Measurement proportionality

For a qualitative analysis it is sufficient to be able to apply a test which has a known sensitivity limit so that negative and positive results may be seen in the right perspective. Where a quantitative analysis is made, however, the relation between measurement and analyte must obey a strict and measurable proportionality only then can the amount of analyte in the sample be derived from the measurement. To maintain this proportionality it is generally essential that all reactions used in the preparation of a sample for measurement are controlled and reproducible and that the conditions of measurement remain constant for all similar measurements. A premium is also placed upon careful calibration of the methods used in a quantitative analysis. These aspects of chemical analysis are a major pre-occupation of the analyst. 

Adequate quality assuranee measures, proportionate to the safety classifieation of structures, systems and components. 

Diffusivity measures the tendency for a concentration gradient to dissipate to form a molar flux. The proportionality constant between the flux and the potential is called the diffusivity and is expressed in m /s. If a binary mixture of components A and B is considered, the molar flux of component A with respect to a reference plane through which the exchange is equimolar, is expressed as a function of the diffusivity and of the concentration gradient with respect to aji axis Ox perpendicular to the reference plane by the fpllqvving relatipn 6 /... 

The existence of this situation (for nonporous solids) explains why the ratio test discussed above and exemplified by the data in Table XVII-3 works so well. Essentially, any isotherm fitting data in the multilayer region must contain a parameter that will be found to be proportional to surface area. In fact, this observation explains the success of Ae point B method (as in Fig. XVII-7) and other single-point methods, since for any P/P value in the characteristic isotherm region, the measured n is related to the surface area of the solid by a proportionality constant that is independent of the nature of the solid. 

We have seen that in a metal the atoms are close-packed, i.e. each metal atom is surrounded by a large number of similar atoms (often 12, or 8). The heat required to break up 1 mole of a metal into its constituent atoms is the heat of atomisation or heat of sublimation. Values of this enthalpy vary between about 80 and 800 kJ. for metals in their standard states these values indicate that the bonds between metal atoms can vary from weak to very strong. There is a rough proportionality between the m.p. of a metal and its heat of atomisation. so that the m.p. gives an approximate measure of bond strength. 

Sensitivity is the change in signal per unit change in the amount of analyte and is equivalent to the proportionality constant, k, in equations 3.1 and 3.2. If ASa is the smallest increment in signal that can be measured, then the smallest difference in the amount of analyte that can be detected is... 

The proportionality between the concentration of chromophores and the measured absorbance [Eqs. (6.8) and (6.9)] requires calibration. With copolymers this is accomplished by chemical analysis for an element or functional group that characterizes the chromophore, or, better yet, by the use of isotopically labeled monomers. 

The solute molecular weight enters the van t Hoff equation as the factor of proportionality between the number of solute particles that the osmotic pressure counts and the mass of solute which is known from the preparation of the solution. The molecular weight that is obtained from measurements on poly disperse systems is a number average quantity. 

This shows that Schlieren optics provide a means for directly monitoring concentration gradients. The value of the diffusion coefficient which is consistent with the variation of dn/dx with x and t can be determined from the normal distribution function. Methods that avoid the difficulty associated with locating the inflection point have been developed, and it can be shown that the area under a Schlieren peak divided by its maximum height equals (47rDt). Since there are no unknown proportionality factors in this expression, D can be determined from Schlieren spectra measured at known times. 

Friction and Adhesion. The coefficient of friction p. is the constant of proportionality between the normal force P between two materials in contact and the perpendicular force F required to move one of the materials relative to the other. Macroscopic friction occurs from the contact of asperities on opposing surfaces as they sHde past each other. On the atomic level friction occurs from the formation of bonds between adjacent atoms as they sHde past one another. Friction coefficients are usually measured using a sliding pin on a disk arrangement. Friction coefficients for ceramic fibers in a matrix have been measured using fiber pushout tests (53). For various material combinations (43) ... 

Open-Loop versus Closed-Loop Dynamics It is common in industry to manipulate coolant in a jacketed reacdor in order to control conditions in the reacdor itself. A simplified schematic diagram of such a reactor control system is shown in Fig. 8-2. Assume that the reacdor temperature is adjusted by a controller that increases the coolant flow in proportion to the difference between the desired reactor temperature and the temperature that is measured. The proportionality constant is K. If a small change in the temperature of the inlet stream occurs, then depending on the value or K, one might observe the reactor temperature responses shown in Fig. 8-3. The top plot shows the case for no control (K = 0), which is called the open loop, or the normal dynamic response of the process by itself. As increases, several effects can be noted. First, the reactor temperature responds faster and faster. Second, for the initial increases in K, the maximum deviation in the reactor temperature becomes smaller. Both of these effects are desirable so that disturbances from normal operation have... 

