Straight line Surface

A variety of experimental data has been found to fit the Langmuir equation reasonably well. Data are generally plotted according to the linear form, Eq. XVn-9, to obtain the constants b and n from the best fitting straight line. The specific surface area, E, can then be obtained from Eq. XVII-10. A widely used practice is to take to be the molecular area of the adsorbate, estimated from liquid or solid adsorbate densities. On the other hand, the Langmuir model is cast around the concept of adsorption sites, whose spacing one would suppose to be characteristic of the adsorbent. See Section XVII-5B for an additional discussion of the problem. 

Flere is the volume of gas required to saturate the monolayer, V the total volume of gas adsorbed, P the sample pressure, P the saturation vapour pressure and C a constant related to the enthalpy of adsorption. The resulting shape of the isothemi is shown plotted in figure Bl.26.6 for C = 500. A plot of P/V(P - Pq) against P/Pq should give a straight line having a slope (C - )/y C and an intercept The BET surface area is... 

The polymers described so far have relatively flexible main chains which can result in complex confonnations. In some cases, tliey can double back and cross over tliemselves. There are also investigations on polymers which are constrained to remain in a confonnation corresponding, at least approximately, to a straight line, but which have amphiphilic properties tliat ensure tliat tliis line is parallel to tire water surface. Chiral molecules are one example and many polypeptides fall into tliis class [107]. Another example is cofacial phtlialocyanine polymers (figure C2.4.9). 

Fig. 1.7 Variation of the value of (pg as the centre of the adsorbed atom moves along a straight line parallel to the surface of a solid and distant Co from it. (---------) For a real surface (-----) for an ideal surface.

The t and a.-methods, the nature of which was explained in Chapter 2, may be used to arrive at a value of the micropore volume. If the surface of the solid has standard properties, the t-plot (or a,-plot) corresponding to the isotherm of the nonporous powder in Fig. 4.11(a) will be a straight line passing through the origin (cf. curve (i) of Fig. 4.11(6)) and having a slope proportional to the specific surface of the powder. For the microporous powder which yields the isotherm (iii).of Fig. 4.11(a), the t-plot (or Oj-plot) will have the form of curve (iii) of Fig. 4.11(6) the linear branch of this curve will be parallel to curve (i), since it corresponds to the area of the outside of the particles which is identical with that of the nonporous parent particles. 

If the isotherm of G on the solid under test is identical in shape with that on the reference sample, then its x,-plot will be a straight line passing through the origin, and having a slope equal to the ratio >4(test solid)/>4(reference solid). Since >4(reference solid) is known, the specific surface of the test solid is obtainable at once. 

Earlier we noted that a response surface can be described mathematically by an equation relating the response to its factors. If a series of experiments is carried out in which we measure the response for several combinations of factor levels, then linear regression can be used to fit an equation describing the response surface to the data. The calculations for a linear regression when the system is first-order in one factor (a straight line) were described in Chapter 5. A complete mathematical treatment of linear regression for systems that are second-order or that contain more than one factor is beyond the scope of this text. Nevertheless, the computations for... 

The obtained nonpenetration condition (3.185) is local as compared to (3.173), (3.176) since this condition is considered only at the curve Tc. Let us recall that we have assumed that the angle between the crack surface and the axis is small. By this assumption, the small deflection x — Px has been neglected in (3.173), (3.176). It is of importance to deduce (3.177) from (3.185). Indeed, if is transformed into a vertical crack, then Cy is a straight line, a y) = 0, and from (3.185) we obtain the nonpenetration condition... 

Make the nozzle point upward at an angle such that a straight line projected from the nozzle would hit the liquid surface A to % of the way across the tank diameter. The idea is to promote top to bottom turnover. However, tilt the nozzle slightly to the left to promote a slight swirl effect. Aim the nozzle at a point about V3 of a radius off-center. 

Fig. 10. A schemalic Zisman plot for a given solid specimen. When the cosine of the static advancing contact angle is plotted against the surface tension for a series of apolar liquids against a test solid, a straight line results. Its extrapolation to cost = 1 yields the critical surface tension of the solid.

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