Measurable quantity

The relationship between the shape-dependent quantity H and the experimentally measurable quantity S originally was determined empirically [66], but a set of quite accurate XjH versus S values were later obtained by Niederhauser and Bartell [67] (see also Refs. 34 and 68) and by Stauffer [69],... 

The potential corresponding to the reversible overall process is the measurable quantity Vobs- If we know the work function for R, that is, the potential for e (in R) = e (in air), then Vobs - r is E for the process... 

The measurable quantity in a three dimensional scattering experiment is the differential cross section da (9)/dQ. This is defined as... 

Each physically measurable quantity has a corresponding operator. The eigenvalues of the operator tell the values of the corresponding physical property that can be observed... 

In addition to operators eorresponding to eaeh physieally measurable quantity, quantum meehanies deseribes the state of the system in terms of a wavefunetion F that is a funetion of the eoordinates qj and of time t. The funetion F(qj,t)p = gives the probability density for observing the eoordinates at the values qj at time t. For a many-partiele system sueh as the H2O moleeule, the wavefunetion depends on many eoordinates. For the H2O example, it depends on the x, y, and z (or r,0, and ([)) eoordinates of the ten... 

The operators F eorresponding to all physieally measurable quantities are Hermitian this means that their matrix representations obey (see Appendix C for a deseription of the bra I > and kef < notation used below) ... 

In quantum meehanies, physieally measurable quantities are represented by hermitian operators. Sueh operators R have matrix representations, in any basis spanning the spaee of funetions on whieh the R aet, that are hermitian ... 

Using the fact that the quantum mechanical coordinate operators q = x, y, z as well as the conjugate momentum operators (pj = px, Py, Pz are Hermitian, it is possible to show that Lx, Ly, and L are also Hermitian, as they must be if they are to correspond to experimentally measurable quantities. 

The digits in a measured quantity, including all digits known exactly and one digit (the last) whose quantity is uncertain. 

As with the rate of polymerization, we see from Eq. (6.37) that the kinetic chain length depends on the monomer and initiator concentrations and on the constants for the three different kinds of kinetic processes that constitute the mechanism. When the initial monomer and initiator concentrations are used, Eq. (6.37) describes the initial polymer formed. The initial degree of polymerization is a measurable quantity, so Eq. (6.37) provides a second functional relationship, different from Eq. (6.26), between experimentally available quantities-n, [M], and [1]-and theoretically important parameters—kp, k, and k. Note that the mode of termination which establishes the connection between u and hj, and the value of f are both accessible through end group characterization. Thus we have a second equation with three unknowns one more and the evaluation of the individual kinetic constants from experimental results will be feasible. 

We shall see in Sec. 9.9 that D is a measurable quantity hence Eq. (9.79) provides a method for the determination of an experimental friction factor as well. Note that no assumptions are made regarding the shape of the solute particles in deriving Eq. (9.79), and the assumption of ideality can be satisfied by extrapolating experimental results to c = 0, where 7=1. 

Since f is a measurable quantity for, say, a protein, and since the latter can be considered to fail into category (3) in general, the friction factor provides some information regarding the eilipticity and/or solvation of the molecule. In the following discussion we attach the subscript 0 to both the friction factor and the associated radius of a nonsolvated spherical particle and use f and R without subscripts to signify these quantities in the general case. Because of Stokes law, we write... 

This result, called the Clausius-Mosotti equation, gives the relationship between the relative dielectric constant of a substance and its polarizability, and thus enables us to express the latter in terms of measurable quantities. The following additional comments will connect these ideas with the electric field associated with electromagnetic radiation ... 

By an assortment of thermodynamic manipulations, the quantities dn/dp and [N (d G/dp )o] can be eliminated from Eq. (10.48) and replaced by the measurable quantities a, /3, and dn/dT the coefficients of thermal expansion, isothermal compressibility, and the temperature coefficient of refractive index, respectively. With these substitutions, Eq. (10.48) becomes... 

In an EXAFS experiment the measurable quantity is the absorption coefficient a of Equation (8.16). If Qq is the absorption coefficient in the absence of EXAFS, deduced from the steeply falling background shown in Figure 8.32, then x k), the fractional change of a due to EXAFS, is given by... 

