Ratios, Fractions, and Percents

When you analyze data, you may be asked to compare measured quantities. Or, you may be asked to determine the relative amounts of elements in a compound. Suppose, for example, you are asked to compare the molar masses of the diatomic gases, hydrogen (H2) and oxygen (O2). The molar mass of hydrogen gas equals 2.00 g/mol the molar mass of oxygen equals 32.00 g/mol. The relationship between molar masses can be expressed in three ways a ratio, a fraction, or a percent. 

Q The mass of one lime would be one-twelfth the mass of one dozen limes. In a crystal of table salt (sodium chloride), each sodium ion is surrounded by chloride ions, yet the ratio of sodium ions to chloride ions is one one. The formula for sodium chloride is NaCI. 

Notice that the ratio 1 16 is the smallest integer (whole number) ratio. It is obtained by dividing both numbers in the ratio by the smaller number, and then rounding the larger number to remove the digits after the decimal. The ratio of the molar masses is one to sixteen. In other words, the ratio indicates that the molar mass of diatomic hydrogen gas is sixteen times smaller than the molar mass of diatomic oxygen gas. 

Ratios are often expressed as fractions in simplest form. A fraction is a quotient of two numbers. To express the comparison of the molar masses as a fraction, place the molar mass of hydrogen over the molar mass of oxygen as follows  

In this case, the simplified fraction is calculated by dividing both the numerator (top of the fraction) and the denominator (bottom of the fraction) by 2.00. This fraction yields the same information as the ratio. That is, diatomic hydrogen gas has one-sixteenth the mass of diatomic oxygen gas. 

Notice that the ratio 1 16 is the smallest integer (whole number) ratio. 

The comparison between molar mass of hydrogen gas and the molar mass of oxygen gas described on the previous page can also be expressed as a percent by taking the fraction, converting it to decimal form, and multiplying by 100 as follows. 

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Immediately quench the reaction by the addition of sodium sulfite (Na2S03) to provide a 2X molar excess over the initial amount of periodate added. Purify the oxidized antibody by gel filtration using a desalting resin. The chromatography buffer is 0.1 M sodium phosphate, 0.15M NaCl, pH 7.2. To obtain efficient separation between the oxidized antibody and excess periodate, the sample size applied to the column should be at a ratio of no more than 5 percent sample volume to the total column volume. Collect 0.5 ml fractions and monitor for protein at 280 nm. 

A PERCENT IS a ratio that represents a part-to-a-whole as a number out of 100. Fractions are used to represent a part-to-a-whole, but fractions can have many different denominators. A percent is like a fraction whose denominator is always 100. The relationship 50 out of 100 is written as 50%. The relationship 1 out of 2 can also be written as 50%. That is because the fraction is equal to Fractions and decimals can be written as percents, and vice versa, as we ll see in Chapter 11. 

Downcomer layout The total tower area of a 6.5-foot tower is 33.2 ft8. In the top section, downcomer area is 8.6 ft2, i.e., the downcomer is about 3.8/33.2 11,5 percent of tower area. This is a relatively small fraction, and there is little incentive to reduce it further. On the other hand, in the bottom section the downcomer occupies 3.5/33.2 = 25.6 percent of the tower area. There is an incentive to reduce this area by using a sloped downcomer. A 2 1 top-to-bottom area ratio is appropriate (1). Therefore, downcomer top area in the bottom section is 8,5 ft2, while downcomer bottom area is 3.5/2 = 4.25 ft . In summary... 

Solvent Crystallization. Two processes, one utilizing acetone (Armour) and the other employing methanol (Emery), are well known. Using the latter, fatty acid is dissolved in 90 percent aqueous methanol in a 1 2 acid/ solvent ratio by the application of heat. The resulting solution is then cooled to H 5°C in a multi-tubular crystallization chamber equipped with scrapers for efficient heat transfer. The crystallized fatty acids are removed by filtration. The filter cake is melted and stripped of any residual solvent to yield the refined stearin fraction, and then the liquid stearin is converted to flakes or powder by a variety of processes, for example, chill roller, and the like. The mother liquor from the filtration is stripped to obtain the olein fraction. The separated stearin and olein fractions have a variety of commercial applications in both the chemical and food processing industries. 

Novolaks were prepared using a phenol-to-formaldehyde molar ratio of 4 1 with 5 mole percent of H2SO4 added as a catalyst. Typically, 47 g liquid phenol (91.7% assay), 3 g paraformaldehyde, and 30 mL water plus the required acid catalyst were added to a three-neck, 250-mL round bottom flask. The flask was fitted with a reflux condenser and stirrer. The mixture was refluxed for 2 to 4 hours with the oil bath at 115 °C then, the mixture was neutralized with 50% (w/w) NaOH and the excess phenol removed by steam distillation for 5 to 6 hours. The remaining viscous oily residue was washed repeatedly with boiling water. A novolak with the P/N fraction was prepared as described above with 1 1 by volume phenol and P/N fraction and half of the amount of formaldehyde. Initial wood-gluing testing with this novolak indicates wood failure rather than glueline failure. 

