Deviations unity

The standardized height distribution 4> (i) is the height distribution scaled to make its standard deviation unity. All this is conventional statistics. 

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... 

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. 

Figure C 1.5.10. Nonnalized fluorescence intensity correlation function for a single terrylene molecule in p-terjDhenyl at 2 K. The solid line is tire tlieoretical curve. Regions of deviation from tire long-time value of unity due to photon antibunching (the finite lifetime of tire excited singlet state), Rabi oscillations (absorjDtion-stimulated emission cycles driven by tire laser field) and photon bunching (dark periods caused by intersystem crossing to tire triplet state) are indicated. Reproduced witli pennission from Plakhotnik et al [66], adapted from [118].

The more a deviates from unity for a given pair of ions, the easier it will be to separate them. If the selectivity coefficient is unfavorable for the separation of two ions of the same charge, no variation in the concentration of H+ (the eluant) will improve the separation. 

In Fig. 2.4, the location of the point of inflection thus calculated is plotted for different values of c. Clearly, the value of n/n at the point of inflection may deviate considerably from unity. At the one value of c = 9 the value of n/n is actually equal to unity and the point of inflection then coincides with the point corresponding to the monolayer capacity but for values of c... 

When the value of c exceeds unity, the value of n can be derived from the slope and intercept of the BET plot in the usual way but because of deviations at low relative pressures, it is sometimes more convenient to locate the BET monolayer point , the relative pressure (p/p°) at which n/n = 1. First, the value of c is found by matching the experimental isotherm against a set of ideal BET isotherms, calculated by insertion of a succession of values of c (1, 2, 3, etc., including nonintegral values if necessary) into the BET equation in the form ... 

This result shows that the square root of the amount by which the ratio M /M exceeds unity equals the standard deviation of the distribution relative to the number average molecular weight. Thus if a distribution is characterized by M = 10,000 and a = 3000, then M /M = 1.09. Alternatively, if M / n then the standard deviation is 71% of the value of M. This shows that reporting the mean and standard deviation of a distribution or the values of and Mw/Mn gives equivalent information about the distribution. We shall see in a moment that the second alternative is more easily accomplished for samples of polymers. First, however, consider the following example in which we apply some of the equations of this section to some numerical data. 

If the molecular species in the liquid tend to form complexes, the system will have negative deviations and activity coefficients less than unity, eg, the system chloroform—ethyl acetate. In a2eotropic and extractive distillation (see Distillation, azeotropic and extractive) and in Hquid-Hquid extraction, nonideal Hquid behavior is used to enhance component separation (see Extraction, liquid—liquid). An extensive discussion on the selection of nonideal addition agents is available (17). 

The mean value of each of the distributions is obtained from these high, modal, and low values by the use of Eq. (9-101). If the distribution is skewed, the mean and the mode will not coincide. However, the mean values may be summed to give the mean value of the (NPV) as 161,266. The standard deviation of each of the distributions is calculated by the use of Eq. (9-75). The fact that the (NPV) of the mean or the mode is the sum of the individual mean or modal values implies that Eq. (9-81) is appropriate with all the A s equal to unity. Hence, by Eq. (9-81) the standard deviation of the (NPV) is the root mean square of the individual standard deviations. In the present case s° = 166,840 for the (NPV). 

In systems that exhibit ideal liquid-phase behavior, the activity coefficients, Yi, are equal to unity and Eq. (13-124) simplifies to Raoult s law. For nonideal hquid-phase behavior, a system is said to show negative deviations from Raoult s law if Y < 1, and conversely, positive deviations from Raoult s law if Y > 1- In sufficiently nonide systems, the deviations may be so large the temperature-composition phase diagrams exhibit extrema, as own in each of the three parts of Fig. 13-57. At such maxima or minima, the equihbrium vapor and liqmd compositions are identical. Thus,... 

The solvent and the key component that show most similar liquid-phase behavior tend to exhibit little molecular interactions. These components form an ideal or nearly ideal liquid solution. The ac tivity coefficient of this key approaches unity, or may even show negative deviations from Raoult s law if solvating or complexing interactions occur. On the other hand, the dissimilar key and the solvent demonstrate unfavorable molecular interactions, and the activity coefficient of this key increases. The positive deviations from Raoult s law are further enhanced by the diluting effect of the high-solvent concentration, and the value of the activity coefficient of this key may approach the infinite dilution value, often aveiy large number. 

Selectivity. The relative separation, or selectivity, Ot of a solvent is the ratio of two components in the extraction-solvent phase divided by the ratio of the same components in the feed-solvent phase. The separation power of a hquid-liquid system is governed by the deviation of Ot from unity, analogous to relative volatility in distillation. A relative separation Ot of 1.0 gives no separation of the components between the two liquid phases. Dilute solute concentrations generally give the highest relative separation factors. 

Compressibility of Natural Gas All gases deviate from the perfect gas law at some combinations of temperature and pressure, the extent depending on the gas. This behavior is described by a dimensionless compressibility factor Z that corrects the perfect gas law for real-gas behavior, FV = ZRT. Any consistent units may be used. Z is unity for an ideal gas, but for a real gas, Z has values ranging from less than 1 to greater than 1, depending on temperature and pressure. The compressibihty faclor is described further in Secs. 2 and 4 of this handbook. 

