Isothermal determination, discontinuous

Different models determine A in different ways. Nation exhibits a water-uptake isotherm as shown in Figure 7. The dashed line in the figure shows the effects of Schroeder s paradox, where there is a discontinuous jump in the value of A. Furthermore, the transport properties have different values and functional forms at that point. Most models used correlate A with the water-vapor activity, since it is an easily calculated quantity. An exception to this is the model of Siegel et al., ° which assumes a simple mass-transfer relationship. There are also models that model the isotherm either by Flory—Huggins theory" or equilibrium between water and hydrated protons in the membrane and water vapor... 

The common method of determining the gross calorific value of coal is with either an adiabatic calorimeter (ASTM D-2015 this test method was discontinued without replacement in 2000 but is still in use in many laboratories) or an isothermal bomb calorimeter (ASTM D-3286). In these methods, a weighed sample is burned in an oxygen bomb covered with water in a container surrounded by a jacket. 

Therefore>one end of the path of the boundary discontinuity must lie on the isotherm. The outlet concentrations ut and v can be determined by combining Equations 16-19. 

The discontinuous procedure is the conventional technique. It is also referred to as the point-by-point procedure. The adsorptive is introduced in successive amounts and at each stage die measurement is made only when adsorption equilibrium is attained. In this way each point on the adsorption isotherm is determined. The original BET technique (Figure 3.1) and all conventional manometric measurements (Figure 3.2) are based on the discontinuous procedure. This is still the procedure most often used with adsorption balances (Figures 3.10-3.12). In principle, it takes account of the requirement that each experimental point of the adsorption isotherm should correspond to thermodynamic equilibrium. 

The results can be presented in the form of a continuous curve of differential enthalpies of adsorption A versus na, as shown in Figure 3.16b, with a resolution which is much higher than that obtained by the discontinuous procedure (Figure 3.16a). If the adsorption calorimeter cannot be easily connected to a well-calibrated and well-temperature-controlled adsorption sonic nozzle set-up, or when the adsorption isotherm is difficult to determine (e.g. if very small amounts are adsorbed), there remains the possibility of determining, separately, the adsorption isotherm by any of the discontinuous or continuous procedures described in Sections 3.3.1 or 3.3.2. A simple procedure can be applied which does not require the gas flow rate calibration ... 

Discontinuous procedures for determining the adsorption isotherms 67 The discontinuous manometric procedure 67... 

Rouquerol and co-workers have recently described the experimental determination of entropies of adsorption using isothermal adsorption microcalorimetry by a slow and constant introduction of adsorbate under quasiequilibrium conditions (77) or by discontinuous introduction of the adsorbate in an open system (72). 

Rouquerol et al. (11, 12) have recently described the experimental determination of entropies of adsorption by applying thermodynamic principles to reversible gas-solid interactions. Theoretically, the entropy change associated with the adsorption process can only be measured in the case of reversible heat exchange. The authors showed how isothermal adsorption microcalorimetry can be used to obtain directly and continuously the integral entropy of adsorption by a slow and constant introduction of adsorbate under quasi-equilibrium conditions (11) or by discontinuous introduction of the adsorbate in an open system (12). 

In the following section (Section 7.3, we will analyze the trajectory of a stable concentration discontinuity injected into the column, i.e., as one of the boxmdaries of a rectangular injection profile. This derivation could be done directly in the case of a Langmuir isotherm [1,2,31] (see Section 7.3.3). It can also be derived as a particular case of a more general solution. First, however, in a last subsection, we determine the time it takes for a concentration shock to form on a simple continuous profile. 

A number of distinct regions are immediately apparent on examining the isotherm. When the monolayer is compressed, it can pass through several different phases, which are identified as discontinuities in the isotherm, as shown in Figure 5.6. The phase behaviour of the monolayer is mainly determined by the physical and chemical properties of the amphiphile, the subphase temperature and the subphase composition. Various monolayer states exist depending on the length of the hydrocarbon chain and the magnitude of cohe-... 

An alternative is the isothermal dilution calorimeter. For an endothermic system, one component is slowly injected into the second component with the simultaneous addition of electrical energy sufficient to maintain the calorimeter isothermal. The addition is discontinued at any desired composition and the excess enthalpy determined from the initial amount of substance of component B in the vessel, the amount of substance of component A injected, and the electrical energy added to maintain isothermal conditions. The apparatus is normally designed so that the entire composition range can be covered in two experimental runs. For exothermic systems it is possible to add electrical energy... 

