Equilibrium concentrations approximation method

Note that, in more complex problems, especially if the equation to be solved exceeds the abilities of the quadratic formula (cubed values and higher), it may be best to find the equilibrium concentrations by a method of successive approximations. You would select a consistent set of concentration values near where it is guessed the answer will be. Then, the calculations for K are performed and repeated until a sufficiently precise result is reached. 

The use of metal ions at trancer concentrations, if possible by employing radioactive tracers, enables one to carry out such extractions where CM < [HA] or [B], thereby permitting the approximation, that the equilibrium concentrations of either extractants are the same as their initial concentrations. Besides, use of radioactive tracers makes the distribution measurements convenient and fast. Such a condition makes the quantitative interpretation of the extraction data much simpler when the slope-analysis, Job s and the curve-fitting methods, described below, are employed. 

This method allows for the accurate determination of K i only within the -1000 to +1000 region or approximately within six orders of magnitude span. These experiments could be complicated by solubility and equilibration kinetics and the properties of a substance. For example, if a studied compound has a property of nonionic surfactant, it will be mainly accumulated at the water-organic interface, and shaking of this two-phase system will create a stable emulsion difficult for analytical sampling. The ultracentrifugation at speed of 14,000 rpm for 15-20 min can be enough in most cases to separate two phases. Actual equilibration of the system is tested by several measurements of the equilibrium concentration at different time intervals. 

The method of successive approximations is often faster to apply than the quadratic formula. Keep in mind that the accuracy of a result is limited both by the accuracy of the input data (values of and initial concentrations) and by the fact that solutions are not ideal. It is pointless to calculate equilibrium concentrations to any degree of accuracy higher than 1% to 3%. 

It is worth noting here that the exact solution of a set of nonlinear equations for more complicated equilibria is often unachievable. In such cases, the approximation method implying a simplification of the overall electroneutrality condition using the only pair of predominant defects can be useful. This approach can be illustrated on the basis of the above example of a Si crystal. As the equilibrium constants (Equations (3.15-3.17)) are functions of temperature, the concentrations of different defects can alter in different ways, depending on the value of the pre-exponential factor K° and the enthalpy of the defects reaction, AH . As a result, it is possible to choose a temperature range where the overall electroneutrality condition (Equation (3.18)) can be approximated by pairing the predominant defects. In this case, two possible approximations can be suggested ... 

Isotherms having an inflection point correspond to ranges of both favorable and unfavorable equilibrium. Since separation operations only involve the part of the isotherm which lies below the feed-concentration, the occurrence of an inflection point corresponding to a higher concentration has no effect. If the inflection point occurs at a lower concentration, fixed-bed separation calculations must be carried out by approximate methods to be described below. 

Write the acid ionization constant (A J in terms of the equilibrium concentrations. First solve for x by the approximate method. If the aprroximation is not valid, use the quadratic equation or the method of successive approximation to solve for x. 

Instead of using the quadratic equation, we may use the method of successive approximations. In this procedure, we will first neglect c compared to the initial concentrations to simplify calculations, and calculate an initial value of jc. Then we can use this first estimate of x to subtract from Ca and Cb to give an initial estimate of the equilibrium concentration of A and B, and calculate a new x. The process is repeated until x is essentially constant. 

Michaelis et al. 16, 17) have estimated Ei and > for several molecules by analyzing the potentiometric titration curve. This method can be applied only to redox systems which give a large equilibrium concentration of semiquinone, usually for systems for which > 0.01. Direct estimation of semiquinone concentration by ESR (electron spin resonance) is about 10 times more sensitive for the Kg estimation than analysis of titration curves. Some values are shown in Table I. The Kg for NADH (reduced form of nicotineamide adenine dinucleotide) is too small to be measured even by ESR, and only an approximate upper limit is given. 

Vaporization in a Foreign Gas In addition to the case of vaporization in vacuum, Langmuir derived two other equations for the rate of vaporization in a foreign gas [23]. In a foreign gas environment, the vaporization rate is limited by the diffusion of molecules from the near-surface layer. The thickness of this layer is approximately equal to the mean free-path length, and the molecules in the layer are in their equilibrium concentration. These assumptions are validated by methods of statistical mechanics. Invoking the Pick s first law for one-dimensional diffusion and the Clapeyron-Mendeleev equation, the molecular flux is... 

