Simultaneous analysis equations

As discussed earlier, the simultaneous analysis of samples containing two analytes requires the isolation of two precipitates. As shown in Example 8.2, conservation of mass can be used to write separate stoichiometric equations for each precipitate. These equations can then be solved simultaneously for both analytes. 

Minimizing the cycle time in filament wound composites can be critical to the economic success of the process. The process parameters that influence the cycle time are winding speed, molding temperature and polymer formulation. To optimize the process, a finite element analysis (FEA) was used to characterize the effect of each process parameter on the cycle time. The FEA simultaneously solved equations of mass and energy which were coupled through the temperature and conversion dependent reaction rate. The rate expression accounting for polymer cure rate was derived from a mechanistic kinetic model. 

Two differential spectrophotometric methods were used by Chatterjee et al. for the simultaneous analysis of diloxanide furoate and metronidazole in pharmaceutical formulations [24]. The first method involved measurement of the absorbance of a methanolic solution of the two drugs at 259 and 311 nm. Since the absorbance of diloxanide furoate at 311 nm is zero, the concentration of metronidazole is directly measured, and a simple equation based on absorbance ratios is used to calculate the concentration of diloxanide furoate. The second method was a differential spectrophotometric determination based on pH-induced spectral changes, on changing from an acidic to an alkaline solution. A marked bathochromic shift was exhibited by metronidazole, while diloxanide furoate showed a slight hypsochromic shift. The wavelength of maximum absorption difference for diloxanide furoate was 267 nm, where metronidazole did not absorb. Similarly, diloxanide furoate did not interfere with metronidazole at when measured at 322 nm. 

A simultaneous analysis of the equation of heat conduction, taking into account the release of heat resulting from the chemical reaction, and the diffusion equation which governs the concentrations of the reacting substances and reaction products, and also allowing for variation of these concentrations due to the chemical reaction, leads to conclusions which are very important for subsequent study.1... 

Recently, another PMR method for the simultaneous analysis of vitamin B5 and Bj, utilising simpler equation for calculation, has been developed (136). The mean percent recovery of vitamin Bg and B in their dosage forms is 97.42 + 2.99 and 97.57 1.52, respectively. 

RAFA and GRAM are limited to simultaneous analysis of one standard and one unknown. One way around this limitation is the direct trilinear decomposition (DTLD) [29], Aside from being a rapid method for solving Equation 12.1, DTLD also provides an excellent starting solution for least-squares and iterative methods. 

A number of modifications and generalizations of the analysis presented here can be explored [14], [40], [45]. A reaction order of zero has been selected here for simplicity of development. By working with two simultaneous differential equations in the reaction zone, the formulation is easily extended to an arbitrary reaction order n. For 0 < n < 1, a modified version... 

There are also other numerical methods for solving differential equations, which we do not discuss. The numerical methods can be extended to sets of simultaneous differential equations such as occur in the analysis of chemical reaction mechanisms. Many of these sets of equations have a property called stiffness that makes them difficult to treat numerically. Techniques have been devised to handle this problem, which is beyond the scope of this book. ... 

Results of Dynamic Analysis. Solving the simultaneous differential equations 1.-3. under the given initial and boundary conditions by the Laplace transformation method, analytical solutuons U.-8. are obtained in the form of dimensionless concentrations ... 

A 98% conversion efficiency has been achieved at 1.5 atmospheric pressure of CO2 and 10% excess of water, while the yield was 85% only when stoichiometric amounts of water and carbon dioxide were used. No isotopic fractionation has been noted when 5 to 10% excess of water has been used in the reaction in equation 5. Doubly labelled methane has been used in simultaneous analysis of and both in the atmosphere and in environmental carbon dioxide as well as in analysis of and H-labelled organic compounds. Proportional counters filled with methane possess very good counting properties and there is no need to add any inactive counting gas. and tritium-labelled methanes have also been produced" by pyrolysis of sodium acetate in molten sodium hydroxide (equation 6) ... 

ABSTRACT. This paper describes a methodology for the sensitivity analysis and optimization of planar constrained mechanical systems. Direct differentiation methods and finite difference techniques have been used in the design sensitivity calculations. The optimization process is developed within the framewoik of mathematical programming techniques. The sensitivity equations were constructed symbolically and subsequently integrated in the dynamic analysis equations of motion and solved simultaneously. 

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