Integral form of the equation

The solution of the initial value problem for the differential equation (9.26) is formally expressed by [16] 

This function provides the chain survival probability, which is the probability for a given active chain with end-to-end vector r at time t to remain active until time t with end vector r. 

Thanks to the affiness assumption, r is uniquely related to r through the equation 

The first term in (9.41) gives the number of chains that, being active in the initial state, remain active until time t. The function m(r,t) in the second term is the number of active chains created at r in a unit time 

The second term in (9.41) therefore gives the number of active chains that are created at time t with end-to-end vector r and remain active until t. Since this term depends on the space integral of xlr r, t ) through Ve(t ), (9.41) is not an explicit solution of the initial value problem, but gives an integral equation for (r,r). 

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El Shahawi concluded that polyurethanes were preferred over carbon because of, among other properties, ease of regeneration. To support this, they determined the H values of a number of pesticides by determining partition coefficient as a function of temperature. The partitioning of a solute is in part defined by the vant Hoff equation. The integrated form of the equation is ... 

The experimental studies of three-component systems based on phase equilibria follow the same principles and methods discussed for two-component systems. The integral form of the equations remains the same. The added complexity is the additional composition variable the excess chemical potentials become functions of two composition variables, rather than one. Because of the similarity, only those topics that are pertinent to ternary systems are discussed in this section of the chapter. We introduce pseudobinary systems, discuss methods of determining the excess chemical potentials of two of the components from the experimental determination of the excess chemical potential of the third component, apply the set of Gibbs-Duhem equations to only one type of phase equilibria in order to illustrate additional problems that occur in the use of these equations, and finally discuss one additional type of phase equilibria. 

Making these modifications, the integral form of the equation of state becomes ... 

The model formulation used with moving meshes is of the arbitrary-Lagrangian-Eulerian (ALE) type. The integral form of the equations governing the incompressible Newtonian fluid in a time-varying control-volume V(t) is written as ... 

Equation 22.2 is known as the differential form of the first-order rate law. The integrated form of the equation is... 

Johnson studied the stability of alumina steaming for various times up to 1,000 h at constant temperature and water pressure. The data were fitted to the integrated form of the equation... 

The TMR is the time taken for the reaction system to reach its maximum self-heating rate under completely adiabatic conditions. Because it assumes that no heat is lost from the reactor to the surroundings it represents the worst possible case. The TMR is based on an integrated form of the equation... 

The use of differential reaction rates to analyze mixtures of alcohols for the primary and secondary hydroxyl contents is possible based on the differences in reaction rate of different alcohols with acetic anhydride. A linear plot for second-order reactions makes possible the analysis of mixtures containing the same functional group. The integrated form of the equation describing second-order reactions is... 

Since Cao does not appear in the integrated form of the equation,... 

The absorption of light by a compound depends on its chromophor, the wavelength of the light and the thickness of the sample. Bouguer derived the relationship between absorption and the thickness of the sample. The integrated form of the equation is shown in Equation [1] where is the intensity of the incident radiation and I the intensity of the transmitted radiation. The factor a related to the absorptivity of the chromophor and b is a measure of the sample thickness. 

Wasserman [29] also developed a method for calculating MWD that is based on the double reptation model. However, whereas Mead [23] chose to use the integral form of the equation and employ various mathematical transforms to manipulate it, Wasserman used discrete variables and numerical techniques. Thus, he writes the double reptation relationship as ... 

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