The bulk phase diagrams of pure hydrocarbons and mixtures are well known from the experiments. In the work by Sage et al. [3], the bubble point pressures of methane + n-butane mixtures are determined experimentally from the discontinuity of isothermal compressibility of constant-composition mixture at the point of phase transition. The composition of vapor phase is determined in that work from the residual specific volume of gas. Later experiments employ phase recirculation techniques [4] to achieve vapor-Uquid equilibrium [5, 6], and the phase compositions are analyzed by more advanced methods such as gas chromatography. 

The discontinuity in the local isotherm is nicely utilized to determine the solution for the distribution function F(E). This is done as follows. Substituting the local isotherm (eq. 6.6-5) into the observed fractional loading equation (6.6-4), we... 

The kinetics of melt-crystallization of PLA has been analyzed by a number of research groups [14, 35, 39, 71-75]. Isothermal bulk crystallization rates were determined in a wide temperature range from 70 to 165 C [71,72]. The maximum crystaUization rate is observed around 100 C, and the most peculiar behavior is a discontinuity in the phase change kinetics around 110-120°C, an example of which is shown in Figure 5.10. Figure 5.10a reports the half-time of crystallization of PLA as function of the isothermal crystallization temperature. The data set shows a broad minimum around 108 C and a step/discontinuity around 116-118°C, indicated by the arrow. The sudden variation in crystallization rate... 

In some instances the enthalpy curves (e.g. for toluene + hexane mixtures) were linear indicating a constant value of A hf in most, however, a break was detected and interpreted as a jump in A h at a certain surface coverage. What is not explained is how A 2 can exhibit a break when the analysis is based on the assumption that Aw 2 is directly proportional to x 2. Nor is it clear in the case, for example, of the acetone + nitrobenzene system how a smooth curve of x" against x l can give rise to a sharp break in AG". Attempts to relate various breaks and discontinuities in the thermodynamic curves to the presence of two types of adsorption site cannot have any validity. Without access to the original data it is difficult to assess this work, which contains some potentially useful experimental results. In any case it is important to remember, as stressed by the author that the thermodynamic quantities of adsorption are calculated from individual isotherms, which depend strongly on the method of their determination . 

The melting of a crystalline substance is a first-order transition, accompanied by an isothermal increase in enthalpy, the latent heat of fusion. Due to the impossibility of obtaining a pure crystalline phase in polymers, melting occurs over a range, and the enthalpy-temperature curve is S-shaped rather than discontinuous. To calculate the heat of fusion from calorimetric measurements entails extrapolation of the curve for the crystalline form to the melting point, which reduces the accuracy of the measurement. Since the quantity required is the heat of melting one mole of crystalline units, the calorimetric heat must be corrected for the fraction of amorphous polymer present, and this introduces a further uncertainty in the determination. 

The pressures are determined in two ways. One involves the measuring and plotting of the PV isotherm, the dew point and bubble point being indicated by the discontinuities in the curve at the beginning and the end of condensation. The other employs a glass or quartz equilibrium cell, and the conditions are determined visually. 

Corrin et al. [97] determined the adsorption isotherm of sodium dodecyl sulfate and potassium myristate on ash-free graphite. Calculations based on two extreme assumptions concerning the concentration of solvent in the surface region yielded speciFic surface values which differed by less than the experimental error. Hie isotherm of sodium dodecyl sulfate exhibited a discontinuity at the critical... 

In the reported examples, an isothermal microcalorimeter with a discontinuous (stepwise) introduction of the adsorptive was fmitfully employed. An alternative, suitable way to collect the experimental data required for determining the enthalpy and entropy changes accompanying the adsorption process is to follow the procedure implying the slow-and-constant adsorptive introduction which was extensively described in Ref. [99]. 

Fig. 2.1 Principle of accurate determination of thermal reaction power during an isothermal, discontinuous reaction [based on the same measuring principle of the calorimeter, this system of intermediate thermostat—controlled heater, base thermostat (controlled heat sink)—was replaced recently [54] by a new type of intermediate thermostat metal, bordering controlled Peltier elements, thermostat (controlled heat sink)]...

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