Using Ratios

We encounter and use ratios regularly. We discuss the speed of our cars in miles per hour and buy our fruit at 1.09 a pound. Most of us use such ratios intuitively. If we look at the way we do this, we can deduce a formal set of rules to apply to chemical observations and situations. Let s say your grocery bill indicates that you paid 4.45 for a 5.0-pound bag of apples. How much did you pay per pound Although you may not think about it, you obtain the answer by creating an appropriate ratio and completing some simple arithmetic. 

The notation per means for each of the specified units. Price per pound is the cost of me pound. Miles per hour tell the distance traveled in me hour. This is useful because one is a convenient number in multiplication. We obtain this type of information by creating and simplifying a ratio. 

With the information we have about the bag of apples, we have another option to create a ratio. We could obtain the number of pounds of apples per dollar. 

For this example, such a manipulation may not be particularly useful—the store probably doesn t sell 1.1 pound bags of apples. Still, it provides useful insight. When we form the ratio and carry out the indicated arithmetic, the result tells us how much of the numerator is equivalent to one unit of the denominator. So, in general, given two equivalent quantities A = B, we can write either of two ratios  

Example Problem 1.4 shows how this type of manipulation fits into a problem solving strategy with non-chemistry examples before we look at it in a chemistry context. 

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The molecular weight distribution of LLDPE resins is usually characterized in industry by the ratios of melt indexes measured in the same apparatus using different loads (2.16, 10.16, and 21.6 kg). The commonly used ratios are melt flow ratio, MFR) and I q/I2. Both of these ratios... 

The dimensionless relations are usually indicated in either of two forms, each yielding identical resiilts. The preferred form is that suggested by Colburn ran.s. Am. In.st. Chem. Eng., 29, 174—210 (1933)]. It relates, primarily, three dimensionless groups the Stanton number h/cQ, the Prandtl number c Jk, and the Reynolds number DG/[L. For more accurate correlation of data (at Reynolds number <10,000), two additional dimensionless groups are used ratio of length to diameter L/D and ratio of viscosity at wall (or surface) temperature to viscosity at bulk temperature. Colburn showed that the product of the Stanton number and the two-thirds power of the Prandtl number (and, in addition, power functions of L/D and for Reynolds number <10,000) is approximately equal to half of the Fanning friction fac tor//2. This produc t is called the Colburn j factor. Since the Colburn type of equation relates heat transfer and fluid friction, it has greater utility than other expressions for the heat-transfer coefficient. 

With the exception of fillers, the raw materials used in two-component urethanes are all liquids. The two components have an isocyanate side and a polyol side. The raw materials are combined in various ways in order to produce with liquids that are combined at a 1 1 volume ratio, preferably, thus keeping the dispensing equipment as simple as possible. Fixed ratios of 3 2 or 1 2 are also commonly used. Ratios other than these are possible, but require the use of a... 

In healthcare settings, the risk of harmful effects is higher with intravenous epinephrine than with epinephrine administered through other routes of injection [2, 6]. Dosing errors have been attributed to the common practice of using ratios such as 1 10,000 to express the epinephrine concentrations therefore, use of mass concentration such as 0.1 mg in 1 ml is recommended [23]. 

Now set up and complete a table of amounts. Because O2 is limiting, it will be completely consumed, so its change is -8.00 mol. Knowing this, we calculate all other changes in amounts using ratios of stoichiometric coefficients ... 

Broman D, Naf C, Rolfif C, Zebuhr Y, Fry B, Hobbie J. 1992. Using ratios of stable nitrogen isotopes to estimate bioaccumulation and flux of polychlorinated dibenzo-p-dioxins (PCDDs) and dibenzofurans (PCDFs) in two food chains from the northern Baltic. Environ Toxicol Chem 11 331-345. 

A second experiment similar to the one above was set up except that 3 ml of extract was used from leachates prepared by using ratios of 5, 10 and 15 g of tops and roots to 100 ml of distilled water. 

To regulate the air flow rate with respect to the fuel gas flow rate, we can use ratio control. Fig. 10.5 illustrates one of the simplest implementations of this strategy. Let s say the air to fuel gas flow rates must be kept at some constant ratio... 

The value of this ratio is characteristic of the reaction order. Table 3.1 contains a tabulation of partial reaction times for various rate expressions of the form r = kCAn as well as a tabulation of some useful ratios of reaction times. By using ratios of the partial reaction times based on experimental data, one is able to obtain a quick estimate of the reaction order with minimum effort. Once this estimate is in hand one may proceed to use a more exact method of determining the reaction rate parameters. 

A measure of the breadth of the molecular mass distribution is given by the ratios of molecular mass averages. The most commonly used ratio Mw/Mn — H, is called the polydispersity index. Wiegand and Kohler discuss the determination of molecular masses (weights) and their distributions in Chapter 6. 

Anionically prepared hydroxy-terminated PBd was reacted with AlEt3 to form the corresponding aluminum alkoxide macroinitiator, capable of initiating the polymerization of L-lactide [117]. Using ratios [PBd-OH]/[AlEt3] between 1 and 6, reaction temperatures between 70 and 120 °C and maintaining the conversion of the lactide polymerization below 90%, products with narrow molecular weight distribution were obtained. 

