Second-order concentration variables

The choice of the y-variable is also important. If one records a series of concentrations, or a quantity proportional to them, then this set is a valid quantity to be fitted by linear least squares. On the other hand, if the equation is rearranged to a form that can be displayed in a linear graph, then the new variable may not be so suitable. Consider the equations for second-order kinetics. The correct form for least-squares fitting is... 

When depends on a alone, the ODE is variable-separable and can usually be solved analytically. If depends on the concentration of several components (e.g., a second-order reaction of the two reactants variety, a = —kab), versions of Equations (1.22) and (1.23) are written for each component and the resulting equations are solved simultaneously. 

Automatic controllers can produce small oscillations of the controlled variable. The effect of sinsoidal variations in concentration, temperature or feed rate on the effluent concentration of a second order reaction in a CSTR will be examined. The unsteady material balance is... 

The general equation of convective diffusion in liquids, equation (15), is a second-order, partial differential equation with variable coefficients. Its solution yields the spatial distribution of c, as a function of time, namely its transient behaviour. On an analytical level, solution of equation (15) into the transient c(t) is possible only for a number of relatively simple systems with well-defined geometry and flow properties. The problem is greatly simplified if the concentration function Cj(x,y,z) is essentially independent of time t, i.e. in the steady-state. Then equation (15) reduces to ... 

For the one-dimensional equation with x as the space variable, the diffusion equation is a partial differential equation of the first order in time and the second order in x. It therefore requires concentration to be known everywhere at a given time (in general =0) and, at any time t>0, concentration, flux, or a combination of both, to be known in two points (boundary conditions). In the most general case, the diffusion equation is a partial differential equation of the first order in time and the second order in the three space coordinates x, y, z. Concentration or flux conditions valid at any time >0 must then be given along the entire boundary. 

For the kinetics of second-order reversible reactions (Reactions 2 to 5 in Table 2-1) under variable temperature, an analytical solution is not available. The evolution of species concentrations may be calculated through numerical methods. Consider the following second-order reversible reaction as an example ... 

The dependence of the reaction rate on the reagent concentration 14 150 152 very complicated variable order with respect to monomer concentration (first order at low and second order at high concentrations) the same order with respect to proton-... 

A general systematic technique applicable to second-order differential equations, of which (11.31) is a particular example, is that of phase plane analysis. We have seen this approach before (chapter 3) in the context of systems with two first-order equations. These two cases are, however, equivalent. We can replace eqn (11.31) by two first-order equations by introducing a new variable g, which is simply the derivative of the concentration with respect to z. Thus... 

The application of this rate law to the simulation of electrochemical behavior requires two dimensionless input parameters ktf and KC. When these are supplied, three-dimensional chronoamperometric or chronocoulometric working surfaces [34] are generated. These working surfaces both indicate first-order behavior when KC is large and second-order behavior when KC is small. Intermediate values of KC produce the variable reaction orders between one and two that are observed experimentally when the bulk olefin concentration is varied. Appropriate curve fitting of the experimental i(t,C) data to the simulation results in the evaluation of k and K details appear in the referenced work. 

Weber stated that the change in concentration of the three components chosen, resulting from the variation of one reaction variable, can be represented by an equation of the second order ... 

This treatment leads to a system of stiff, second-order partial differential equations that can be solved numerically to yield both transient and steady-state concentration profiles within the layer (Caras et al., 1985a). Because the concentration profile changes most rapidly near the x = L boundary an ordinary finite-difference method does not yield a stable solution and is not applicable. Instead, it is necessary to transform the distance variable x into a dummy variable y using the relationship... 

On the contrary, the introduction of a second-order kinetics in this model produces the best results for all the measured variables, as shown in Fig. 3.6 in particular, the errors on concentrations of phenol and product are reduced to about 1.6 and 0.4 percent, respectively, while the errors on the specific thermal power are very small. 

In addition to the molecular weight of the free polymer, there axe other variables, such as the nature of the solvent, particle size, temperature, and thickness of adsorbed layer which have a major influence on the amount of polymer required to cause destabilization in mixtures of sterically stabilized dispersions and free polymer in solution. Using the second-order perturbation theory and a simple model for the pair potential, phase diagrams relat mg the compositions of the disordered (dilute) and ordered (concentrated) phases to the concentration of the free polymer in solution have been presented which can be used for dilute as well as concentrated dispersions. Qualitative arguments show that, if the adsorbed and free polymer are chemically different, it is advisable to have a solvent which is good for the adsorbed polymer but is poor for the free polymer, for increased stability of such dispersions. Larger particles, higher temperatures, thinner steric layers and better solvents for the free polymer are shown to lead to decreased stability, i.e. require smaller amounts of free polymer for the onset of phase separation. These trends are in accordance with the experimental observations. 

A) The first question can be done in your head, but you may be asked to show your work, so it is useful to be able to solve the problem both ways. The quick inspection method is to look at what happens to the rate when only one substance changes. For example, in experiment 2 the concentration of A doubles while the concentration of B stays the same. The reaction rate increases by 4, which indicates the reaction is second order for A. In experiment 3, the concentration of B doubles while A remains constant. This change produces no change in reaction rate, meaning that the reaction is zero order for B. By inspection, the rate equation becomes Rate = k[A]2. Using rates equations, you can also determine the rate law. This requires solving for the variables m and n in the equation ... 

When ux = uy = uz = 0, indicating no convective motion of the gas, Eq. 10.15 reverts to the pure diffusion case. The terms ux, uy, and uz are not necessarily equal, nor are they usually constant, since convective velocities decrease as a surface is approached. Equation 10.15 thus represents a second-order partial differential equation with variable coefficients. These types of equations are usually quite difficult to solve. However, often it is sufficient to consider only the steady-state solution, i.e., the case where dc/dt = 0, indicating that the concentration at any point within the system is not changing with time. Then Eq. 10.15 becomes... 

Partial differential equations, such as Pick s second law (in which the concentration is a function of both time and space), are generally more difficult to solve than total differential equations, in which the dependent variable is a function of only one independent variable. An example of a total differential equation (of second order ) is... 

Several equivalent forms may be written by considering equilibria involving Br2, Brj etc. Sourisseau found a third-order initial reaction, altering to second-order after some hours (at 20 °C). Variable reaction order with changing pH is also reported, and implied by Skrabal s mechanisms. Thus, the results of one study , indicate an order between two and three at pH about 7, falling with rising pH to the value two. Some of the uncertainty was removed by analysis for bromite during reactions at pH values between 8.0 and 9.15 . Its concentration rose to a maximum and then declined slowly, in the manner of a reactive intermediate. It is not likely that reaction (2) could lead to such behaviour. The successive equations (4) and (5) proposed... 

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