Equality solving problem, factorization

A mole of a substance has a mass (in grams) equal to its formula mass, called the molar mass, which can be used as a factor in solving problems. 

Notice that each conversion factor equals 1. That is because the two quantities divided in any conversion factor are equivalent to each other— as in this case, where 4 quarters equal 1 dollar. Because conversion factors are equal to 1, they can be multiplied by other factors in equations without changing the validity of the equations. You can use conversion factors to solve problems through dimensional analysis. Dimensional analysis is a mathematical technique that allows you to use units to solve problems involving measurements. When you want to use a conversion factor to change a unit in a problem, you can set up the problem in the following way. 

Chapter 2. Chemistry and Measurements, looks at measurement and emphasizes the need to understand numerical relationships of the metric system. Significant numbers are discussed in the determination of final answers. Prefixes from the metric system are used to write equalities and conversion factors for problem-solving strategies. Density is discussed and used as a conversion factor. 

This is a relationship between unknown field g and two measured quantities, namely, the distance 5 and time t, provided that we neglect terms proportional to the square of the coefficient a and those of higher order. Besides, this equation contains three unknown parameters, namely, the position of the mass. so at the moment when we start to measure time, the initial velocity, vo, at this moment and, finally, the rate of change of the gravitational field, a, along the vertical. Thus, in order to solve our problem and find the field we have to perform measurements of the distance. s at four instants. If so is known, the number of these measurements is reduced by one. In modern devices the coefficient of the last term on the right hand side of Equation (3.14) has a value around 100 pGal and it is defined by calculations as a correction factor s(vo, g, t, a). In the case when we can let so equal to zero, it is sufficient to make measurements at two instances only. 

Presenting these various problems does not imply that they cannot be solved. Rather, it indicates that more work will be required before we have an adequate theory of factorization that is based on the symmetry of the simplex. It is a matter of individual judgment whether the advantages to be gained in pursuing this objective warrant the effort, or whether we should adopt another approach. This alternative scheme (Sect. IV), which primarily factors stereoisomerism rather than chirality, avoids ambiguity without recourse to a factorization rule, requires no symmetrization by the equalization of ligands, and is not restricted to the simplex. 

You first write the two numbers in terms of the same unknown or variable. If the first number is x, then the other number is 7 - x. How did I pull the 7 - x out of my hat Think about two numbers having a sum of 7. If one of them is 5, then the other is 7 - 5, or 2. If one of them is 3, then the other number is 7 - 3, or 4. Sometimes, when you do easy problems in your head, it s hard to figure out how to write what you re doing in math speak. So, if the two numbers are x and 7 - x, then you have to square each of them, add them together, and set the sum equal to 29. The equation to use is x2 + (7 - x)2 = 29. To solve this equation, you square the binomial, combine like terms, subtract 29 from each side, factor the quadratic equation, and then set each of the factors equal to 0. 

Measurement of power quality requires the use of proper instrumentation to suit the application. The user of the instrument must be well trained in the use and care of the instrumentation. The engineer should be knowledgeable in the field of power quality. Most importantly, the engineer should be safety conscious. All these factors are equally important in solving power quality problems. As indicated earlier, power quality work requires patience, diligence, and perseverance. It is very rare that the solution to a problem will present itself accidentally, although it does happen occasionally. Power quality work has its rewards. One that the author cherishes the most is the satisfaction of knowing that he has left clients happier than when he first met them. 

SOLUTION This problem can be solved multiple ways. The simplest method is to consider what happens after one half-life the Cp drops by a factor of 0.5. In other words, once one half-life has elapsed from the time of an IV bolus, Cp will be equal to 0.5CP°. After two half-lives, Cp drops by a factor of 0.25 (0.52). Continuing to five half-lives, we see that Cp has dropped by a factor of 0.0325 (0.5s). Therefore, after five half-lives, Cp from an IV bolus will be 0.0325Cp. 