In this expression, p(H) is referred to as the prior probability of the hypothesis H. It is used to express any information we may have about the probability that the hypothesis H is true before we consider the new data D. p(D H) is the likelihood of the data given that the hypothesis H is true. It describes our view of how the data arise from whatever H says about the state of nature, including uncertainties in measurement and any physical theory we might have that relates the data to the hypothesis. p(D) is the marginal distribution of the data D, and because it is a constant with respect to the parameters it is frequently considered only as a normalization factor in Eq. (2), so that p(H D) x p(D H)p(H) up to a proportionality constant. If we have a set of hypotheses that are exclusive and exliaus-tive, i.e., one and only one must be true, then... 

This measurement ean be aeeomplished by using a meehanieal system or various types of eleetronie systems. All of these systems are expensive and in many eases require repeated ealibration. The meehanieal system (Figure 19-14) is a three-gear, phase-related system whieh measures the displaeement between two gears and the proportionate shaft twist. A third gear is situated so that any variations other than shaft twist will oeeur in the first two gears. This signal is used to eliminate errors eaused by these variations. 

It should be noted that low-loss spectra are basically connected to optical properties of materials. This is because for small scattering angles the energy-differential cross-section dfj/dF, in other words the intensity of the EEL spectrum measured, is directly proportional to Im -l/ (E,q) [2.171]. Here e = ei + iez is the complex dielectric function, E the energy loss, and q the momentum vector. Owing to the comparison to optics (jqj = 0) the above quoted proportionality is fulfilled if the spectrum has been recorded with a reasonably small collection aperture. When Im -l/ is gathered its real part can be determined, by the Kramers-Kronig transformation, and subsequently such optical quantities as refraction index, absorption coefficient, and reflectivity. 

The simplest mode of IGC is the infinite dilution mode , effected when the adsorbing species is present at very low concentration in a non-adsorbing carrier gas. Under such conditions, the adsorption may be assumed to be sub-monolayer, and if one assumes in addition that the surface is energetically homogeneous with respect to the adsorption (often an acceptable assumption for dispersion-force-only adsorbates), the isotherm will be linear (Henry s Law), i.e. the amount adsorbed will be linearly dependent on the partial saturation of the gas. The proportionality factor is the adsorption equilibrium constant, which is the ratio of the volume of gas adsorbed per unit area of solid to its relative saturation in the carrier. The quantity measured experimentally is the relative retention volume, Vn, for a gas sample injected into the column. It is the volume of carrier gas required to completely elute the sample, relative to the amount required to elute a non-adsorbing probe, i.e. 

In the structure with all the surfactant molecules located at monolayers, the volume fraction of surfactant should be proportional to the average surface area times the width of the monolayer divided by the volume, i.e., Ps (X Sa/V. The proportionality constant is called the surfactant parameter [34]. This is true for a single surface with no intersections. In our mesoscopic description the volume is measured in units of the volume occupied by the surfactant molecule, and the area is measured in units of the area occupied by the amphiphile. In other words, in our model the area of the monolayer is the dimensionless quantity equal to the number of amphiphiles residing on the monolayer. Hence, it should be identified with the area rescaled by the surfactant parameter of the corresponding structure. 

Defining ethane, ethylene and acetylene to have bond orders of 1, 2 and 3, the constant a-is found to have a value of approximately 0.3 A. For bond orders less than 1 (i.e. breaking and fonning single bonds) it appears that 0.6 A is a more appropriate proportionality constant. A Mulliken style measure of the bond strength between atoms A and B can be defined from the density matrix as (note that this involves the elements of the product of the D and S matrices). 

The constant of proportionality is K/fx, where K is the permeability and is Newtonian viscosity. The dielectric properties of the resin are also measured using sensors. These measurements were correlated with viscosity and used as a part of the FRTM control system. 

Figure 54.7 illustrates several rise/run measurements for a constant Angle A . Unless Angle A changes, an increase in rise results in a proportionate increase in run. This relationship allows the alignment calculations to be made without using the theoretical offset value and its corresponding run. 

The secant modulus measurement is used during the designing of a product in place of a modulus of elasticity for materials where the stress-strain diagram does not demonstrate a linear proportionality of stress to strain or E is difficult to locate. 

The diversity and the effectiveness of the means for coping with deviations from proportionality attest the great interest in x-ray emission spectrography and the resourcefulness of the analytical chemist. Only representative references can be cited. The methods used to ensure reliable results in the face of the three classes of deviations are described briefly below. The obvious measure of separating the element to be determined from the matrix is omitted. 

---