Williamson equation

The Williamson equation is useful for modeling shear-thinning fluids over a wide range of shear rates (15). It makes provision for limiting low and high shear Newtonian viscosity behavior (eq. 3), where T is the absolute value of the shear stress and is the shear stress at which the viscosity is the mean of the viscosity limits TIq and, ie, at r = -H... 

As an alternative to the method used in Problem 13.8, the approximate method of Carey and Williamson (equation 13.54) is adopted. 

Equation (60) implies that stress relaxation down to 2,5 MPa at the upper annealing temperature (T = takes 14 minutes, while a decrease to 0.5 MPa takes about 70 minutes. According to the Maxwell equation, the stress relaxation is faster if 14 minutes were required for a 10-fold decrease in initial stress to 2.5 MPa, then for a decrease to 0.5 MPa, only 10 extra minutes are necessary, 24 altogether compared with 70 minutes according to the Adams-Williamson equation. 

For the investigated emulsions, i oo tio Under this condition, the Peeck-Mak-Lean-Williamson equation transforms into the Ferry equation and, therefore, is not suitable for description of the viscosity of extracting emulsions. In the Meter equation, the term (P/Pav) ( oo/T o) approaches 1 at Tjoo i1o- Thus, Equation 8 is transformed into the Ellis equation. Values of P1/2 and the exponential coefficient A for the Ellis model are presented in Figure 8. It should be noted that the value for A is constant and equal to 6 in the equation which describes the rheological curves of the extracting emulsions for the indicated range of dispersed phase content. 

Write equations describing two different ways in which benzyl ethyl ether could be prepared by a Williamson ether synthesis J... 

Write the names of reagents and equations for the preparation of the following ethers by Williamson s synthesis ... 

A rapid one-pot method for converting 1,3-diols into oxetanes by the intramolecular Williamson reaction has recently been described. The monolithium salt is generated by treatment of the diol with one equivalent of butyllithium in cold THF, followed by addition of one equivalent of tosyl chloride to give a monotosylate, which is cyclized by addition of a second equivalent of butyllithium (equation 83). Yields of 70-90% are reported for a variety of alkyl- and aryl-substituted oxetanes (81S550). Another simple method for converting 1,3-diols into oxetanes consists of converting them to cyclic carbonate esters by ester... 

The so-called Williamson synthesis of ethers is by far the most important ether synthesis because of its versatility it can be used to make unsymmetrical ethers as well as symmetrical ethers, and aryl alkyl ethers as well as dialkyl ethers. These reactions involve the nucleophilic substitution of alkoxide ion or phenoxide ion for halide (equation 70).26°... 

A few examples will show the importance of these reactions in synthetic organic chemistry. If X is an alkoxide ion, equation (1) describes the Williamson ether synthesis (2). If X is a cyanide ion, (1) describes a standard nitrile preparation (3). If X is an aromatic nucleus, and the departure of the leaving group is facilitated by A1C13, then (1) describes the Friedel-Crafts alkylation (4). With X as the cyanate ion, (1) describes the preparation of isocyanates (5) and if it is ammonia, (1) describes a common method of preparation of amines (6). In addition, nucleophilic transmethylation is a very... 

Write equations for the preparation of an ether using a Williamson synthesis. 

In Equation 4.1, the factor fiF(0) is included, which is the peak profile function, that describes particle size broadening and other sources of peak broadening. The XRD method can be used as well for the measurement of the crystallite size of powders by applying the Scherrer-Williamson-Hall methodology [4,35], In this methodology, the FWHM of a diffraction peak, p, is affected by two types of defects, that is, the dislocations, which are related to the stress of the sample, and the grain size. It is possible to write [35]... 

Several examples of the Williamson ether synthesis are given in the following equations ... 

The Ellis (46), Carreau (47), and Cross-Williamson (48) models are three-parameter models. The Ellis model gives the viscosity as a function of T o, the shear rate cr, the parameter ai/2, which is the shear rate for which the viscosity is rio/2, and a dimensionless parameter a. The equation is... 

As a natural extension of the methods shown in Section 13.2, IB expressions for the various sources of line broadening can be combined. This is the basis of the Williamson-Hall method, introduced in the late 1940s.Considering Equations (7) and (10), size and strain contributions can be combined as ... 

The two broadening effects can be separated by using a Williamson-Hall plot [174], which combines Equations (4.7) and (4.8) to give ... 

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