Obviously for this method to work the ratio T1IT2 must be appreciably smaller than unity. Provided this condition is met, this method is a simple and reliable way to test for an isokinetic relationship or to detect deviations from such a relationship. Exner shows examples of systems plotted both as log 2 vs. log and as AH vs. A5, demonstrating the inadequacy of the latter plot. Exner has also developed a statistical analysis of the Petersen method this analysis yields p and an uncertainty estimate of p. Exner has applied his statistical methods to 100 reaction series, finding that 78 of them follow approximately valid isokinetic relationships. 

If (A i[X ]/A 2[Y ]) is not much smaller than unity, then as the substitution reaction proceeds, the increase in [X ] will increase the denominator of Eq. (8-65), slowing the reaction and causing deviation from simple first-order kinetics. This mass-law or common-ion effect is characteristic of an S l process, although, as already seen, it is not a necessary condition. The common-ion effect (also called external return) occurs only with the common ion and must be distinguished from a general kinetic salt effect, which will operate with any ion. An example is provided by the hydrolysis of triphenylmethyl chloride (trityl chloride) the addition of 0.01 M NaCl decreased the rate by fourfold. The solvolysis rate of diphenylmethyl chloride in 80% aqueous acetone was decreased by LiCl but increased by LiBr. ° The 5 2 mechanism will also yield first-order kinetics in a solvolysis reaction, but it should not be susceptible to a common-ion rate inhibition. 

Select now a second neutral indicator base C that is weaker than B by roughly an order of magnitude thus, a solvent can be found of such acidity that a significant fraction of both B and C will be protonated, but this will no longer be a dilute aqueous solution, so the individual activity coefficients will in general deviate from unity. For this solution containing low concentrations of both B and C,... 

Equation (4) indicates that the slope n of the log D versus the pH plot corresponds to the number of protons released upon extraction. If the logarithm of the ratio between Fe content in aqueous phase and organic phase is plotted as a function of pH, a linear relation was obtained between pH 3.5-5.4, which deviated from linearity at lower pH values (2.2). The fact that the slope of the curves were very close to unity indicates that only one proton has separated from the ligand (Eq. [5]). 

The heart of the question of non-ideality deals with the determination of the distribution of the respective system components between the liquid and gaseous phases. The concepts of fugacity and activity are fundamental to the interpretation of the non-ideal systems. For a pure ideal gas the fugacity is equal to the pressure, and for a component, i, in a mixture of ideal gases it is equal to its partial pressure yjP, where P is the system pressure. As the system pressure approaches zero, the fugacity approaches ideal. For many systems the deviations from unity are minor at system pressures less than 25 psig. 

Although the Clark plot can be used to visualize the slope relationship between pECS0 and -Log ([B] + 10 pKs), deviation of the slope from unity is better obtained by refitting the data to a power departure version of Equation 6.27 ... 

FIGURE 6.15 Example of application of method of Lew and Angus [10]. (a) Dose-response data, (b) Clark plot according to Equation 6.27 shown, (c) Data refit to power departure version of Equation 6.27 to detect slopes different from unity (Equation 6.28). (d) Data refit to quadratic departure version of Equation 6.27 to detect deviation from linearity (Equation 6.29). 

By measuring the partial molar heat of solution as a function of temperature for infinitely dilute concentrations of Cu, Ag, and Au in liquid tin, Oriani and Murphy51 have determined ACp for the liquid solutes to be 1.0, 0, and 3.0 cal/deg mole respectively. These numbers bear no relationship to the sign of the heat of solution, or to atom-size disparity, but seem to be related to the deviation from unity of the ratio of the masses of the components. 

To compute the variance, we first find the mean concentration for that component over all of the samples. We then subtract this mean value from the concentration value of this component for each sample and square this difference. We then sum all of these squares and divide by the degrees of freedom (number of samples minus 1). The square root of the variance is the standard deviation. We adjust the variance to unity by dividing the concentration value of this component for each sample by the standard deviation. Finally, if we do not wish mean-centered data, we add back the mean concentrations that were initially subtracted. Equations [Cl] and [C2] show this procedure algebraically for component, k, held in a column-wise data matrix. 

The osmotic coefficient is often used as a measure of the activity of the solvent instead of a because a is nearly unity over the concentration range where 7 is changing, and many significant figures are required to show the effect of solute concentration on a. The osmotic coefficient also becomes one at infinite dilution, but deviates more rapidly with concentration of solute than does a. ... 

The extrapolation of the MH values for g = 1 and g = 0.86 g/cm3 yields the limiting values for an ideal polyethylene (PE) crystal (Hc 150-180 MN m-2) and ideally PE amorphous matrix (Ha 1 MN m-2) respectively. It is noteworthy that the extrapolated value obtained for Hc in PE practically coincides with the theoretical value of S0 given in Table I. The experimental MH values given in the literature evidently correspond to materials with g largely deviating from unity. 

At very low pressures, deviations from the ideal gas law are caused mainly by the attractive forces between the molecules and the compressibility factor has a value less than unity. At higher pressures, deviations are caused mainly by the fact that the volume of the molecules themselves, which can be regarded as incompressible, becomes significant compared with the total volume of the gas. 

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