Long time approximation

After substitution of the long-time approximation together with eqn. (34) into eqn. (33), kt will be eliminated, which means that jF is no longer determined by the charge transfer process. This will be discussed further in Sect. 2.2.5. 

The chronocoulometric method offers some advantages over the chronoamperometric method, (a) The measured signal grows with time and hence the later part of the transient, that corresponds better with theory, suffers less from instrumental inaccuracy, (b) The act of integration smoothes random noise in the current signal and hence also improves the accuracy, (c) Even the long-time approximation, eqn. (37b), contains... 

When nuclease-(1-126) is mixed with nuclease-T-(49,50-149) (the mixture of nuclease-T-(49-149), and nuclease-T-(50-149) ], enzymic activity is generated at the level of nuclease-T and the formation of ordered structure is also indicated (80). After incubation of the enzymically active mixture in the presence of pdTp and Ca2+ with trypsin for a fairly long time, approximately 50% of the original enzymic activity remains. The enzymically active complex thus formed is the same as nuclease-T and is composed of the two fragments nuclease-T-(6-48) and nuclease-T-(49-149) [or nuclease-T-(50-149)] (80). Therefore, nuclease-T-(49, 50-149) binds to the nuclease-T-(6-48) portion of nuclease-(1-126) forming an enzymically active structure like nuclease-T, and the redundant residues 49-126 of nuclease-(1-126) extend in a disordered manner from the ordered structure. 

FIGURE 6.14 Fractional release profiles from a slab based on short- and long-time approximations. 

Finally, the same study showed that the short-time solution of Mahon and Oldham (12.24) is 0.1% accurate in the range 0 < T <1, and the long-time approximation in the range T > 0.4. Thus, the two formulae yield accurate current values over the whole time scale without a gap. 

Replacing c(0,t) by —T(t) and, after some rearrangements, a long time approximation is... 

For radially growing drops MacLeod Radke (1994) derived a long time approximation for a drop growing into air. 

Additional short and long time approximations have been summarised by Fainerman et al. [99] based on diffusion-controlled, barrier-controlled and mixed kinetic models. An analysis of the known long time approximations was given by Makievski et al. [15]. They compared the long time approximations given by Hansen and by Joos. While Hansen s approximation [22, 117] yields... 

Note that this time is long compared with our long-time approximation criteria which stated that t should be > 0.1[L /( r D)] 0.01(L /E>) in order to reasonably truncate the series solution to a single term. Thus, our decision to approximate the series with a single term in this example was reasonable. 

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