Mixing statistics

The residuals are comparable to those obtained for GdsGe4. However, the displacement parameter of Sil becomes negative, while the displacement parameter of Si3 is about twice that of other atoms, as shown in Table 7.31. It is unfeasible that some of the Gd atoms are mixed statistically with Sil atoms because of the large difference in their atomic volumes. Refinement of the occupancy of the Si3 site does not result in any defects. As we already explained before, this experimental artifact may be the result of the low scattering ability of Si when compared to that of Gd, coupled with small but unaccounted experimental errors that could be present in the data (see section 7.5 describing the refinement of a related crystal structure of NdsSi4). 

For those who are familiar with the statistical mechanical interpretation of entropy, which asserts that at 0 K substances are nonnally restricted to a single quantum state, and hence have zero entropy, it should be pointed out that the conventional thennodynamic zero of entropy is not quite that, since most elements and compounds are mixtures of isotopic species that in principle should separate at 0 K, but of course do not. The thennodynamic entropies reported in tables ignore the entropy of isotopic mixing, and m some cases ignore other complications as well, e.g. ortho- and para-hydrogen. 

The entropy of mixing of very similar substances, i.e. the ideal solution law, can be derived from the simplest of statistical considerations. It too is a limiting law, of which the most nearly perfect example is the entropy of mixing of two isotopic species. 

By the standard methods of statistical thermodynamics it is possible to derive for certain entropy changes general formulas that cannot be derived from the zeroth, first, and second laws of classical thermodynamics. In particular one can obtain formulae for entropy changes in highly di.sperse systems, for those in very cold systems, and for those associated, with the mixing ofvery similar substances. 

Few populations, however, meet the conditions for a true binomial distribution. Real populations normally contain more than two types of particles, with the analyte present at several levels of concentration. Nevertheless, many well-mixed populations, in which the population s composition is homogeneous on the scale at which we sample, approximate binomial sampling statistics. Under these conditions the following relationship between the mass of a randomly collected grab sample, m, and the percent relative standard deviation for sampling, R, is often valid. ... 

In this problem you will collect and analyze data in a simulation of the sampling process. Obtain a pack of M M s or other similar candy. Obtain a sample of five candies, and count the number that are red. Report the result of your analysis as % red. Return the candies to the bag, mix thoroughly, and repeat the analysis for a total of 20 determinations. Calculate the mean and standard deviation for your data. Remove all candies, and determine the true % red for the population. Sampling in this exercise should follow binomial statistics. Calculate the expected mean value and expected standard deviation, and compare to your experimental results. 

Although PVT equations of state are based on data for pure fluids, they are frequently appHed to mixtures. 7h.e virial equations are unique in that rigorous expressions are known for the composition dependence of the virial coefficients. Statistical mechanics provide exact mixing rules which show that the nxh. virial coefficient of a mixture is nxh. degree in the mole fractions ... 

Most of the assumptions are based on idealized models, indicating the limitations of the mathematical methods employed and the quantity and type of experimental data available. For example, the details of the combinatorial entropy of a binary mixture may be well understood, but modeling requires, in large measure, uniformity so the statistical relationships can be determined. This uniformity is manifested in mixing rules and a minimum number of adjustable parameters so as to avoid problems related to the mathematics, eg, local minima and multiple solutions. 

Although the virial equation itself is easily rationalized on empirical grounds, the mixing rules of Eqs. (4-183) and (4-184) follow rigorously from the methods of statistical mechanics. The temperature derivatives of B and C are given exactly by... 

Sampling of slurries and solids, differs fundamentally from sampling a completely mixed liquid or gas, A hulk quantity of sohds incorporates characteristic heterogenity—that is, a sample Sj differs inherently from a sample S2 when both are taken from a thoroughly mixed load of solids as a result of property variances embodied in solids. In contrast, all individual samples from a completely mixed liquid or gas container are statistically identical. 

There are basically two different computer simulation techniques known as molecular dynamics (MD) and Monte Carlo (MC) simulation. In MD molecular trajectories are computed by solving an equation of motion for equilibrium or nonequilibrium situations. Since the MD time scale is a physical one, this method permits investigations of time-dependent phenomena like, for example, transport processes [25,61-63]. In MC, on the other hand, trajectories are generated by a (biased) random walk in configuration space and, therefore, do not per se permit investigations of processes on a physical time scale (with the dynamics of spin lattices as an exception [64]). However, MC has the advantage that it can easily be applied to virtually all statistical-physical ensembles, which is of particular interest in the context of this chapter. On account of limitations of space and because excellent texts exist for the MD method [25,61-63,65], the present discussion will be restricted to the MC technique with particular emphasis on mixed stress-strain ensembles. 

Exponent in scale-up equation, describing type/degree of mixing required, Eigure 5-32, or number of samples in statistics Number of impellers Number of tube baffles (vertical)... 

Mixing correlation exponent, or empirical constant = Arithmetic mean (statistics)... 

The equilibrium between a compressed gas and a liquid is outside the scope of this review, since such a system has, in general, two mixed phases and not one mixed and one pure phase. This loss of simplicity makes the statistical interpretation of the behavior of such systems very difficult. However, it is probable that liquid mercury does not dissolve appreciable amounts of propane and butane so that these systems may be treated here as equilibria between a pure condensed phase and a gaseous mixture. Jepson, Richardson, and Rowlinson39 have measured the concentration of... 

---