Galego and Arroyo [14] described a simultaneous spectrophotometric determination of OTC, hydrocortisone, and nystatin in the pharmaceutical preparations by using ratio spectrum-zero crossing derivate method. The calculation was performed by using multivariate methods such as partial least squares (PLS)-l, PLS-2, and principal component regression (PCR). This method can be used to resolve accurately overlapped absorption spectra of those mixtures. 

The procedure for cross-multiplication is straightforward and relatively easy. The challenge in solving word problems using ratios is in the set-up of the proportion. Take care to keep all terms in order. Remember that two ratios are being compared and that the order of the ratio set-up has meaning. 

Typical values of the correlation coefficients are 0.5 for two isotopic ratios equal to unity and quickly tend to unity for large ratios. Correlations between errors due to counting statistics are clearly minimized by using ratios with values <1. 

Use ratios to compare the reference reaction with the known enthalpy change AHi) to the reaction with the unknown enthalpy change (AH2). 

Alternatively, you can compare the rate law equation for each experiment using ratios. This method is useful when the relationship between concentration and rate is not immediately obvious from the data. 

By using ratio the number of edge (C7) atoms to the number of (111) face (Cg) atoms (Fig. 12) to define an effective particle size , the selectivity of n-butane hydrogenolysis as a function of particle size for the two surfaces could be plotted and compared to selectivities measured on supported Ir catalysts This comparison is shown in Fig. 13. Clearly, the results on single... 

In practice, many feedforward control systems are implemented by using ratio control systems, as discussed in Chap. 8. Most feedforward control systems are installed as combined feedforward-feedback systems. The feedforward controller takes care of the large and frequent measurable disturbances. The feedback controller takes care of any errors that come through the process because of inaccuracies in the feedforward controller or other unmeasured disturbances. Figure 11.4d shows the block diagram of a simple linear combined fe forward-/ feedback system. The manipulated variable is changed by both the feedforward controller and the feedback controller. 

Show that you can obtain the same result for SAQ 7.4 by using ratios. 

Nitric acid has a 65 25 fertilizer explosive end use ratio. The production of ammonium nitrate makes up nearly all of these two uses. The other 10% is made into miscellaneous compounds adipic acid, nitroglycerin, nitrocellulose, ammonium picrate, trinitrotoluene, nitrobenzene, silver nitrate, and various isocyanates. 

Ammonium nitrate has an 82 18 fertilizer explosive end use ratio. The chief use of ammonium nitrate until after World War II was as an explosive. Although it still accounts for more than 75% of all explosives, its major use is now as a fertilizer because of its high nitrogen content, 33.5% N. 

The commonly used ratios of admixture to cement for the purpose of shrinkage compensation are 9-11 (admixture) to 91-89 (cement). At these ratios the properties of CSA concrete are similar to Portland cement concretes of similar mix proportions. At admixture dosages exceeding 11% however, concrete workability and strength decrease, while expansion and air entrainment increase. When expansion is umestrained and exceeds 0.3%, strength is reduced [74]. 

Note that the systematic error in x/ y is better then the precision of both X (0.3/15=2%) and y (0.3/10=3%). This is why geochemists like to use ratios instead of concentrations, as ratios can reduce systematic errors. 

Similar conclusions regarding the relative indoor and outdoor concentrations have been reached in studies of office and commercial buildings. For example, Ligocki et al. (1993) measured indoor and outdoor concentrations of particles and their components at five museums in southern California. The indoor-to-outdoor ratios of particle mass varied over a wide range, depending to a large extent on the ventilation and filtration systems in use. Ratios varied from 0.16 to 0.96 for particles with diameters less than 2.1 /xm and from 0.06 to 0.3 for coarse particles with diameters greater than this. 

Detonation (and Explosion), Fugacity of Products of. Cook (1958), p 381 describes a method of calculating fugacities and the calcn of equilibrium concentrations using ratios of fugacities... 

The growing success of fluorescent sensors in the biomedical sciences is at least partly due to the ability to cancel out microenvironmental fluctuations such as local optical paths, quenching agents, and degrees of sensor incorporation by using ratioing of two optical channels. Most commonly, this means that two excitation wavelengths are used separately, one of which yields ion-sensitive fluorescence and... 

The second useful ratio is the instantaneous concentration of a given monomer, /i or /2 ... 

However, bed voidage depends on dji) (Dixon, 1988). It can be proved that for dji) values lower than 0.1, the bed voidage can be considered practically constant. Indeed, it is common practice to use ratios of dJD lower than 0.1, and therefore, the bed voidage can be actually considered the same for common fixed beds. For instance, a bed consisting of particles with 2 mm diameter should have a diameter greater that 2 cm, which leads to a dji) ratio with a value of 0.1 maximum. Consequently, the bed voidage is critical only in laboratory experiments. In Figure 3.39, Dixon s correlations are presented for spheres and cylinders, in the case of dji) < 0.4 (Dixon, 1988). For cylinders, dp is equal to the diameter of a sphere of equal volume. 

This disorder must be suspected if THAldo is low (< 5 pg/24 h) without manifestations of disordered cortisol metabolism, such as AME syndrome. A useful ratio is THAldo(xl00) Fs, which is typically <2. 

To aid in grouping the materials, methods were worked out using ratios between the concentrations of pairs of elements and by computer-produced correlation plots between such pairs. Scandium and iron were highly correlated and of a distinct ratio for a given geographic area and clay type. 

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