The Iso-State Method

A relevant question to ask is then if it is possible to construct a candidate AR and then project the same construction onto lower dimensional subspaces, then is it possible to perform the reverse process and assemble a candidate AR from multiple lower dimensional constructions  

The iso-state method attempts to construct a candidate region in IR (where n 2), by a successive assembly process involving smaller two-dimensional subproblems, where each subproblem forms a slice of the larger region. The reasons for using two-dimensional slices, as opposed to higher dimensional slices, are as follows  

Flexibility. Higher dimensional systems can be broken down into a number of smaller and simpler two-dimensional problems. For systems involving exactly two independent reactions, the resulting AR can still be computed with the method. Two-dimensional constructions provide a minimum working dimension from which candidate regions may be determined. 

Simplicity. When sub-constructions are performed in two dimensions specifically, there is no reliance on the standard DSR equation, and the difficulties associated with their use might be avoided. 

2 Basic Idea Construction via two-dimensional slices is intuitive in principle. Candidate ARs are convex polytopes that always reside in the full space defined by 

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It will be useful to explain the constmction process of the iso-state method with a clear picture of the reactor behavior in mind. Figure 8.16 shows a hypothetical construction of a two-dimensional plane in. (The same idea also applies to higher dimensional systems.)... 

ILLUSTRATION Results of the iso-state method, applied to three-dimensional Van de Vusse kinetics... 

Figure 8.17(a) and (b) shows construction results obtained by the iso-state method for three-dimensional Van de Vusse kinetics. These results are those obtained using 25 two-dimensional planes for each component... 

Figure 8.17 (a) Iso-compositional trajectories for planes in Cg-Co space at fixed values of c. The trajectories all lie within a two-dimensional plane. See color plate section for the color representation of this figure.) (b) Final candidate region produced for the Van de Vusse system using the iso-state method. 

Let us dwell now on the dependence of the rate constant on the isotopic composition of the reactant molecules which is usually called kinetic isotope effect. Various types of isotope effects are illustrated in Table 3. Assuming that the transmission coefficient is independent of the isotopic composition, Eq. (11.1) would yield the ratio of the rate constants ki/kg for reactions of molecules with a different isotopic composition. This ratio depends on the symmetry of reactants, their zero-point vibrational energies and effective masses corresponding to motions along the reaction coordinate (for detail see [222, 304]). In the classical limit (E2, EJ < kT), the ratio kj/kg depends on the ratio of effective masses rather than on temperature. In the essentially quantum case (E, EJ kT), the value of kj/kg is influenced mainly by the change in zero-point energies however, the ratio X1/X2 Iso substantially differ from unity [308, 309] as demonstrated for different isotopic variants of reaction H + Hg H2 + H. It is just the difficult calculation of the transmission coefficient that limits the applicability of the transition-state method to the calculation of the isotope effect. 

Several countries have developed their own standard test methods for cellular plastics, and the International Organization for Standards (ISO) Technical Committee on Plastics TC-61 has been developing international standards. Information concerning the test methods for any particular country or the ISO procedures can be obtained in the United States from the American National Standards Institute. The most complete set of test procedures for cellular plastics, and the most used of any in the world, is that developed by the ASTM these procedures are pubUshed in new editions each year (128). There have been several reviews of ASTM methods and others pertinent to cellular plastics (32,59,129—131). 

In the United States, a number of physical tests are performed on siUcon carbide using standard AGA-approved methods, including particle size (sieve) analysis, bulk density, capillarity (wettabiUty), friabiUty, and sedimentation. Specifications for particle size depend on the use for example, coated abrasive requirements (134) are different from the requirements for general industrial abrasives. In Europe and Japan, requirements are again set by ISO and JSA, respectively. Standards for industrial grain are approximately the same as in the United States, but sizing standards are different for both coated abrasives and powders. 

