Group situation description

Where surface-active agents are present, the notion of surface tension and the description of the phenomena become more complex. As fluid flows past a circulating drop (bubble), fresh surface is created continuously at the nose of the drop. This fresh surface can have a different concentration of agent, hence a different surface tension, from the surface further downstream that was created earlier. Neither of these values need equal the surface tension developed in a static, equiUbrium situation. A proper description of the flow under these circumstances involves additional dimensionless groups related to the concentrations and diffusivities of the surface-active agents. 

Data gathered in the lessons were used to produce case studies for each of the groups of students because such case studies yield rich descriptions of events that are presented in a chronological narrative that incorporates the researcher s observations. Due to the possible inclusion of such an interpretation of the data, case studies go beyond simple descriptions of the situation and support the analysis of the phenomenon being studied (Cohen, Manion, Morrison, 2000). In order to discuss the research questions, we browsed the original case studies to identify evidence of how the students dealt with the levels of representation. Whenever it is appropriate, such evidence is included in the later sections of this chapter. 

Oosawa (1971) developed a simple mathematical model, using an approximate treatment, to describe the distribution of counterions. We shall use it here as it offers a clear qualitative description of the phenomenon, uncluttered by heavy mathematics associated with the Poisson-Boltzmann equation. Oosawa assumed that there were two phases, one occupied by the polyions, and the other external to them. He also assumed that each contained a uniform distribution of counterions. This is an approximation to the situation where distribution is governed by the Poisson distribution (Atkins, 1978). If the proportion of site-bound ions is negligible, the distribution of counterions between these phases is then given by the Boltzmann distribution, which relates the population ratio of two groups of atoms or ions to the energy difference between them. Thus, for monovalent counterions... 

The description of conjugated dienes as shown by equation 17 and the associated comparison with butadiene in equation 18 corresponds most closely to the conventional definition. The results are plausible in that groups on one double bond that are c/5-situated relative to the other encourage nonplanarity, cause destabilization and result in lessened conjugation energy. Or so we say. The biggest debit of this approach is that the thermochemistry of the monoenes related by single addition of H2 is often absent. An example... 

Experience suggests that d-type functions are required on second-row and heavier main-group elements even though they are not occupied in the free atoms (discussion is provided in Section II). This situation is very much like that found for alkali and alkaline-earth elements where p-type functions, while not occupied in the ground-state atoms, are required for proper description of bonding in molecules. Here, the absence of p functions leads to descriptions... 

In situations where a complete description of the physical behavior of a system is unknown, scale-up approaches often involve the use of dimensionless groups, as described in Chapter 1. Unlike flow behavior in a blender, the flow behavior of powder through bins and hoppers can be predicted by a complete mathematical relationship. In light of this, analysis of powder flow in a bin or... 

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