in an isothermal reactor ... 

The decompositions of compounds containing the ions [Cr(NHj)5(H20)] or [Cr(NH3)4(H20)2] were studied by Wendlandt et al. [30,31]. In subsequent investigations of the thermal deaquation and anation reactions of [Cr(NH3)5(H20)]X3 (X = Cl", Br, T and NO3 ), Nagase and Yokobayashi [32] supplemented TG and thermochemical studies with isothermal measurements of evolved gas pressure, a-time curves were sigmoid and values of EJkJ mol, based on first-order rate constants, increased slightly in the sequence NOj" (102) < Cl" (110) < Br (124) < r (128). These values differ fi om earher values estimated from non-isothermal data. 

A quantitative, isothermal measurement of the crystallization kinetics, usable for analysis by the Avrami method, is illustrated in Fig. 3.98 by the upper left curve. Similar curves can be generated by dilatometry or adiabatic calorimetry as described in Sects. 4.1 and 4.2. At time zero, one assumes that the isothermal condition has been reached. The dotted segments of the heat-flow response are then proportional to the heat of crystallization evolved in the given time intervals and can be converted directly into the changes of the mass fraction of crystallinity after calibration or normalization to the total heat evolved. An independent crystallinity determination... 

In isothermal measurements, the time required to establish equilibrium conditions (teq) can be determined in a simple experiment illustrated in Figure 3.24. The solvent is thermostated at the desired temperature and stirred at moderate rate. After introducing an excess amount of solid into the solvent, samples of clear solution are removed at different times and analyzed for concentration subsequently. When the solution concentration reaches a constant value, saturation is achieved and the minimal equilibration time (for that temperature) can be derived from the concentration-time plot, as given in Figure 3.24. 

Figure 1. Isothermal measurement of plant metabolic heat rates. Plant tissue samples are placed in the calorimeter ampules, sealed, and heat rates are followed over time. The initial rapid changes in heat rates observed in this figure illustrate thermal equilibration of ampules and samples. This is followed by heat rate values that, when adjusted for any baseline corrections, represent the metabolic heat rate of the tissue preparation. When tissue metabolic rates are stable over the time course of the experiment, constant heat rates are obtained (a). When tissues are unstable or are stressed so that metabolic activity decreases during the calorimetric measurement, a decreasing heat rate is ob.served (b).

It would be difficult to over-estimate the extent to which the BET method has contributed to the development of those branches of physical chemistry such as heterogeneous catalysis, adsorption or particle size estimation, which involve finely divided or porous solids in all of these fields the BET surface area is a household phrase. But it is perhaps the very breadth of its scope which has led to a somewhat uncritical application of the method as a kind of infallible yardstick, and to a lack of appreciation of the nature of its basic assumptions or of the circumstances under which it may, or may not, be expected to yield a reliable result. This is particularly true of those solids which contain very fine pores and give rise to Langmuir-type isotherms, for the BET procedure may then give quite erroneous values for the surface area. If the pores are rather larger—tens to hundreds of Angstroms in width—the pore size distribution may be calculated from the adsorption isotherm of a vapour with the aid of the Kelvin equation, and within recent years a number of detailed procedures for carrying out the calculation have been put forward but all too often the limitations on the validity of the results, and the difficulty of interpretation in terms of the actual solid, tend to be insufficiently stressed or even entirely overlooked. And in the time-honoured method for the estimation of surface area from measurements of adsorption from solution, the complications introduced by... 

The isothermal curves of mechanical properties in Chap. 3 are actually master curves constructed on the basis of the principles described here. Note that the manipulations are formally similar to the superpositioning of isotherms for crystallization in Fig. 4.8b, except that the objective here is to connect rather than superimpose the segments. Figure 4.17 shows a set of stress relaxation moduli measured on polystyrene of molecular weight 1.83 X 10 . These moduli were measured over a relatively narrow range of readily accessible times and over the range of temperatures shown in Fig. 4.17. We shall leave as an assignment the construction of a master curve from these data (Problem 10). 

Softening and cure is examined with the help of a torsional pendulum modified with a braid (65), which supports thermosets such as phenoHcs and epoxies that change from a Hquid to a soHd on curing. Another method uses vibrating arms coupled to a scrim-supported sample to measure storage and loss moduH as a function of time and temperature. An isothermal analytical method for phenoHc resins provides data regarding rate constants and activation energies and allows prediction of cure characteristics under conditions of commercial use (47). 

Both kinetic and equilibrium experimental methods are used to characterize and compare adsorption of aqueous pollutants in active carbons. In the simplest kinetic method, the uptake of a pollutant from a static, isothermal solution is measured as a function of time. This approach may also yield equilibrium adsorption data, i.e., amounts adsorbed for different solution concentrations in the limit t —> qo. A more practical kinetic method is a continuous flow reactor, as illustrated in Fig. 5. 

The reactor in Fig. 5 operates as follows. A feed solution containing a given concentration of pollutant is pumped to the adsorbent module at a fixed volumetric flow rate. The module is kept isothermal by a temperature control unit, such as a surrounding water bath. Finally, the concentration of the outlet solution is measured as a function of time from when the feed was introduced to the adsorbent module. These measurements are often plotted as breakthrough curves. Example breakthrough curves for an aqueous acetone solution flowing... 

Figure 6.11 shows a famous example of the application of isothermal calorimetry. Gordon (1955) deformed high-purity copper and annealed samples in his precision calorimeter and measured heat output as a function of time. In this metal, the heat output is strictly proportional to the fraction of metal recrystallised. 

Fig. 4. Reduced time plots, fr = (f/fo.s), for the isothermal decomposition of ammonium vanadyl oxalate using master data for the Avrami—Erofe ev equation [eqn. (6), n = 2], by the application of a method of analysis [73] described in the text. The circles are experimental measurements and the lines correspond to exact agreement with the equation. (Reproduced, with permission, from Thermochimica Acta.)...

From microscopic measurements of the rates of nucleation and of growth of particles of barium metal product, Wischin [201] observed that the number of nuclei present increased as the third power (—2.5—3.5) of time and that the isothermal rate of radial growth of visible nuclei was constant. During the early stages of reaction, the acceleratory region of the a—time plot obeyed the power law [eqn. (2)] with 6 temperature coefficients of these processes were used by Wischin [201]... 

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