For these reactions, the equilibrium mixture will not have a lot of products present mostly reactants are present at equilibrium. If we define tbe change that must occur in terms of x as the amount (molarity or partial pressure) of a reactant that must react to reach equilibrium, then x must be a small number because fC is a very small number. We want to know the value of x in order to solve the problem, so we don t assume = 0. Instead, we concentrate on the equilibrium row in the ICE table. Those reactants (or products) that have equilibrium concentrations in the form of 0.10 — x or 0.25 + or 3.5 — 3x, etc., is where an important assumption can be made. The assumption is that because K 1, x will be small (x 1) and when we add x or subtract x from some initial concentration, it will make little or no difference. That is, we assume that 0.10 — X 0.10 or 0.25 + x 0.25 or 3.5 — 3x 3.5 we assume that the initial concentration of a substance is equal to the equilibrium concentration. This assumption makes the math much easier and usually gives a value of x that is well within 5% of the true value of x (we get about the same answer with a lot less work). When the 5% rule fails, the equation must be solved exactly or by using the method of successive approximations (see Appendix A1.4). 39. [CO2] = 0.39 M [CO] = 8.6 X 10 M [O2] = 4.3 x 10 M 41. 66.0% 43. a. 1.5 X 10 atmb. Pco = Pci = 1-8 X 10 atm Pcoci2 = 5.0 atm 45. Only statement d is correct. Addition of a catalyst has no effect on the equilibrium position the reaction just reaches equilibrium more quickly. Statement a is false for reactants that are either solids or liquids (adding more of these has no effect on the equilibrium). Statement b is false always. If temperature remains constant, then the value of K is constant. Statement c is false for exothermic reactions where an increase in temperature decreases the value of K. 47. a. no effect b. shifts left c. shifts right 49. H " + OH — H2O sodium hydroxide (NaOH) will react with the H " on the product side of the reaction. This effectively removes H " from the equilibrium, which will shift the reaction... 

Locci et al. describe a method to use xenon to monitor chemical transformations, which they describe as the Spin-Spy methodology. Xenon was added to a solution of a- and p-D-glucose and the change in concentration of these speeies was monitored via the Xe chemical shift as equilibrium concentrations of the interconverting sugars were reached over a period of 300 min. The xenon chemieal shift difference in a 1 M concentration of these two species is approximately 1.3 ppm. Xenon s interaction with liquid crystal environments has also been reported over the past several years." This area was reviewed extensively by Jokisaari in 1994. 

The remainder of this chapter centers upon the calculation of the equilibrium properties of MM-level models. Such models with ion concentrations of up to 10 molecules/cm correspond to ionic solutions with total ionic concentrations up to about 1M. This concentration is roughly a tenth of the ionic concentration in a molten salt it is low enough so that many approximation methods that are quite satisfactory for BO-level models at densities up to a tenth that of the liquid may be used to calculate the measurable properties of MM-level models for the solutions. A typical approximation method of this kind is the HNC integral equation (Section 7). 

The expressions that describe the equilibrium concentrations of solutes in the ultracentrifuge cell have been derived from both classic thermodynamics, in which there are few uncertainties, and from material transport theory, in which approximations exist. Svedberg (47,48) derived expressions from the two approaches and showed that identical results can be deduced from both methods for ideal solutions. 

If we could know the equilibrium fractions a of the species present, it would be a simple matter to find the pH of solutions. However, we usually know only the initial materials, the analytical concentrations. Algebraic methods of approximation will be given in the next chapter. The graphical methods here help in understanding these equilibria and in evaluating the approximate equations to be found later. 

If equal volumes of 0.2000 M RbCl and 0.2000 M HCIO4 are mixed at 25°, what concentrations are left in solution at equilibrium Follow the method in this chapter and continue approximations until three significant figures are obtained. 

Once you know the value of for an acid HA, you can calculate the equilibrium concentrations of species HA, A , and H30 for solutions of different molarities. The general method for doing this was discussed in Chapter 15. Here we illustrate the use of a simplifying approximation that can often be used for weak acids. In the next example, we will look at an amplification of the first question posed in the chapter opening. 

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