Sample Stage Calculation

An aqueous waste stream containing 3.25% by weight phenol is to be extracted with one-third its volume of methylene chloride to produce a raffinate without more than 0.2% phenol. How many stages are required  

The operating line is determined on a solute free basis as follows  

per 100 g of water feed, the amount of phenol removed is  

At a volumetric feed rate of solvent equal to one-third the feed, and a specific gravity of 1.31 for methylene chloride, the weight ratio of solvent to feed is  

The phenol removed from the 100 g of water (3.16 g) must be in the extract, which contains 45.1 g of methylene chloride  

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SPM force curves are acquired by moving the tip toward the sample and recording the cantilever deflection as a function of the so-called Z position. Cantilever deflection is directly proportional to the force exerted on the sample by the tip. If the spring constant (fc) of the cantilever is known, the force can be calculated. The Z position defines the distance from the sample to the piezo, to which the base of the cantilever is attached (Figure 9.13). By convention the closest point of approach by the piezo is designated as zero on the x-axis. Note that for some instruments the piezo is attached to the sample stage and thus moves the sample up toward the tip however, this does not change the analysis. 

Note that for the cases Fig. 6. lc, d, depending on the difference between the 0 and 1 systems, it may be better to go with higher-order stage calculations. For example, for case (d) one can construct an intermediate MB bridging r0 and r, so that the relationship of Tg and r B r are both described by case (c), and then perform overlap sampling calculations between 0 and MB as well as 1 and MB. 

There is usually an implicit assumption in this calculation that the standard deviations are the same in each of the treatment groups. Generally speaking this assumption is a reasonable one to make as the effect of treatment will be to change the mean with no effect on the variability. We will say a little more about dealing at the analysis stage with situations where this is not the case in a later section. The sample size calculation, however, is also easily modified, if needed, to allow unequal standard deviations. 

The power of a study where the primary endpoint is time-to-event depends not so much on the total patient numbers, but on the number of events. So a trial with 1000 patients with 100 deaths has the same power as a trial with only 200 patients, but with also 100 deaths. The sample size calculation for survival data is therefore done in two stages. Firstly, the required number of patients suffering events is... 

Do we need to revisit the sample size calculation at some interim stage in the trial ... 

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