Isobaric data

Application of the algorithm for analysis of vapor-liquid equilibrium data can be illustrated with the isobaric data of 0th-mer (1928) for the system acetone(1)-methanol(2). For simplicity, the van Laar equations are used here to express the activity coefficients. 

For the isobaric data the values of the average absolute deviation in temperature is about 2°C except for the potassium acetate system where the value is very high at about 17.5°C (19). For the latter system, use of parameters that were linear functions of temperature to account for the large temperature variations encountered in the three binary and the ternary systems (up to 61 °C), gave... 

The present author (4) has previously preferred to use raw isobaric data coupled with Barker s method (2) and the Wilson equation (5), but interpolation of the calculated discrete vapor composition values requires smoothing of boiling point-liquid composition data at some stage. 

Figure 5 shows the isobar data of CeOx-C>2 system. At one atmosphere with a 150 torr oxygen partial pressure, the cerium oxide almost is dioxide, CeC>2, and there is no phase transition until the temperature increases up 1200 °C. However, when the oxygen partial pressure is about 10-9 torr the oxygen content of the... 

FIGURE 6 Isobaric data of the PrOx-Oj system at 199 torr oxygen partial pressure. 

FIGURE 16 Isobar data of the Ceo.6Tbo.2Zro.2Ox oxide under 11 and 152 torr oxygen partial pressure. 

In 1966 Hyde et al., showed that the tensimetric isobaric data for the praseodymium oxides indicated a stable phase with a composition PrOi.sis at oxygen partial pressures of 30-205 torr in the temperature range 420-450 °C. This composition has not been prepared as a single phase because it easily decomposes to the e-phase, PrOi.778, during heating from the a-phase, PrOi.833, or on cooling down from PrOi.80- The eutectoid reaction of PrOi.sis always occurs and thus this phase co-exists with the other phases (Hyde et al., 1966). 

Isobaric data reduction is complicated by the fact that both composition and temperature vary from point to point, whereas for isothermal data composition is the only significant variable. (The effect of pressure on liquid-phase properties is assumed negligible.) Because the activity coefficients are strong functions of both liquid composition and T, which are correlated, it is quite impossible without additional information to separate the effect of composition from that of T. Moreover, the Pi sat values depend strongly on T, and one must have accurate vapor-pressure data over a temperature range. 

Table I. Calculated Vapor Compositions from Fit of Isobaric Data Ethanol—Water-Saturated Ammonium Chloride (14)...

There are three sources of error in the calculated vapor composition when these are calculated from boiling point data random error in each experimental observation systematic error in one or more of the observations and the model is imperfect (this is particularly true for isobaric data because use is made of the Gibbs-Duhem equation which was derived for constant temperature and pressure). In the present work we shall assume that the only error in the data is caused by randomness. 

To check that the method can be used for isobaric data a set of perfect data are generated and random errors added to x, y, T, and tt in turn and all together to see what effect they have on our standard procedure. For large samples we expect 68% of the sample values to lie within one standard deviation of the perfect value of the selected variable. In the case of small samples, e.g., twelve data, error bounds are calculated using binomial probabilities for each of the above variables so that, with probability of 0.95, we expect 41-95% of the sample observations to lie within one standard deviation of the perfect value of the selected variable (the normal distribution is assumed). Twelve is a common number of data points with salt-saturated solutions and this shows the desirability of taking more experimental observations. 

Abstract The present study demonstrates, by means of broadband dielectric measurements, that the primary a- and the secondary Johari-Goldstein (JG) /3-processes are strongly correlated, in contrast with the widespread opinion of statistical independence of these processes. This occurs for different glassforming systems, over a wide temperature and pressure range. In fact, we found that the ratio of the a- and P- relaxation times is invariant when calculated at different combinations of P and T that maintain either the primary or the JG relaxation times constant. The a-P interdependence is quantitatively confirmed by the clear dynamic scenario of two master curves (one for a-, one for P-relaxation) obtained when different isothermal and isobaric data are plotted together versus the reduced variable Tg(P)/T, where Tg is the glass transition temperature. Additionally, the a-P mutual dependence is confirmed by the overall superposition of spectra measured at different T-P combinations but with an invariant a-relaxation time. 

We see that equation (9.21) is approximately valid up to the isobaric value of the friction coefficient, CjgH = I, where the prediction that (P2/pilc o J = p /pf = 0.5283 is confirmed well by the direct data. But there is a growing divergence for values of significantly greater than I and at the associated low valve pressure ratios. Here it should be noted that all globe valves operate with > 1 (see Figure 9.2). To improve the match, we note first that the isobaric data point, (pr/p, i,ok,) = Pn/P at C/, = I, will be reproduced for any value of q by an equation of... 

Fig. 8.1, Pressure versus volume for water, showing the 700° C and 600° C isotherms. The effects of various types of adiabatic expansion are shown, starting from point A at 700°C, 2000 bars. A-B isentropic A-C isoenergetic (AC/ = 0) A-D isenthalpic A-E isobaric. Data in Table 8.1.

PTxy data contain the maximum of information. Isobaric data are preferable in design because incorporate the temperature effect. Indeed, the activity coefficients are much more dependent on temperature than pressure. Isothermal data are preferred in scientific studies. Vapour composition data are less accurate than liquid composition. 

The van Laar interaction constants and A, are, in theory, only constant for a particular binary pair at a given temperature. In practice, they are frequently computed from isobaric data covering a range of temperature. The van Laar theory expresses the temperature dependence of A,7 to be... 

If, in accordance with equation (72a), log yjyz is plotted against a , the areas above and below the ar-axis must be equal if the isothermal data are thermodynamically reliable. As an example. Fig. 55 shows this relationship for the system chloroform-ethanol at 45 °C. The difference between the areas amounts to only 1.6%. Heriiigton [127] extended this method to ternary systems and isobaric data. 

FIGURE 14.21 Initial thermal expansion coefficient of glassy PS and its CPNC versus MMT content. The value was calculated from Eq. (14.1) using isobaric data at 303 to 323 K. 

Xi empirical fitting parameters to isothermal-isobaric data... 

The main problem with modeling the MMA-H2O system is the absence of reliable phase equilibrium data. The only data available in hterature on the MMA-water system consisted of mutual solubility data of water in MMA at standard pressure and different temperatures [45]. In this study, the interaction parameters are fitted to isobaric data. Because of this restriction, the parameters are fitted to one single set of data points, i.e. the mutual solubility of water and MMA at a certain temperature. This procedure provides interaction parameters that are temperature and pressure dependent. However the effect of pressure on the mixing of MMA has been neglected, as the compressibility of this liquid-liquid system is generally assumed to be negligible. A similar temperature dependency can be observed for the Stryjek-Vera parameters. 

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