Deviation from

Figures 3 and 4 show fugacity coefficients for two binary systems calculated with Equation (10b). Although the pressure is not large, deviations from ideality and from the Lewis rule are not negligible.

The virial equation is appropriate for describing deviations from ideality in those systems where moderate attractive forces yield fugacity coefficients not far removed from unity. The systems shown in Figures 2, 3, and 4 are of this type. However, in systems containing carboxylic acids, there prevails an entirely different physical situation since two acid molecules tend to form a pair of stable hydrogen bonds, large negative... 

Figure 4-4. Representation of vapor-liquid equilibria for a binary system showing moderate positive deviations from Raoult s law.

Figure 4 shows experimental and predicted phase equilibria for the acetonitrile/benzene system at 45°C. This system exhibits moderate positive deviations from Raoult s law. The high-quality data of Brown and Smith (1955) are very well represented by the UNIQUAC equation. 

Figure 5 shows the isothermal data of Edwards (1962) for n-hexane and nitroethane. This system also exhibits positive deviations from Raoult s law however, these deviations are much larger than those shown in Figure 4. At 45°C the mixture shown in Figure 5 is only 15° above its critical solution temperature. Again, representation with the UNIQUAC equation is excellent.

Figure 7 shows a fit of the UNIQUAC equation to the iso-baric data of Nakanishi et al. (1967) for the methanol-diethyl-amine system this system also exhibits strong negative deviations from Raoult s law. The UNIQUAC equation correctly re-... 

Figure 4-9. Vapor-liquid equilibria for a binary system where one component dimerizes in the vapor phase. Activity coefficients show only small deviations from liquid-phase ideality.

Moderate errors in the total pressure calculations occur for the systems chloroform-ethanol-n-heptane and chloroform-acetone-methanol. Here strong hydrogen bonding between chloroform and alcohol creates unusual deviations from ideality for both alcohol-chloroform systems, the activity coefficients show... 

The results shown in Table 2 indicate that UNIQUAC can be used with confidence for multicomponent vapor-liquid equilibria including those that exhibit large deviations from ideality. 

For a real vapor mixture, there is a deviation from the ideal enthalpy that can be calculated from an equation of state. The enthalpy of the real vapor is given by... 

Figure 3 presents results for acetic acid(1)-water(2) at 1 atm. In this case deviations from ideality are important for the vapor phase as well as the liquid phase. For the vapor phase, calculations are based on the chemical theory of vapor-phase imperfections, as discussed in Chapter 3. Calculated results are in good agreement with similar calculations reported by Lemlich et al. (1957). ... 

An additional advantage derived from plotting the residuals is that it can aid in detecting a bad data point. If one of the points noticeably deviates from the trend line, it is probably due to a mistake in sampling, analysis, or reporting. The best action would be to repeat the measurement. However, this is often impractical. The alternative is to reject the datum if its occurrence is so improbable that it would not reasonably be expected to occur in the given set of experiments. 

We have repeatedly observed that the slowly converging variables in liquid-liquid calculations following the isothermal flash procedure are the mole fractions of the two solvent components in the conjugate liquid phases. In addition, we have found that the mole fractions of these components, as well as those of the other components, follow roughly linear relationships with certain measures of deviation from equilibrium, such as the differences in component activities (or fugacities) in the extract and the raffinate. 

By contrast with ideal models, practical reactors must consider many factors other than variations in temperature, concentration, and residence time. Practical reactors deviate from the three idealized models but can be classified into a number of common types. 

The spectroscopic methods, NMR and mass spectrometry for predicting cetane numbers have been established from correlations of a large number of samples. The NMR of carbon 13 or proton (see Chapter 3) can be employed. In terms of ease of operation, analysis time (15 minutes), accuracy of prediction (1.4 points average deviation from the measured number), it is... 

The above equation is valid at low pressures where the assumptions hold. However, at typical reservoir temperatures and pressures, the assumptions are no longer valid, and the behaviour of hydrocarbon reservoir gases deviate from the ideal gas law. In practice, it is convenient to represent the behaviour of these real gases by introducing a correction factor known as the gas deviation factor, (also called the dimensionless compressibility factor, or z-factor) into the ideal gas law ... 

