Isobaric condition

These are equivalent to the dusty gas model equations, but are valid only for isobaric conditions, and this fact severely limits the capability of the model to represent Che behavior of systems with chemical reaction. To see this we need only remark that (8,7) and (3.8) together imply that ... 

They then compared measured and predicted fluxes for diffusion experiments in the mixture He-N. The tests covered a range of pressures and a variety of compositions at the pellet faces but, like the model itself, they were confined to binary mixtures and isobaric conditions. Feng and Stewart [49] compared their models with isobaric flux measurements in binary mixtures and with some non-isobaric measurements in mixtures of helium and nitrogen, using data from a variety of sources. Unfortunately the information on experimental conditions provided in their paper is very sparse, so it is difficult to assess how broadly based are the conclusions they reached about the relative merits oi their different models. 

For a process carried out under isothermal and isobaric conditions... 

Figure 6.8 Separation of Triton X-114 by SFC using prograMmed elution on a 10 cm x 2 mm I.D. Nucleosil column, 3 micrometer packing, at 170 C with UV detection at 278 nm. The separation on the left was performed under isobaric conditions at 210 bar with a mobile phase of carbon dioxide -t- methanol (2 + 0. 5) ml/min. The separation in the center was obtained using a ccmt. sition gradient from 0.025 to 0.4 ml/mln over 8 min with other conditions as above. The separation on the right was obtained using a pressure program from 130 to 375 bar over 8 min with the same mobile phase used for the isobaric sepeuration. (Reproduced with permission from ref. 57. Copyright Preston Publications, Inc.)...

In the kinetic study, data were obtained from the amount of hydrogen consumed by the reaction over time using an automated gas uptake control system. The apparatus is designed to maintain isothermal and isobaric conditions while monitoring H2 consumption. Procedures used were as described previously.3"5... 

This additive relationship may be applied to any other property of the system (i.e., viscosity, surface potential, etc.) under isothermal and isobaric conditions with the same result (Gaines, 1966). 

A gas-phase hydrogenation reactioa C2H (A) + H2 - C2H6, is conducted under isothermal and isobaric conditions in a CSTR The feed, consisting of equimolar amounts of each reac-... 

The dehydrogenation of ethane (A) to ethene (B) is conducted in a 0.5-m3 PFR. The reaction is first-order with respect to A, with a rate constant of 15.2 min-1 at 725°C. The feed contains pure ethane at 725°C, 400 kPa, and a flow rate of 1. 0 kmol min-1. Compare the conversion predicted if isothermal, isobaric conditions are assumed with that if the pressure drop is accounted for with isothermal flow. The diameter of the reactor tube is 0.076 m, and the viscosity of the gas is 2.5 X 10-5 Pa s. 

It should be noted that we have here considered the system at constant pressure. If we are not considering the system at isobaric conditions, the invariant equilibrium becomes univariant, and a univariant equilibrium becomes divariant, etc. A... 

At a given temperature and pressure eqs. (4.7) and (4.8) must be solved simultaneously to determine the compositions of the two phases a and P that correspond to coexistence. At isobaric conditions, a plot of the composition of the two phases in equilibrium versus temperature yields a part of the equilibrium T, x-phase diagram. 

It is sometimes convenient to fix the pressure and decrease the degrees of freedom by one in dealing with condensed phases such as substances with low vapour pressure. The Gibbs phase rule for a ternary system at isobaric conditions is Ph + F = C + 1=4, and there are four phases present in an invariant equilibrium, three in univariant equilibria and two in divariant phase fields. Finally, three dimensions are needed to describe the stability field for the single phases e.g. temperature and two compositional terms. It is most convenient to measure composition in terms of mole fractions also for ternary systems. The sum of the mole fractions is unity thus, in a ternary system A-B-C ... 

The Gibbs energy, G, is often a more appropriate variable at isobaric conditions. For condensed systems, G can be assumed to be equivalent to A because their difference, the termpV, is usually negligible. The Gibbs energy of condensed phases can therefore in most cases be approximated as... 

Under isobaric conditions (k2 = k2 H2 J. many hydrogenations exactly follow this model. The classical example is the asymmetric hydrogenation of prochiral dehydroamino acid derivatives with Rh or Ru catalysts [21]. 

By contrast, the measurement of the hydrogen consumption under normal pressure is relatively simple. The elementary structure of many such measuring devices is similar, and is based principally on the fact that the pressure drop is balanced by reduction in the reaction volume or by supply of the consumed gas, thus ensuring isobaric conditions. An appropriate device for monitoring major gas consumptions is described in [38]. 

For hydrogenations under normal pressure and isobaric conditions, we use a device which registers gas consumption automatically (Fig. 10.3). Possible error sources resulting from such gas consumption measurements and possibilities of their minimization will be discussed. 

The result of the described methodical solution to monitor gas-consuming reactions at reduced partial pressure under isobaric conditions is shown in Figure 10.8 for the catalytic hydrogenation of COD with a cationic Rh-complex. The slope of the measured straight lines corresponds to the maximally obtainable rate (Vsat = k2 [E]0 = k 2 [H2] [E]0) [42 b], which is directly proportional to the hydrogen concentration in solution and at validity of Henry s law to the hydrogen partial pressure above the reaction solution. The experiments prove that the dilution factor of the gas phase can adequately be found in the rate constant (Further examples can be found in [47].)... 

The common further treatment of the approach - assumption of steady-state conditions for the intermediate substrate complexes, consideration of the catalyst balance ([catalyst]0=[solvent complex] + [IRe] + [Isi] + [IIRe] + [Hsi]) and of the stoichiometry of the hydrogenation - provides the rate of hydrogen consumption under isobaric conditions (Eq. (13)) [57f]. A more general derivation can be found in [59]. 

At constant temperature and pressure, AUT is the observed energy dissipated or absorbed by the reaction. A negative value of AUr means that energy is dissipated by the reaction. If the reaction occurs under isobaric conditions... 

Under isobaric conditions, a small amount of energy is consumed or released by expansion or contraction of the system, and AHr is the observed (i.e., net) heat dissipated or absorbed. 

If the reaction occurs under isobaric conditions (constant pressure), energy can be replaced by enthalpy (AHr) and the former equation can be described for constant volume conditions as follows ... 

Differential scanning calorimetry has been used to measure89,90 the enthalpy of reaction for the displacement of a nitrogen donor ligand, L, from tungsten complexes [W(CO)6 L ] by carbon monoxide under isobaric conditions. The reaction is described by the equation... 

In the specific case of binary (C = 2) systems, V = 4 — P. If, however, we consider systems at constant pressure (isobaric conditions), as is usual when studying solidification, melting processes, the number of variables decreases by one and the variance decreases to V = C — P + 1, that is, for a two-component system, V = 3 — P. 

Swelling experiments are usually conducted in a solution. In this case, the isobar condition is automatically attained, i.e. II = 0. The phase diagram is then calculated from Eq. (2.12) by imposing the condition, IT = 0... 

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