Positive and Negative Deviations

Let us now briefly describe some broader phenomenological aspects of the P-x diagrams for binary solutions, ranging from the ideal solution limit to extreme nonideal deviations of either positive or negative sign. 

The limiting linear behavior of the ideal P-xeq liquid diagram was previously described in Section 7.3.1. The corresponding P-xeap vapor curve can be added to give the full P-xB diagram for an ideal solution as follows  

Ideal solution behavior is observed only when the solute and solvent molecules have similar sizes and intermolecular interactions, as in benzene/toluene or hexane/octane solutions. 

As shown in (7.53), the P-xsap boundary always curves below the linear P-x boundary, in such a manner that the vapor phase is always enriched in the more volatile component, as physical intuition would suggest  

Sidebar 7.10 describes the mathematical relationship between. Vb1 and a bup for an ideal 

Sidebar 7.10 describes the mathematical relationship between xBq and Xbp for an ideal solution, showing how (7.54a, b) are achieved in this simple case. However, the physically reasonable relationships (7.54a, b) between coexisting liquid and vapor compositions are also satisfied in more general nonideal solutions described below. 

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Calibration curves showing positive and negative deviations from Beer s law. 

These equations are ealled the moment equations, beeause we are effeetively taking moments of the data about a point to measure the dispersion over the whole set of data. Note that in the varianee, the positive and negative deviates when squared do not eaneel eaeh other out but provide a powerful measure of dispersion whieh... 

We are now in a position to examine the relative accuracies of a variety of different model chemistries by considering their performance on the G2 molecule set. The following table lists the mean absolute deviation from experiment, the standard deviation and the largest positive and negative deviations from experiment for each model chemistry. The table is divided into two parts the first section lists results for single model chemistries, and the remaining sections present results derived from... 

In our discussion of (vapor + liquid) phase equilibria to date, we have limited our description to near-ideal mixtures. As we saw in Chapter 6, positive and negative deviations from ideal solution behavior are common. Extreme deviations result in azeotropy, and sometimes to (liquid -I- liquid) phase equilibrium. A variety of critical loci can occur involving a combination of (vapor + liquid) and (liquid -I- liquid) phase equilibria, but we will limit further discussion in this chapter to an introduction to (liquid + liquid) phase equilibria and reserve more detailed discussion of what we designate as (fluid + fluid) equilibria to advanced texts. 

This weighted sum of absolute values in e(x) was also discussed in Section 8.4 as a way of measuring constraint violations in an exact penalty function. We proceed as we did in that section, eliminating the nonsmooth absolute value function by introducing positive and negative deviation variables dpt and dnt and converting this nonsmooth unconstrained problem into an equivalent smooth constrained problem, which is... 

These results shown in Figures 1 and 2 demonstrate the similarity of the effects of short-range forces on the properties of nonelectrolytes and concentrated electrolytes. One finds both positive and negative deviations from ideality and these effects may be ascribed to the difference between the intermolecular potential energy of attraction of unlike species to the mean of the corresponding potentials for pairs of like molecules. Previous discussion of these systems has focused on the hydration of the positive ion as the dominant effect, but we see in Figure 1 that... 

Deviations that arise probabilistically and have two characteristics (a) the magnitude of these errors is more typically small, and (b) positive and negative deviations of the same magnitude tend to occur with the same frequency. Random error is normally distributed, and the bell-shaped curve for frequency of occurrence versus magnitude of error is centered at the true value of the measured parameter. See Statistics (A Primer)... 

Figure 9.38, for example, shows the application of the chemical mass balance approach to the fine particle fraction of particles collected at a location in Philadelphia (Dzubay et at., 1988 Olmez et al., 1988 Gordon, 1988). If the set of equations (II) fitted the data perfectly, the sum of the contributions of the various sources would be 100% for each element. Clearly, from the top frame, this is not the case for a number of elements, and both positive and negative deviations from 100% can be seen. However, the contributions of several sources are clear Si and Fe from soil, Ni, V, and Ca from oil-fired power plants, Ti from a paint pigment plant, La, Ce, and Sm from a catalytic cracker, K, Zn, and Sn from an incinerator, Sb from an antimony roaster, and Pb and Br from motor vehicles. 

