The Dispersion Equation

The total intensity of radiation scattered into a solid angle of 4tr during a Raman process in which a molecular system, initially in a state G I, makes a transition to a final state If), is given by 

II is the intensity of the incident radiation of wavenumber pl- is the wavenumber of the scattered radiation Ps = l - gf. where the plus sign refers to anti-Stokes and the minus sign to Stokes scattering. Pqf is the wavenumber of the molecular transition 1G I F . ( po)g F is the polarisability or scattering tensor for the process I G - -1 F in which the polarisations of the incident and scattered radiation are represented by the indices a and p respectively. (o,p =x,y,z). 

In the usual experimental set up, the scattered radiation is collected in some small solid angle around an observation direction at 90° to that of the incident radiation. A schematic diagram of a typical laser Raman set-up involving 90° collection optics is shown in Fig. I, together with the definition of the depolarisation ratio 

Pi Ii/I for linearly polarised incident radiation. In this case, Eq. (1) must be multiplied by the factor (3/8 rr) (1 + pj)/(l + 2pi). (5a). If only the parallel component of scattered radiation is measured the factor is (3/8 tt) (1 + 2 px) . (I = Pi I ). For naturally polarised (unpolarised) or circularly polarised incident light the relevant factors are given in terms of the depolarisation ratio, p , for natural polarisation, where 

The polarisability tensor is given by a second-order perturbation expression known as the Kramers-Heisenberg dispersion formula (4) 

---

With these two-point boundary conditions the dispersion equation, Eq. (23-50), may be integrated by the shooting method. Numerical solutions for first- and second-order reaciions are plotted in Fig. 23-15. 

C. In their first series of experiments, six data sets were obtained for (H) and (u), employing six solvent mixtures, each exhibiting different diffusivities for the two solutes. This served two purposes as not only were there six different data sets with which the dispersion equations could be tested, but the coefficients in those equations supported by the data sets could be subsequently correlated with solute diffusivity. The solvents employed were approximately 5%v/v ethyl acetate in n-pentane, n-hexane, n-heptane, -octane, -nonane and n-decane. The solutes used were benzyl acetate and hexamethylbenzene. The diffusivity of each solute in each solvent mixture was determined in the manner of Katz et al. [3] and the values obtained are included... 

Table 2. Experimental Values for the Dispersion Equation Coefficients Obtained by a Curve Fitting Procedure...

In summary, it can be said that all the dispersion equations that have been developed will give a good fit to experimental data, but only the Van Deemter equation, the Giddings equation and the Knox equation give positive and real values for the constants in the respective equations. 

The form of the solution of the dispersion equation (11.61) depends on the sign of the determinant D = q + Pl, i.e., on the values of the characteristic parameters g and P. The latter are determined by the physical properties of the liquid and its vapor, as well as the values of the Peclet number. This allows us to use g and P as some general characteristics of the problem considered here. 

Then the dispersion equation for the problem considered here I form ... 

Substitution of Equation A7 into A6 gives the dispersion equation relating the disturbance growth rate and wavelength ... 

There are several closed form approximate solutions to both the general and first-order forms of the dispersion equations (11.2.9 and 11.2.10). For example, Levenspiel and Bischoff... 

Solve the dispersion equation for several values of Pe for both closed... 

The dispersion equation and its equivalent as two first order equations... 

Solve the dispersion equation for a second order reaction for several... 

Second order dispersion reaction, using P5.08.06 or P5.08.10, or integrating the dispersion equation. 

The reaction is second order and attains 95% conversion. Find R = kC0t. The dispersion equation is... 

The solution of the dispersion equation along a stream line is given in problem P5.08.08. That Equation (10) can be written in the form... 

Equating determinant of the system (35) to zero, we get the dispersion equation for RPA eigenvalues... 

It is seen that an excellent fit is obtained with the Van Deemter equation with an Index of determination, for that particular fit, of 0.999885 However, the results from testing all the data to each of the dispersion equations need to be known in order to identify the equation that, overall, shows the best fit. 

Irnrr and retention time for droplrts to coalesce A pipe diameter is chosen that is large enough to present coalesced 1 droplets from shearing in accordance with the dispersion equation discussed in the previous installment. In effect, the unit grows a droplet distribution curve in the inlet stream that can then be treated in the tank or flume. Fig. 1 shows an ( installation using SP packs in a series of tanks Fig. 2 shows an installation in a series of compartments in a horizontal flume such as a barge bull or a pit. 

Neglecting radial effects, the dispersion equation for dispersed tracer with bulk flow in a packed bed is ... 

P5.08.06. FIRST AND SECOND ORDER REACTIONS. TABLES AND GRAPHS The dispersion equation... 

---