Measured quantities, definitions

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 time difference (delay) between the measured quantity and the measurement result is called the inertial error. A definition- is the error due to iner tia (mechanical, thermal, etc.) of the parts of a measuring instrument. In ventilation equipment the critical component in the measuring chain, from the dynamic point of view, is often the sensor or the measuring transducer (probe). 

Depending on the type of relationships between the measured quantity and the measurand (analytical quantity) it can be distinguished (Danzer and Currie [1998]) between calibrations based on absolute measurements (one calibration is valid for all1 on the basis of the simple proportion y = b x, where the sensitivity factor b is a fundamental quantity see Sect. 2.4 Hula-nicki [1995] IUPAC Orange Book [1997, 2000]), definitive measurements (b is given either by a fundamental quantity complemented by an empirical factor or a well-known empirical (transferable) constant like molar absorption coefficient and Nernst factor), and experimental calibration. 

In general, a thermometer is called primary if a theoretical reliable relation exists between a measured quantity (e.g. p in constant volume gas thermometer) and the temperature T. The realization and use of a primary thermometer are extremely difficult tasks reserved to metrological institutes. These difficulties have led to the definition of a practical temperature scale, mainly based on reference fixed points, which mimics, as well as possible, the thermodynamic temperature scale, but is easier to realize and disseminate. The main characteristics of a practical temperature scale are both a good reproducibility and a deviation from the thermodynamic temperature T which can be represented by a smooth function of T. In fact, if the deviation function is not smooth, the use of the practical scale would produce steps in the measured quantities as function of T, using the practical scale. The latter is based on ... 

For the most part, in this book we use SI dimensions and units (SI stands for le systeme international d unites). A dimension is a name given to a measurable quantity (e.g., length), and a unit is a standard measure of a dimension (e.g., meter (for length)). SI specifies certain quantities as primary dimensions, together with their units. A primary dimension is one of a set, the members of which, in an absolute system, cannot be related to each other by definitions or laws. All other dimensions are secondary, and each can be related to the primary dimensions by a dimensional formula. The choice of primary dimensions is, to a certain extent, arbitrary, but their minimum number, determined as a matter of experience, is not. The number of primary dimensions chosen may be increased above the minimum number, but for each one added, a dimensional constant is required to relate two (or more) of them. 

Electrochemical interfaces are sometimes referred to as electrified interfaces, meaning that potential differences, charge densities, dipole moments, and electric currents occur. It is obviously important to have a precise definition of the electrostatic potential of a phase. There are two different concepts. The outer or Volta potential ij)a of the phase a is the work required to bring a unit point charge from infinity to a point just outside the surface of the phase. By just outside we mean a position very close to the surface, but so fax away that the image interaction with the phase can be ignored in practice, that means a distance of about 10 5 — 10 3 cm from the surface. Obviously, the outer potential i/ a U a measurable quantity. 

The mathematical definition of the steady-state combustion efficiency in Eq (3) is reformulated into practical and measurable quantities. 

If many products are formed, we sometimes have many possible definitions such as one based on carbon products, hydrogen products, a particular functional group, etc. It is also common to base these on weights or volumes if these are measured quantities in a particular process. Another and quite different definition is to take the ratio of particular desired and undesired products, a quantity that goes from zero to infinity this definition becomes ambiguous for other than simple reaction sets, and we will not use it here. 

Lavoisier had formed a clear, consistent, and suggestive mental picture of chemical changes. He thought of a chemical reaction as always the same under the same conditions, as an action between a fixed and measurable quantity of one substance, having definite and definable properties, with fixed and measurable quantities of other substances, the properties of each of which were definite and definable. 

There was, then, a certain degree of accuracy in the alchemical description of the processes we now call chemical changes, as being the removal of the outer properties of the things which react, and the manifestation of their essential substance. But there is a vast difference between this description and the chemical presentment of these processes as reactions between definite and measurable quantities of elements, or compounds, or both, resulting in the re-distribution, of the elements, or the separation of the compounds into their elements, and the formation of new compounds by the re-combination of these elements. 

Quantities, their symbols, units of measure and definitions... 

The remaining errors in the data are usually described as random, their properties ultimately attributable to the nature of our physical world. Random errors do not lend themselves easily to quantitative correction. However, certain aspects of random error exhibit a consistency of behavior in repeated trials under the same experimental conditions, which allows more probable values of the data elements to be obtained by averaging processes. The behavior of random phenomena is common to all experimental data and has given rise to the well-known branch of mathematical analysis known as statistics. Statistical quantities, unfortunately, cannot be assigned definite values. They can only be discussed in terms of probabilities. Because (random) uncertainties exist in all experimentally measured quantities, a restoration with all the possible constraints applied cannot yield an exact solution. The best that may be obtained in practice is the solution that is most probable. Actually, whether an error is classified as systematic or random depends on the extent of our knowledge of the data and the influences on them. All unaccounted errors are generally classified as part of the random component. Further knowledge determines many errors to be systematic that were previously classified as random. 

Known Variables — Controllable In any experimental situation, cettain conditions are held constant during the course of the experiment. A batch of copolymer may be made with a measured quantity of catalyst, at a given reaction temperature, and reacted for a certain definite time. The vessel is free of water, die monomers are charged in a definite proportion, and any other conditions that may affect the result are held fixed. The next portion of the experiment may involve one of these conditions controlled at a new level while the others remain fixed. Proper design of experimental programmes presupposes ability to control the important factors so that the variation due to these factors can be calculated. In Chapter VI, experimental designs for various situations are coveted. 