The absorption curves given by coal macerals approached the horizontal (magnetic field strength) axis more slowly than a Gaussian distribution curve. Shape analysis (16) showed that over much of the curve, the form closely approximated a Lorentzian distribution curve, but both positive and negative deviations were found in the wings of the curves (that is, in various examples, the curves approached the axis either somewhat more or somewhat less rapidly... 

By signal averaging in repetitive experiments. This technique is useful when the noise is truly random so that positive and negative deviations from the true signal are equally likely. The S/N ratio increases then with the square root of the number of experiments... 

HF/6-31G /MP2(fc)/6-31G, respectively, the mean of nine errors for each method is 3.0/2.3/1.9. Errors are given in the Errors column as HF/3-21G< )/HF/6-31G /MP2/6—31G A minus sign means that the calculated value is less than the experimental. The numbers of positive and negative deviations from experiment and the average errors (arithmetic means of the absolute values of the errors) are summarized at the bottom of the Errors column. Calculations are by the author, references to experimental measurements are given for each measurement. Some molecules have calculated minima at other dihedrals in addition to those given here, e.g. FCH2CH2F at 180°. Errors are presented HF/3-21G( VHF/6-31G /MP2/6-31G. a[lg], pp.151, 152... 

Dipole moments are in Debyes the computational levels are arranged, from left to right, in what is conventionally considered lowest to highest. Calculations are by the author experimental values are taken from reference [lg], pp 326, 329, 332,335. For each level is given the number of positive and negative deviations and the arithmetic mean of the absolute values of the deviations. 

Table 7.9 UV spectra (as transition energies in eV) of acetone, acetaldehyde, and formaldehyde, calculated by time-dependent DFT, using Gaussian 98 [78]. The results of using MP2/6-311+G [110] and (calculations by the author) AMI geometries are compared both sets of calculations are single-point B3P86/6-311++G. For each molecule only 6 transitions, all singlets, are shown. The number of positive and negative deviations from experiment and the mean absolute errors are given...

Figure 1 Positive and negative deviations for ideal behavior.

Figure 39.4 shows that if = 1 then a = X (i.e. the effective mole fraction corresponds to the actual mole fraction as made up) then we have ideal behaviour. As discussed before (Frame 33, Figure 33.2) both positive and negative deviations from ideality can occur in real liquid mixtures. 

Summing over positive and negative deviations separately, we obtain... 

The probability that a measurement is more than 1 standard deviation from the mean will be twice this, because both positive and negative deviations are possible and the curve is symmetrical, and is equal to 0.317 32. Put another way, around a third of all measurements will fall outside 1 standard deviation from the mean. 

PAL has been used to study both miscible and immiscible polymer blends [41, 61, 67-70], PAL results have shown both positive and negative deviations from additivity of free volume with blend composition. In the case of multi phase systems, PAL data analysis is complicated by the fact that Ps may diffuse between the different blend phases. 

Bell has examined some of the reasons for expecting gross deviations from the Bronsted relation and has suggested that both positive and negative deviations may occur if the transition state represents a structure considerably different in its charge distribution than the base of the conjugate acid (or vice versa). 

In terms of intermolecular forces, what gives rise to positive and negative deviations from Raoult s law ... 

If Pi exceeds the value given by equation (52a), then the p -X vapor-pressure curve is said to exhibit a positive deviation on the other hand, cases in which Pi < Xy are described as negative deviations. Molecular interpretations for the causes of positive and negative deviations are discussed in [18]. Typical vapor-pressure curves exhibiting positive deviations are shown in Figure A.l. 

Positive and negative deviations from the line would cancel out, possibly making a terrible fit look good. 

From Table 26-1, for a gaussian distribution of errors the probability of an error greater than a is 0.3174 (that is, 1 — 0.8413 + 0.1587) of an error greater than 2a, 0.0456 and of an error greater than 3a, 0.0026. In each case, positive and negative deviations are equally probable. 

FIGURE 11.10 In an ideal solution, a graph of solvent vapor pressure Pi versus mole fraction of solvent Xi is a straight line. Nonideal solutions behave differently examples of positive and negative deviations from the ideal solution are shown. The vapor pressure of pure solvent is P°. 

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