A more sophisticated and increasingly popular method of on-condItion maintenance is to monitor the performance of equipment on-line. For example, a piece of rotating equipment such as a turbine may be monitored for vibration and mechanical performance (speed, inlet and outlet pressure, throughput). If a base-line performance is established, then deviations from this may indicate that the turbine has a mechanical problem which will reduce its performance or lead to failure. This would be used to alert the operators that some form of repair is required. 

From downhole pressure drawdown and build-up surveys the reservoir permeability, the well productivity index and completion skin can be measured. Any deviation from previous measurements or from the theoretically calculated values should be investigated to determine whether the cause should be treated. 

The P-radiation was shown to cause the positive deviation from the background level (approx. 3 times) before the earthquake. 

Finally, under the heading Specific pressure equipment requirements specific requirements are set out for equipment with a risk of overheating, for piping and, last but not least, specific quantitative requirements which set out a series of safety factors for certain pressure equipment. These latter provisions apply as a general rule which means that a manufacturer or a harmonised standard may deviate from these factors if it can be demonstrated that appropriate measures have been taken to achieve an equivalent level of safety. 

Perhaps the best discussions of the experimental aspects of the capillary rise method are still those given by Richards and Carver [20] and Harkins and Brown [21]. For the most accurate work, it is necessary that the liquid wet the wall of the capillary so that there be no uncertainty as to the contact angle. Because of its transparency and because it is wet by most liquids, a glass capillary is most commonly used. The glass must be very clean, and even so it is wise to use a receding meniscus. The capillary must be accurately vertical, of accurately known and uniform radius, and should not deviate from circularity in cross section by more than a few percent. 

The deviation of Gibbs monolayers from the ideal two-dimensional gas law may be treated by plotting xA// 7 versus x, as shown in Fig. III-15c. Here, for a series of straight-chain alcohols, one finds deviations from ideality increasing with increasing film pressure at low x values, however, the limiting value of unity for irAfRT is approached. 

Marlow and Rowell discuss the deviation from Eq. V-47 when electrostatic and hydrodynamic interactions between the particles must be considered [78]. In a suspension of glass spheres, beyond a volume fraction of 0.018, these interparticle forces cause nonlinearities in Eq. V-47, diminishing the induced potential E. 

We conclude with the matter of adsorbate-adsorbate interactions these give rise to deviations from Henry s law behavior. These may be expressed in the form of a virial equation, much as is done for imperfect gases. Following Steele [8], one can write... 

In general, it seems more reasonable to suppose that in chemisorption specific sites are involved and that therefore definite potential barriers to lateral motion should be present. The adsorption should therefore obey the statistical thermodynamics of a localized state. On the other hand, the kinetics of adsorption and of catalytic processes will depend greatly on the frequency and nature of such surface jumps as do occur. A film can be fairly mobile in this kinetic sense and yet not be expected to show any significant deviation from the configurational entropy of a localized state. 

Much surface work is concerned with the local atomic structure associated with a single domain. Some surfaces are essentially bulk-temiinated, i.e. the atomic positions are basically unchanged from those of the bulk as if the atomic bonds in the crystal were simply cut. More coimnon, however, are deviations from the bulk atomic structure. These structural adjustments can be classified as either relaxations or reconstructions. To illustrate the various classifications of surface structures, figure A1.7.3(a ) shows a side-view of a bulk-temiinated surface, figure A1.7.3(b) shows an oscillatory relaxation and figure A1.7.3(c) shows a reconstructed surface. 

In general, the phonon density of states g(cn), doi is a complicated fimction which can be directly measured from experiments, or can be computed from the results from computer simulations of a crystal. The explicit analytic expression of g(oi) for the Debye model is a consequence of the two assumptions that were made above for the frequency and velocity of the elastic waves. An even simpler assumption about g(oi) leads to the Einstein model, which first showed how quantum effects lead to deviations from the classical equipartition result as seen experimentally. In the Einstein model, one assumes that only one level at frequency oig is appreciably populated by phonons so that g(oi) = 5(oi-cog) and, for each of the Einstein modes. is... 

It is detemrined experimentally an early study was the work of Andrews on carbon dioxide [1], The exact fonn of the equation of state is unknown for most substances except in rather simple cases, e.g. a ID gas of hard rods. However, the ideal gas law P = pkT, where /r is Boltzmaim s constant, is obeyed even by real fluids at high temperature and low densities, and systematic deviations from this are expressed in tenns of the virial series ... 

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