Measurements exact numbers

Students often cannot discriminate between exact numbers and measurements. Exact numbers are infiniteiy precise, regardiess of the aigebraic form in which they are written. Exact numbers are often encountered in definitions of units. 

The most reliable estimates of the parameters are obtained from multiple measurements, usually a series of vapor-liquid equilibrium data (T, P, x and y). Because the number of data points exceeds the number of parameters to be estimated, the equilibrium equations are not exactly satisfied for all experimental measurements. Exact agreement between the model and experiment is not achieved due to random and systematic errors in the data and due to inadequacies of the model. The optimum parameters should, therefore, be found by satisfaction of some selected statistical criterion, as discussed in Chapter 6. However, regardless of statistical sophistication, there is no substitute for reliable experimental data. 

Temperature and pressure represent two of the major parameters measured and evaluated in a monitoring system. All gas turbine engines are equipped with sensors of this type however, the exact number as well as their location varies considerably among manufacturers. 

Integers and exact numbers In multiplication or division by an integer or an exact number, the uncertainty of the result is determined by the measured value. Some unit conversion factors are defined exactly, even though they are not whole numbers. For example, 1 in. is defined as exactly 2.54 cm and the 273.15 in the conversion between Celsius and Kelvin temperatures is exact so 100.000°C converts into 373.150 K. 

The count of a rubber thread is the number of threads which measure exactly one inch across when placed side by side. See Cloth Count and Yam Count. 

Six Sherman rats (gender not specified) were exposed to graded concentrations of aniline vapor for 4 h and observed for 14 d post-exposure (Carpenter et al. 1949). The concentration that killed approximately half of the rats (exact number not stated) was 250 ppm. Concentrations were based upon empirical calculation and were not measured. An 8-h exposure to 550 ppm killed 82% of an unreported number of rats (Comstock and Oberst 1952, as cited in Oberst et al. 1956). Methemoglobinemia was the only pathologic change cited no further details were reported. 

We deal with two types of numbers in chemistry—exact and measured. Exact values are just that—exact, by definition. There is no uncertainty associated with them. There are exactly 12 items in a dozen and 144 in a gross. Measured values, like the ones you deal with in the lab, have uncertainty associated with them because of the limitations of our measuring instruments. When those measured values are used in calculations, the answer must reflect that combined uncertainty by the number of significant figures that are reported in the final answer. The more significant figures reported, the greater the certainty in the answer. 

A farmer needs to add exactly 4 quarts of weed killer to the fertilizer mix that he s preparing to spread over his field. Unfortunately, he has only a 3-quart container and a 5-quart container, not a 4-quart container. He can t just guess. The accuracy of his measurement is most important. How can he measure exactly 4 quarts with the two containers he has (There are several possible solutions, but the farmer wants the one that takes the fewest number of steps.)... 

The exact number of clay stacks having the above three categories of platelet stacking were measured by taking at least six different HRTEM images for each type of nanocomposite sample and the average distribution of the clay platelets were noted down (Fig. 38). This has been represented as the extent of exfoliation (B) in Table 9. It is apparent that as the level of exfoliation increases, the number of clay platelets per stack decreases and their effective surface area contribution increases. 

The turnover frequency, N, (commonly called the turnover number) defined, as in enzyme catalysis, as molecules reacting per active site in unit time, can be a useful concept if employed with care. In view of the problems in measuring the number of active sites discussed in 1.2.4, it is important to specify exactly the means used to express Q in terms of active sites. A realistic measure of such sites may be the number of surface metal atoms on a supported catalyst but in other cases estimation on the basis of a BET surface area may be the only readily available method. Of course, turnover numbers (like rates) must be reported at specified conditions of temperature, initial concentration or initial partial pressures, and extent of reaction. 

Some numbers are exact. These include k (3.14159. . . ), numbers arising from counting (e.g., the number of experimental determinations of an observed measurement), and numbers which involve a definition (the mass of one atom of 12C is exactly 12 u and the conversion of cm to m involves exactly 10 2 m/cm). 

An exact number is a small number that can be reproducibly determined by counting or one that is defined to a particular value. Exact numbers have infinite precision and significant figures. Exact numbers are not obtained using measuring devices. 

Exact numbers have no uncertainty and contain an infinite number of significant figures. These relationships are definitions. They are not measurements. 

Before you look more closely at matter, you need to know how much you can depend on measurements. How can you recognize when a measurement is trustworthy How can you tell if it is only an approximation For example, there are five Great Lakes. Are you sure there are five Is there any uncertainty associated with the value five in this case What about the number of millilitres in 1 L, or the number of seconds in 1 min Numbers such as these—numbers that you can count or numbers that are true by definition—are called exact numbers. You are certain that there are five Great Lakes (or nine books on the shelf, or ten students in the classroom) because you can count them. Likewise, you are certain that there are 1000 mL in 1 L, and 60 s in 1 min. These relationships are true by definition. 

Solution Measure the halfwidth of each peak and multiply this by the height of the peak. The exact numbers you calculate will depend on the scale of your ruler, but the three products should be essentially identical. ... 

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