Integration time initial, equation

In a curve-fitting method the concentration of a reactant or product is monitored continuously as a function of time, and a regression analysis is used to fit an appropriate differential or integral rate equation to the data. Eor example, the initial concentration of analyte for a pseudo-first-order reaction, in which the concentration of a product is followed as a function of time, can be determined by fitting a rearranged form of equation 13.12... 

Integrate state equations forward with the given initial conditions until M=Mf. Let tf be the time when M=Mf. 

This is the integrated rate equation for a first-order reaction. When dealing with first-order reactions it is customary to use not only the rate constant, k for the reaction but also the related quantity half-life of the reaction. The half-life of a reaction refers to the time required for the concentration of the reactant to decrease to half of its initial value. For the first-order reaction under consideration, the relation between the rate constant k and the half life t0 5 can be obtained as follows ... 

Equation (43) can be integrated and evaluated at the obvious initial conditions (time = 0, radius = r0) and final conditions (time = t, radius = r) as... 

In this equation, In denotes the natural logarithm, [A]0 designates the concentration of A at some initial time, arbitrarily considered to be t = 0, and [A]f is the concentration of A at any time f thereafter. (See Appendix A.2 for a review of logarithms.) The ratio [A]f/[A]0 is the fraction of A that remains at time t. Thus, the integrated rate law is a concentration-time equation that makes it possible to calculate the concentration of A at any time t or the fraction of A that remains at any time t. The integrated rate law can also be used to calculate the time required for the initial concentration of A to drop to any particular value or to any particular fraction of its initial concentration (Figure 12.6a). Worked Example 12.5 shows how to use the integrated rate law. 

Since the change in solute concentration is negligitolegn be approximated toyb, the weight of undissolved solute crystals at time 0. This is a reasonable approximation when the initial amount of solute is less than one-twentietiy D) of its solubility. Integration of Equation 17.23 underthese conditions gives the cubic root law, which can be written as follows ... 

By integrating these equations over time, the position of any fluid particle with a given initial condition can be predicted at any time t. This approach is different from the common approach in fluid mechanics where flow is described in terms of velocity fields. However, by following the motion of the fluid we obtain a physically more profound understanding of mixing. 

Its time evolution after instantaneous photoexcitation satisfies the integral kinetic equation (3.737) with the initial condition... 

Half-life (t,/2) Time required for the concentration of a reactant to fall to one half of its initial value. Calculated by substituting the value 0.5[A]0 for [A] in the integrated rate equation and solving for t. ... 

Integration of equation (1.6) (initial condition x = 0 at t = 0) gives an expression relating the time of growth of the ApBq layer to its total thickness ... 

Numerical integration of equations (2) and (3) with initial values for X,Y on the limit cycle and with one of the rate constants oscillating as in equation (4) or (5) may result in a transition of the X,Y trajectory across the separatrix towards the stationary state. The occurrence of a transition is dependent on the parameters g, u) and 0. For extremely small amplitude perturbations (g - -0), the trajectory continues to oscillate close to the limit cycle. As g is increased, however, transitions may occur. The time taken for a transition is then primarily a function of the frequency of the perturbation. The time from the onset of the oscillating perturbation to the time at which the trajectory attains the lower steady state (At) is plotted in Figure 3 as a function of with all other parameters held constant. The arrow marks the minimum value for At which occurs when the frequency of the external perturbation exactly equals that of the unperturbed limit cycle itself. The second minimum occurs at the first harmonic of the limit cycle. Qualitatively similar results are obtained when numerical integration is carried out with differing values for g and 0. 

And if the initial concentration (time, t = 0) of monomer is cQ then we can integrate this equation as shown in Equations 4-14. 

The integrated rate equations in kinetics give the concentration c of reactant after time t in terms of the rate constant k and initial concentration cq. State what type of plot would give a straight line and the value of the resultant gradient and intercept for ... 

III. Free Diffusion Junction.—The free diffusion type of boundary is the simplest of all ir. practice, but it has not yet been possible to carry out an exact integration of equation (41) for such a junction. In setting up a free diffusion boundary, an initially sharp junction is formed between the two solutions in a narrow tube and unconstrained diffusion is allowed to take place. The thickness of the transition layer increases steadily, but it appears that the liquid junction potential should be independent of time, within limits, provided that the cylindrical symmetry at the junction is maintained. The so-called static junction, formed at the tip of a relatively narrow tube immersed in a wdder vessel (cf. p. 212), forms a free diffusion type of boundary, but it cannot retain its cylindrical symmetry for any appreciable time. Unless the two solutions contain the same electrolyte, therefore, the static type of junction gives a variable potential. If the free diffusion junction is formed carefully within a tube, however, it can be made to give reproducible results. ... 

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