The physical sciences use a problem-solving approach called dimensional analysis. Dimensional analysis requires conversion factors. A conversion factor is a numerator and a denominator that are equal to each other. Some conversion factors are... 

There are a variety of problem-solving strategies that you will use as you prepare for and take the AP test. Dimensional analysis, sometimes known as the factor label method, is one of the most important of the techniques for you to master. Dimensional analysis is a problem-solving technique that relies on the use of conversion factors to change measurements from one unit to another. It is a very powerful technique but requires careful attention during setup. The conversion factors that are used are equalities between one unit and an equivalent amount of some other unit. In financial terms, we can say that 100 pennies is equal to 1 dollar. While the units of measure are different (pennies and dollars) and the numbers are different (100 and 1), each represents the same amount of money. Therefore, the two are equal. Let s use an example that is more aligned with science. We also know that 100 centimeters are equal to 1 meter. If we express this as an equation, we would write ... 

The correct answer is (C). The total [OH ] of the solution will equal the hydroxide ion concentration of NaOH, added to that of Ba(OH)2. The biggest misleading factor here is that the barium hydroxide will donate two hydroxide ions to solution. That means the problem will be solved as ... 

If M is a diagonal matrix (experimental errors, but no correlations), M-1 is called the weight matrix, and the solution (Eqs. 21) is that for the weighted, uncorrelated least-squares problem. If M is introduced in Eqs. 21 as the unity matrix, Eqs. 21 solve the equally (unity-) weighted, uncorrelated problem. In this special case, the covariance matrices Eqs. 21b and 21e) depend on the measurements only through the common factor a2 (Eq. 21c). 

Perhaps you still can t picture the size of a 6 ml sample of water. Think, instead, of a 1-liter bottle of water. Let s calculate the number of moles of water molecules in 1 liter of water. One liter is equal to 1000 ml. We already said that one mole of water occupies about 18 ml. We need to divide 1000 ml by 18 ml to determine how many moles of water molecules there are in one liter of water. Let s solve the problem using the factor-label method, which you ll remember from Chapter 2. 

The phenomenon of anomalous scattering is extensively used in modem macromolecular crystallography to solve the phase problem. To understand how this is done, we need to return to the simple picture of X-rays reflecting from Bragg planes, where it makes no difference which side of the plane is the reflecting surface . This leads to two structure factors Fhki and F h differing only in the sign of their phase. The phase — a complex number - drops out because we measure intensities (/= F2 see above) and I k,i and are equal. 

The next thing to determine are the given and unknown substances. In this problem, ethane is the given substance and carbon dioxide is the unknown. Our task will be to convert from moles of ethane to moles of carbon dioxide. We will do this using the mole ratio, which is based on the equality of 2 moles C2Hg = 4 moles CO2. In simple terms, the ratio tells you that you will always produce twice as many moles of CO2 as the number of moles of ethane. To solve the problem, we will use the mole ratio as the conversion factor to change units from ethane to carbon dioxide. 

But there is a faster way to solve this problem, eliminating the need for the conversion from grams Fe to mole Fe. A conversion factor in terms of g Fe and atoms Fe can be assembled from the following equality that allows a one-step solution to the problem ... 

There is a lesson here Choose or develop the most efficient conversion factor to solve the problem. If you re converting grams to atoms, find the equality that relates grams and atoms for the conversion factor. If it s mole to atoms, find the connection between mole and atoms. Use the units g Fe, mole Fe, and atoms Fe to guide the correct use of the conversion factor. 

First, convert kilogram to gram 1.000 kg of NaCl equals 1,000 g of NaCl. A 2.5%(m/v) solution of sodium chloride contains 2.5 g of NaCl in 100 mL of solution. This relationship provides the conversion factor needed to solve the problem. The answer has two significant figures ... 

The conversion factor in Example 1.5 may be written as 4 qt/1 gal or 1 gal/ 4 qt, because both are equal to 1. However, only the first factor, 4 qt/1 gal, will give us the units we need to solve the problem. If we had set up the problem incorrectly, we would obtain... 

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