Proportionality, direct inverse

Most of the calculations that are to be made in a course in college chemistry involve simple direct proportionality or inverse proportionality. These calculations can be carried out correctly and easily by using arguments similar to the foregoing. 

The best results seem reasonable predictions, but somewhat unsafe, and therefore as mentioned in the previous sections logical proportionality (directly or inversely) ate imported to make the results more dependable. 

As seen in the figures, the quantity of initiator required to yield a given conversion varies directly with the frequency factor for an initiator with a specified activation energy and inversely with the activation energy for an initiator of specified frequency factor. The relationships do not show a linear proportionality between the reactor conversion and square root of the inlet initiator concentration. 

Direct and indirect competition formats, illustrated in Figure 1, are widely used for both qualitative and quantitative immunoassays. Direct competition immunoassays employ wells, tubes, beads, or membranes (supports) on to which antibodies have been coated and in which proteins such as bovine semm albumin, fish gelatin, or powdered milk have blocked nonspecific binding sites. Solutions containing analyte (test solution) and an analyte-enzyme conjugate are added, and the analyte and antibody are allowed to compete for the antibody binding sites. The system is washed, and enzyme substrates that are converted to a chromophore or fluorophore by the enzyme-tracer complex are added. Subsequent color or fluorescence development is inversely proportionate to the analyte concentration in the test solution. For this assay format, the proper orientation of the coated antibody is important, and anti-host IgG or protein A or protein G has been utilized to orient the antibody. Immunoassays developed for commercial purposes generally employ direct competition formats because of their simplicity and short assay times. The price for simplicity and short assay time is more complex development needed for a satisfactory incorporation of the label into the antibody or analyte without loss of sensitivity. 

The reduction of silver chloride, precipitated in the presence of excess chloride ion, yielded the S-shaped curve typical of an autocatalyzed reaction (James, 25). The initial reaction rate, measured in terms of the reciprocal of the time required to complete 5 % of the total reaction, varied directly as the hydroxylamine concentration and inversely as the chloride ion concentration when the latter was relatively large. The specific surface of the freshly prepared precipitate, as measured by dye adsorption, decreased with aging, and the reaction rate decreased proportionately. 

The proportionality constant, jx, is the viscosity. When the velocity gradient is cm/s cm, or simply s , we call it the shear rate because of the units of inverse time. Often, the lowercase Greek letter gamma with an overdot, y, is used to represent the shear rate, dvxidy. The shear stress of x-directed moment in the y direction, Xyx, typically has units of g/cm s. Hence, viscosity has cgs units of g/cm s, or poise. In the SI system, the unit of Pa s is often used. Note that 1 Pa s = 10 poise. In this chapter we will have a lot to say about viscosity and how it varies for different materials in their molten or liquid states. 

Mathematically, this means that the volume is directly proportional to the inverse pressure, or V -jL To remove the proportionality sign and replace it with an equal sign, V must be multiplied by a proportionality constant, k. 

You will note that the dependence of volume on the absolute temperature is a direct proportionality, whereas the volume is inversely proportional to the pressure. The nature of these two relations is illustrated in Figure 8-3. 

It should be noted that the relationship between the final signal output and concentration of the analyte (dose-response) may be one of direct or inverse proportionality, and is dependent on the specific assay format. In addition, a number of different reporter enzymes may be used (e.g., horseradish peroxidase, alkaline phosphatase, p-galactosidase), along with a number of different signaling systems (e.g., substrates that yield chromogenic or fluorescent or chemiluminescent products, activation of signaling enzymes, amplification by biotin-avidin system or polymerase chain reaction). 

The second parameter Moi also scales <5 of dipolar compounds and can be evaluated from the absorption spectrum using the Strickler-Berg treatment [338]. Reorganization of this relation results in Eq. (26) [338-340], showing the inverse proportionality between (M01)2 and excitation energy Em, but a direct relation exists between (Mqi)2 and the integral over the absorption band. 

The impact of incorrect definitions provided at pre-university level remains throughout the courses, confirming that the internalisation of the first definitions encountered reaches deeper levels than subsequent attempts to correct them. The most important ones concern the proportionality concepts [40]. Students interpret as direct proportionality any case in which the dependent variable increases as the independent variable increases (52), and as inverse proportionality any case in which the dependent variable decreases as the independent variable increases (53), regardless of the type of dependence, even when they refer explicitly to the equation expressing the dependence (54) and even the absence of dependence may be viewed as direct proportionality (55) ... 

Henry s Law states that the amount of gas dissolved by a given liquid, with which, it does not combine chemically, is directly proportional to the partial pressure of the gas if the pressure of a gas is doubled then the amount of gas physically dissolved in the solution is doubled. The constant which converts the proportionality to an equality in the Henry s Law equation is called the Henry s Law constant this constant is the solubility coefficient of the gas in the particular solution. The solubility coefficient varies with the nature of the gas and liquid, the presence of solutes in the liquid, and inversely with the temperature. Thus at a constant pressure, but under hypothermic conditions, more gas can be dissolved in a given amount of fluid (tissue). 

When analysing the results of a neutron experiment we shall need to relate the number of neutrons seen in a detector of small area, dA, as it is positioned at different scattering angles around the sample, see Fig. 2.1. The detector subtends a small solid angle, di2, at the sample. The number of neutrons per unit time, or flux, scattered to the detector, Jf, is directly proportional to the incident neutron flux, Ji. It is also proportional to the size of the detector but inversely proportional to the square of the distance, d, between the sample and the detector, also df2= 6A / d. The differential cross section, dcr/d/2, is the constant of proportionality. 

The constants of proportionality here contain the molecular correlation time, Tc, in addition to a number of known physical constants, and B is now taken to signify inversion rather than saturation of B. In principle, if tc were known, this would directly provide a measure of rAB- Whilst it is possible to determine this (such as from relaxation time measurements) this is rarely done in practice, and it is more common to use a known internal distance as a reference and avoid the need for such laborious measurements. If the NOE between reference nuclei X and Y of internuclear separation rxy is also measured then ... 

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