Rate third-order variation with

Thirdly one needs a drastic step to turn this integral equation into a differential equation. This is the Markov approximation , which comes in two varieties. The first variety consists in replacing ps(t — t) with ps(t). The error is of relative order rc/rm (where l/rm is the unperturbed rate of change due to S s) and of absolute order a2T2/rm. In this approximation one may as well omit the S s in the exponent of (4.13) and the result is the same as (3.19). The second variety takes the zeroth order variation of ps into account by setting ps(t — r) = e T sps(t). The result is the same as was obtained in (3.14) by means of the interaction representation, and the only requirement is arc <[Pg.444]

Figure 6. The variation of the third-order rate constant with different concentrations of toiuene.

The results at other conversions are shown in the third column of Table 4-4. Although there is some variation from point to point, there is no significant trend. Hence the differential method also confirms the validity of a second-order rate equation. The variation is due to errors associated with the measurement of slopes of the curve in Fig. 4-1. 

The data are represented by the theoretical expression for a reaction first order with respect to HbSH, with the asymptotic value of Xmoiai equal to that for ferrohemoglobin. To evaluate XHbSH the susceptibility values over the nearly linear portion of the curve were extrapolated to zero time for four solutions, with pH 5.1, 5.7, 5.7, and 7.0, respectively, the values 2240, 2110, 2260 and 1930.10" , average 2140 10" , were obtained. A value for the dissociation constant of the substance is not provided by our data. The rate constant k = —d(ln[HbSH])/d< has the approximate value 5 10" (with t measured in minutes), the observed values of fe 10 being 5.0 at pH 5.08, 12.0 at pH 5.73, 6.2 at pH 7.02, and 3.0 at pH 5.73. (The third eiqieriment was made with phosphate buffer, the others with acetate buffers thefourth was made with 0.1 ml., the others with 0.5 ml. of 4/sodium hydrosulfide solution added.) It is seen that no more than about 2-fold variation was observed over the pH range 5.1 to 7.0, and that the rate seems to increase with increase in the concentration of hydrosulfuric acid. 

Fig. 2.13. The solid circles show the variation with temperature of the experimental effective third-order rate coefficients (i.e. Ai2nd/lHe)) of the benzene dimerization. The open triangle for T = 123.0 K represents an upper limit to the rate coefficient at this temperature. The open circles show the calculated third-order rate coefficients.

We shall in this chapter discuss the methods employed for the optimization of the variational parameters of the MCSCF wave function. Many different methods have been used for this optimization. They are usually divided into two different classes, depending on the rate of convergence first or second order methods. First order methods are based solely on the calculation of the energy and its first derivative (in one form or another) with respect to the variational parameters. Second order methods are based upon an expansion of the energy to second order (first and second derivatives). Third or even higher order methods can be obtained by including more terms in the expansion, but they have been of rather small practical importance. 

---