They are similar in that they both are ratios of each component to the total and in that the total is specified in each case (100% for percent composition and 1 mol for mole fraction). They differ in the units used to calculate them—moles for mole fraction and masses for percent composition. 

To apply the mass balance to the Sound, the details of the sand distribution were studied in a small area along the north shore. A series of short cores were taken along the tracks shown in Fig. 3. The top 20 cm of these samples was wet sieved and the shell removed with dilute HCl. The material that passes a 1-mm screen but is retained on a 0.064-mm screen is defined as sand. The volume of material retained on the fine screen was measured after a standard settling time (1 hr) and was compared to the initial volume of sediment to define the sand-mud ratio. The weight-percent sand was found to be about 1.2 times the sand-mud ratio. Harbors along the north shore in the study area have mud floors with a lower sand content than the offshore sediments. South of the harbors, however, the curves of constant sand fraction run nearly north and south... 

The incorporation of hydrophobic monomer into the polymers was determined by a comparison of the absorptivity of the hydrophobe unit in the polymer to the corresponding model. With the assumption that the absorptivity of the hydrophobe in the polymer ik the same as that of the model, the ratio of polymer absorptivity to model gives the weight fraction of hydrophobe (jch) in the polymer (equation 3). Converting Xh to mole fraction and dividing by the feed content in mole percent of hydrophobic monomer (My) results in the hydrophobe incorporation. The expressions for incorporation into RAM and HRAM polymers are given by equations 4 and 5, respectively. 

This solution is prepared by dissolving 10 g of salt in 100 mL of solution. The mass units in a % w/v solution are grams, but the volume units are milliliters of solution, not grams of solution. Therefore the units do not cancel, and the % w/v is not a true fraction or percent. Rather than writing, say, 5% NaCl w/v, it would be more meaningful to express the concentration as 5 g NaCl/100 mL (of solution). Mathematically, it is handled in the same way as % w/w or % v/v problems, either by the factor-label or ratio methods. 

Having validated the mechanism on ammonia-oxygen flames, the yield of NO from nitrogen doped CH4-air flames was examined. Both NH3 and NO doping were investigated. Only post-flame NO concentrations were measured. These are compared with calculations of the full kinetics and with adiabatic equilibrium calculations. The calculated profiles indicate the complexity of the NO dynamics in these flames. The temperature and major species profiles in the undoped flames had been studied in earlier work( ). Three near stoichiometric methane-air flames having initial equivalence ratios(0) of 0.8, 1.0 and 1.2 are diluted with less than 5 volume percents of NH3 or NO. In this section NO concentration is expressed both as a mole fraction and as a fraction of the total nitrogen concentration ... 

The p chart is used to present percent defective. A sample of 100 to 1000 items would be examined for each data point and the percent defective would be calculated and plotted on the chart. This calculation involves taking the number of defectives divided by the number in the sample and then multiplying by 100 to get the percent. The graph can be in fractions or percent. The p chart is one of the control charts for attributes that can have a variable sample size due to the ratio calculation from the data. The thing to remember is that when making a p chart the sample size should not vary much more than 25 percent from the largest sample size to the smallest sample size. 

The test method details procedures for the production of a liquified gas, distillate fractions and residuum of standardized quality on which analytical data can be obtained, and the determination of yields of the above fractions by both mass and volume. From the above information a graph of temperature versus mass-percent distilled can be produced. This distillation curve corresponds to a laboratory technique which is defined at 15/5 (15 theoretical plate column, 5 1 reflux ratio) or TBP (true boiling point). 

The problem of explosion of a vapor cloud is not only that it is potentially very destructive but also that it may occur some distance from the point of vapor release and may thus threaten a considerable area. If the explosion occurs in an unconfined vapor cloud, the energy in the blast wave is generally only a small fraction of the energy theoretically available from the combustion of all the material that constitutes the cloud. The ratio of the actual energy released to that theoretically available from the heat of combustion is referred to as the explosion efficiency. Explosion efficiencies are typically in the range of 1 to 10 percent. A value of 3 percent is often assumed. 

The effect of pulsating flow on pitot-tube accuracy is treated by Ower et al., op. cit., pp. 310-312. For sinusoidal velocity fluctuations, the ratio of indicated velocity to actual mean velocity is given by the factor /l + AV2, where X is the velocity excursion as a fraction of the mean velocity. Thus, the indicated velocity would be about 6 percent high for velocity fluctuations of 50 percent, and pulsations greater than 20 percent should be damped to avoid errors greater than 1 percent. Tne error increases as the frequency of flow oscillations approaches the natural frequency of the pitot tube and the density of the measuring fluid approaches the density of the process fluid [see Horlock and Daneshyar, y. Mech. Eng. Sci, 15, 144-152 (1973)]. 

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