Another method consisted of charging the shothole with a definite quantity of ammonium nitrate and pouring a measured quantity of fuel into it through a funnel. 

According to the modern convention, measurable quantities are expressed in SI (System Internationale) units and replace the centimetre-gram-second (cgs) system. In this system, the unit of length is a metre (m, the unit of mass is kilogram (kg) and the unit of time is second (s). All the other units are derived from these fundamental units. The unit of thermal energy, calorie, is replaced by joule (1 J = 107 erg) to rationalize the definition of thermal energy. Thus, Planck s constant... 

The velocity was measured manometrically, definite quantities of nitric oxide and oxygen being allowed to stream into a vacuous vessel, which was provided with a bromnaphthalene manometer, protected from the oxides of nitrogen by an air-buffer. 

Notice that the dose has a strict definition of energy per unit mass of the absorber and, in principle, can be measured for a given radiation at a certain energy in a specific material. The equivalent dose is a relative unit in that a radiation weighting factor is applied to a measured quantity. The dose can be measured from ionization in an electronic radiation detector the equivalent dose must take into account the type of radiation causing the ionization. 

And if i ye0o< is replaced with measurable quantity which is the definition Aristotle provides of to p ye0oq just a few lines prior, then these definitions become ... 

Because Aristode s definition of body can be illuminated by focussing on the nature of peyeGoq, in the remainder of this chapter, I will leave peyeGoq in the definition of body rather than what could in principle replace it, namely measurable quantity . Hence, the definition of a body that emerges from book V of the Metaphysics is ... 

The partial derivatives in this equation have definite physical meanings and are measurable quantities. For liquids they are related to two commonly tabulated properties ... 

Quantity Definition SI units or their derived equivalents Equipment use to measure the quantity... 

You would never express the mass of a lump of gold, like the one in Figure 5.11, in atomic mass units. You would express its mass in grams. How does the mole relate the number of atoms to measurable quantities of a substance The definition of the mole pertains to relative atomic mass, as you learned in section 5.1. One atom of carbon-12 has a mass of exactly 12 u. Also, by definition, one mole of carbon-12 atoms (6.02 x 1023 carbon-12 atoms) has a mass of exactly 12 g. 

Dimensions. The fundamental measurables of a unit system in physics—those which are defined through operational definitions. All other measurable quantities in physics are defined through mathematical relations to the fundamental quantities. Therefore any physical measurable may be expressed as a mathematical combination of the dimensions. See operational definitions. 

When we write F = ma, we are expressing a relation between measurable quantities, one which holds underspecified conditions, qualifications and limitations. There s more to it than the equation. One must, for example, specify that all measurements are made in an inertial frame, for if they aren t, this relation isn t correct as it stands, and must be modified. Many physical laws, including this one, also include definitions. This equation may be considered a definition of force, if m and a are previously defined. But if F was previously defined, this may be taken as a definition of mass. But the fact that this relation can be experimentally tested, and possibly be shown to be false (under certain conditions) demonstrates that it is more than a mere definition. 

One can argue that for the application of 1) and 2) only trends in orbital nature and orbital populations matter so that any quantum chemical method (HF or DFT) or any way of calculating charges (Mulliken or natural orbital populations) suffices for this purpose as long as it is applied in a consistent way. One can further argue that the silylium cation character of a given silyl ion has only to be determined for the gas phase. Once this has been done, other molecular properties of the ion in question can be calculated and compared with the corresponding measured values for either gas or solution phase. In this way, a purely theoretical definition of the silylium ion character could be adjusted and extended to measurable quantities. 

It follows also from Eq. (1-3) that there exist electron states having discrete or definite values for energy (or, states with discrete values for any other observable). This can be proved by construction. Since any measured quantity must be real, Eq. (1-3) suggests that the operator 0 is Hermitian. We know from mathematics that it is possible to construct eigenstates of any Hermitian operator. However, for the Hamiltonian operator, which is a Hermitian operator, eigenstates are obtained as solutions of a differential equation, the time-independent Schroedinger equation. 

In cases where comparisons have been made, theoretical data obtained by digital simulations are always in agreement with those from analytical solutions of the diffusion-kinetic equations within the limit of experimental error of quantities which can be measured. A definite advantage of simulation over the other calculation techniques is that it does not require a strong mathematical background in order to learn and to use the technique. A very useful guide for the beginner has recently appeared (Britz, 1981). 

Either Eq. 5F or 6F can be considered to be a definition of the transfer coefficient a. Equation 5F relates it directly to the measured quantity (ar /alog i), while Eq. 6F can be regarded as a formal definition, indicating that the transfer coefficient a, is simply the reciprocal Tafel slope in dimensionless form. 

While gas phase work on the h5q5erpolarizability of small molecules has been relatively free of problems concerned with the definitions of measured quantities and their formal relationship to computed quantities, the same cannot be said about solution studies of rather larger organic species. It is the latter that possess the very large nonlinear response functions that are of greatest interest. The prototype system for such studies has been 4-nitroaniline (pNA) and this review is mainly concerned with the relation between the measurements, in vacuo and in solution, of the hyperpolarizabilities of pNA and the closely related molecule, MNA (2-methyl, 4-nitroaniline) to ab initio and DFT calculations of these quantities. 

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