True solutions

We have in mind trajectory calculations in which the time step At is large and therefore the computed trajectory is unlikely to be the exact solution. Let Xnum. t) be the numerical solution as opposed to the true solution Xexact t)- A plausible estimate of the errors in X um t) can be obtained by plugging it back into the differential equation. 

The second type is a stable dispersion, or foam. Separation can be extremely difficult in some cases. A pure two-component system of gas and liquid cannot produce dispersions of the second type. Stable foams can oe produced only when an additional substance is adsorbed at the liquid-surface interface. The substance adsorbed may be in true solution but with a chemical tendency to concentrate in the interface such as that of a surface-active agent, or it may be a finely divided sohd which concentrates in the interface because it is only poorly wetted by the liquid. Surfactants and proteins are examples of soluble materials, while dust particles and extraneous dirt including traces of nonmisci-ble liquids can be examples of poorly wetted materials. 

Equations (22-86) and (22-89) are the turbulent- and laminar-flow flux equations for the pressure-independent portion of the ultrafiltra-tion operating curve. They assume complete retention of solute. Appropriate values of diffusivity and kinematic viscosity are rarely known, so an a priori solution of the equations isn t usually possible. Interpolation, extrapolation, even precuction of an operating cui ve may be done from limited data. For turbulent flow over an unfouled membrane of a solution containing no particulates, the exponent on Q is usually 0.8. Fouhng reduces the exponent and particulates can increase the exponent to a value as high as 2. These equations also apply to some cases of reverse osmosis and microfiltration. In the former, the constancy of may not be assumed, and in the latter, D is usually enhanced very significantly by the action of materials not in true solution. 

Process Description Microfiltration (MF) separates particles from true solutions, be they liquid or gas phase. Alone among the membrane processes, microfiltration may be accomplished without the use of a membrane. The usual materi s retained by a microfiltra-tion membrane range in size from several [Lm down to 0.2 [Lm. At the low end of this spectrum, very large soluble macromolecules are retained by a microfilter. Bacteria and other microorganisms are a particularly important class of particles retained by MF membranes. Among membrane processes, dead-end filtration is uniquely common to MF, but cross-flow configurations are often used. 

The pumped-discharge case is generally more difficult to solve because of the uncertainty in deahng with negative numerical results. As a final answer, a negative value could indicate that the pump has completely emptied the tank however, as an intermediate value, it could mean that it is not a true solution. A simple check is to try a different initial estimate and see if the intermediate negative results disappear. 

However, its cooling properties can be utilized when other materials are added to improve machining performance. One of the most common water-based cutting fluids is the so-called soluble oil, which, in fact, is not a true solution but an oiTin-water emulsion in which very fine droplets of oil are suspended in water. Such a fluid has very effective cooling power and the petroleum oil and its additives provide its lubricating and protective properties. 

A. Thiocyanate method Discussion. Iron(III) reacts with thiocyanate to give a series of intensely red-coloured compounds, which remain in true solution iron(II) does not react. Depending upon the thiocyanate concentration, a series of complexes can be obtained these complexes are red and can be formulated... 

This represents the extent to which the approximation Xa fails to satisfy the equations, while 8 represents the error, which is the amount by which the approximation Xa deviates from the true solution Most methods depend upon the use of a sequence of matrices Ca chosen in such a way that... 

These characteristics are typically classified as a hydrophile-lipophile balance (HLB value). For example, hydrophilicity may be denoted within a range of 2 to 20, with true solutions being obtained at HLB values >14 and poor dispersibility occurring at HLB values <6. Oil-in-water emulsification requires a high HLB value surfactant, while water-in-oil emulsification needs a low HLB value surfactant. 

Colloids will reflect or scatter light while true solutions do not this is known as the Tyndall effect. 

After an initial startup period, Equations (2.25) and (2.26) become reasonable approximations of the true solutions. See Figure 2.3 for the case of kf = ks = lOkf. The approximation becomes better when there is a larger difierence between kf and the other two rate constants. 

FIGURE 2.3 True solution versus approximation using the quasi-steady hypothesis. 

A hydrogel is formed by a water-soluble polymer that has been lightly crosslinked. Hydrogels swell as they absorb water but they do not dissolve. The volume expansion is limited by the degree of crosslinking. The minimum number of crosslinks needed to form a three-dimensional matrix is approximately 1.5 crosslinks per chain, and this yields the maximum expansion possible without separation of the chains into a true solution. Thus, a hydrogel may be more than 95% water and, in that sense, has much in common with living soft tissues. 

Metals are insoluble in common liquid solvents but can dissolve in each other (like dissolves like). A mixture of substances with metallic properties is called an alloy. Some alloys are true solutions, but microscopic views show that others are heterogeneous mixtures. Brass, for instance, is a homogeneous solution of copper (20 to 97%) and zinc (80 to 3%), but common plumber s solder is a heterogeneous alloy of lead (67%) and tin (33%). When solder is examined under a microscope, separate regions of solid lead and solid tin can be seen. When brass is examined, no such regions can be detected. 

Particles whose dimensions are between 1 nanometer and 1 micrometer, called colloids, are larger than the t3/pical molecule but smaller than can be seen under an optical microscope. When a colloid is mixed with a second substance, the colloid can become uniformly spread out, or dispersed, throughout the dispersing medium. Such a dispersion is a colloidal suspension that has properties intermediate between those of a true solution and those of a heterogeneous mixture. As Table 12-3 demonstrates, colloidal suspensions can involve nearly any combination of the three phases of matter. Gas-gas mixtures are the exception, because any gas mixes uniformly with any other gas to form a true solution. 

Shown in Figure 8 is essentially the same plot as Figure 7, but one of the expansions is made using the first order theory. This figure shows that although the first order theory is not adequate for an accurate description of the x2 surface even for a 5% variation around the true solution, the first order theory nevertheless results in a solution which is closer to the true solution compared with its starting point and that the method of iteration may be applied to improve the accuracy of the solution. 

A colloid is a suspension of particles with diameters between 1 nm and 100 nm. The particles are charged and can be subjected to cataphoresis (electrophoresis). They are subject to Brownian movement and have a large amount of surface activity. Their properties lie between those of true solutions and coarse suspensions. 

A state of subdivision of matter with a particle size between 10"7 and 10 5 cm (1 nm to 100 nm). The properties of colloids lie between those of true solutions and coarse suspensions. 

A true solution consists of molecules or ions of a solid substance uniformly distributed throughout a liquid. A rubber solution is colloidal, i.e., intermediate between a true solution and a coarse suspension. Rubber solutions are used as adhesives and in the manufacture of rubber products by the dipping process. 

Do the trace elements remain in true solution, to be diluted throughout the ocean by mixing ... 

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