Two particular test methods have become very widely used. They are the Vicat softening point test (VSP test) and the heat deflection temperature under load test (HDT test) (which is also widely known by the earlier name of heat distortion temperature test). In the Vicat test a sample of the plastics material is heated at a specified rate of temperature increase and the temperature is noted at which a needle of specified dimensions indents into the material a specified distance under a specified load. In the most common method (method A) a load of ION is used, the needle indentor has a cross-sectional area of 1 mm, the specified penetration distance is 1 mm and the rate of temperature rise is 50°C per hour. For details see the relevant standards (ISO 306 BS 2782 method 120 ASTM D1525 and DIN 53460). (ISO 306 describes two methods, method A with a load of ION and method B with a load of SON, each with two possible rates of temperature rise, 50°C/h and 120°C/h. This results in ISO values quoted as A50, A120, B50 or B120. Many of the results quoted in this book predate the ISO standard and unless otherwise stated may be assumed to correspond to A50.)... 

There are two types of external documents, those in the public domain and those produced by specific customers. In some cases the issues of both types of documents are stated in the contract and therefore it is important to ensure that you possess the correct version before you commence work. Where the customer specifies the issue status of public domain documents that apply you need a means of preventing their withdrawal from use in the event that they are revised during the term of the contract. Where the issue status of public domain documents is not specified you may either have a free choice as to the issue you use or, as is more likely, you may need to use the latest issue in force. Where this is the case you will need a means of being informed when such documents are revised to ensure that you can obtain the latest version. The ISO 9000 series for instance is reviewed every five years, so could well be revised at five-year intervals. With national and international legislation the situation is rather different as these can change at any time. You need some means of alerting yourself to changes that affect you and there are several methods from which to choose ... 

Probably the ISO method alone will be accepted by all Member States. 

The RA 25 contract has been extended by CEC and renumbered RA 100 in order to continue this exercise with the ISO 1182 apparatus. However, the major part of the work of the RA 100 group will be the preparation of unified test reports for each of the test methods, which will ensure that all the necessary details are recorded to enable a product to be classified in any of the EEC Member States. Two new laboratories, i.e. the French Laboratoire National d Essai and the Danish Dantest, joined the previous seven. 

The ISO definition of validation is confirmation by examination and provision of objective evidence that the particular requirements of a specified intended use are fulfilled [15]. Method validation is needed to confirm the fitness for purpose of a particular analytical method, that is, to demonstrate that a defined method protocol, applicable to a specified type of test material and to a defined concentration rate of the analyte —the whole is called the analytical system — is fit for a particular analytical purpose [4]. This analytical purpose reflects the achievement of analytical results with an acceptable standard of accuracy. An analytical result must always be accompanied by an uncertainty statement, which determines the interpretation of the result (Figure 6). In other words, the interpretation and use of any measurement fully depend on the uncertainty (at a stated level of confidence) associated with it [8]. Validation is thus the tool used to demonstrate that a specific analytical method actually measures what it is intended to measure and thus is suitable for its intended purpose [11,55,56]. 

Limit of detection (LOD) sounds like a term that is easily defined and measured. It presumably is the smallest concentration of analyte that can be determined to be actually present, even if the quantification has large uncertainty. The problem is the need to balance false positives (concluding the analyte is present, when it is not) and false negatives (concluding the analyte is absent, when it is really present). The International Union of Pure and Applied Chemistry (IUPAC) and ISO both shy away from the words limit of detection, arguing that this term implies a clearly defined cutoff above which the analyte is measured and below which it is not. The IUPAC and ISO prefer minimum detectable (true) value and minimum detectable value of the net state variable, which in analytical chemistry would become minimum detectable net concentration. Note that the LOD will depend on the matrix and therefore must be validated for any matrices likely to be encountered in the use of the method. These will, of course, be described in the method validation document. 

When ASTM, followed by ISO and others, started conducting systematic interlaboratory trials to obtain precision data for test methods, the true state of affairs became apparent15. For many standards the variability was worse than realised and in some cases was so bad as to question whether the tests were worth doing at all or whether specifications based on them could be considered valid. The general advance of the quality movement prompted these investigations and have ensured that reproducibility has continued to occupy one of the top spots for attention in recent years. 

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