Another property, used to compare the flammabiUty of textile fibers, is the limiting oxygen index (LOI). This measured quantity describes the minimum oxygen content (%) in nitrogen necessary to sustain candle-like burning. Values of LOI, considered a measure of the intrinsic flammabiUty of a fiber, are Hsted in Table 2 in order of decreasing flammabiUty. 

The external surface area of the filler can be estimated from a psd by summing the area of all of the equivalent spheres. This method does not take into account the morphology of the surface. It usually yields low results which provide Htde information on the actual area of the filler that induences physical and chemical processes in compounded systems. In practice, surface area is usually determined (5) from the measured quantity of nitrogen gas that adsorbs in a monolayer at the particle surface according to the BET theory. From this monolayer capacity value the specific surface area can be determined (6), which is an area per unit mass, usually expressed in m /g. 

Box Foa.ms. A measured quantity of the reaction mixture can be placed ia an open-topped crate or box and allowed to foam ia a free rise mode. The block is removed after gelling and is cut iato end use pieces after curiag. 

Injection Molding. Matched metal molds are used in the fabrication of plastic closures, specialty packages, and botde preforms. In conventional injection mol ding the plastic resin is melted in an extmder which forces a measured quantity or shot into a precision-machined chilled mold after which the nozzle of the extmder is withdrawn. 

Rotational Molding. Hodow articles and large, complex shapes are made by rotational mol ding, usuady from polyethylene powder of relatively low viscosity (57—59). The resin is in the form of a fine powder. A measured quantity is placed inside an aluminum mold and the mold is heated in an oven and rotated at low speed. The resin sinters and fuses, coating the inside of the mold. The mold is then cooled by water spray and the part solidifies, dupHcating the inside of the mold. 

Gas streams can be analy2ed for ammonia by bubbling a measured quantity of the gas through a boric acid solution to absorb the ammonia. The solution is then titrated against sulfuric acid. This analysis is applicable only if other constituents in the gas stream do not react with boric acid. 

The median particle diameter is the diameter which divides half of the measured quantity (mass, surface area, number), or divides the area under a frequency curve ia half The median for any distribution takes a different value depending on the measured quantity. The median, a useful measure of central tendency, can be easily estimated, especially when the data are presented ia cumulative form. In this case the median is the diameter corresponding to the fiftieth percentile of the distribution. 

T abular. A typical distributioa as measured by modem iastmmeatatioa can iaclude size information on tens of thousands and even millions of iadividual particles. These data can be Hsted ia a computer and then sorted iato a series of successive size iatervals, keeping track of the measured quantity, such as number, surface area, or mass, within each group. For narrow size distributions it may be sufficient to group the data ia linear iatervals, such as 0—1, 1—2, 2—3 p.m, etc, and then Hst the iatervals as a perceat value of the whole. 

Each of the two laws of thermodynamics asserts the existence of a primitive thermodynamic property, and each provides an equation connecting the property with measurable quantities. These are not defining equations they merely provide a means to calculate changes in each property. 

Thermodynamics provides a set of differential equations that interrelate the properties of PTT systems. Most of these properties are abstract, but the equations provide a limited number of connections with measurable quantities. One of the functions of thermodynamics is to maximize the return in useflil information for any investment in experiment. 

From the definition of a partial molar quantity and some thermodynamic substitutions involving exact differentials, it is possible to derive the simple, yet powerful, Duhem data testing relation (2,3,18). Stated in words, the Duhem equation is a mole-fraction-weighted summation of the partial derivatives of a set of partial molar quantities, with respect to the composition of one of the components (2,3). For example, in an / -component system, there are n partial molar quantities, Af, representing any extensive molar property. At a specified temperature and pressure, only n — 1) of these properties are independent. Many experiments, however, measure quantities for every chemical in a multicomponent system. It is this redundance in reported data that makes thermodynamic consistency tests possible. 

The obvious next step is to ehminate the partial-differential coefficients in favor of measurable quantities. 

Equations (4-34) and (4-35) are general expressions for the enthalpy and entropy of homogeneous fluids at constant composition as functions of T and P. The coefficients of dT and dP are expressed in terms of measurable quantities. 

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