Problem Solving Using Unit Conversion

246 Write the equality and conversion factors for each of the following pairs of units  

Each day, we make choices about what we do or what we eat, often without thinking about the risks associated with these choices. We are aware of the risks of cancer from smoking or the risks of lead poisoning, and we know there is a greater risk of having an accident if we cross a street where there is no light or crosswalk. 

A basic concept of toxicology is the statement of Paracelsus that the dose is the difference between a poison and a cure. To evaluate the level of danger from various substances, natural or synthetic, a risk assessment is made by exposing laboratory animals to the substances and monitoring the health effects. Often, doses very much greater than humans might ordinarily encounter are given to the test animals. 

The process of problem solving in chemistry often requires the conversion of a given quantity in one unit to a needed quantity that has a different unit. We will show the problem-solving process by analyzing the problem to 

Suppose you need to convert 18.2 mm to meters. By separating the problem into units, you would identify the given as millimeters and the needed as meters. 

Now write a plan that will convert the given unit to the needed unit 

Now look for the connection between millimeters and meters. From Table 2.6, use the equality that relates meters and millimeters, which is 1 m = 1000 mm. From this equality, write two conversion factors. Be sure to include the units in all the quantities of the 

To set up the calculation, select the conversion factor that cancels the given unit, which is millimeter. Complete the calculation and ronnd off to give the needed answer with the correct number of significant figures. 

---

In this section, we will introduce one of the two common methods for solving problems. (You will see the other method in Chapter 5.) This is the Unit Conversion Method. It will be very important for you to take time to make sure you fully understand this method. You may need to review this section from time to time. The Unit Conversion Method, sometimes called the Factor-Label Method or Dimensional Analysis, is a method for simplifying chemistry problems. This method uses units to help you solve the problem. While slow initially, with practice it will become much faster and second nature to you. If you use this method correctly, it is nearly impossible to get the wrong answer. For practice, you should apply this method as often as possible, even though there may be alternatives. 

Most of the applications of artificial intelligence in chemistry so far have not involved numerical computation as a primary goal. Yet there are aspects of the AI approach to problem-solving which have relevance to computation. In scientific computation, one could view the knowledge base as the set of equations, input variable values, and unit conversions relevant to the problem, and the inference engine the numerical method used to solve the equations. This paper describes such a software system,... 

Dimensional analysis is a technique for solving problems that involve units or conversions that is taught in many engineering schools. It is a very useful technique in some areas of the emergency services, especially in EMS, where drug and fluid administration rates need to be calculated. 

The dimensional-analysis method and the use of ballpark checks are techniques that will help you solve problems of many kinds, not just unit conversions. Problems sometimes seem complicated, but you can usually sort out the complications by analyzing the problem properly ... 

The conversion factor problem-solving technique has been used throughout this book, especially in the units on moles and stoichiometry. These problem solutions are generally in a format like this ... 

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 ... 

Dimensional analysis often uses conversion factors to solve problems that involve units. A conversion factor is a ratio of equivalent values. 

You will see many different types of unit conversions in this chapter, but they can all be worked using the same general procedure. To illustrate the process, we will convert 2 teaspoons to milliliters and solve the problem of how much medicine to give the little boy described above. 

To convert one unit to another, we must set up a conversion factor or series of conversion factors that relate two units. The proper use of these conversion factors is referred to as the factor-label method. This method is used either to convert from one unit to another within the same system or to convert units from one system to another. It is a very useful problem-solving tool. 

Am. We examine the balanced equation N2 + SHj -> 2NH3. To solve this type of problem, first convert the mass of nitrogen to moles, then solve for the corresponding numbers of moles of hydrogen or ammonia using the balancing coefficients as unit conversion factors. Then finally convert back to mass, as follows ... 

Many kinds of numerical problems in everyday life as well as in science can be solved by extending the use of conversion factors and cancellation of units beyond unit conversion. The use of density expressed as mass per unit volume is a simple example of such an extension. Density provides the connection between mass and volume. Given that the density of lead is 11.4 g/cm , you can find the mass in grams of a piece of lead of known volume or the volume of a piece of lead of known mass. If the known information is that a piece of lead has a volume of 25.0 cm and the unknown information is its mass, the problem is set up and solved as follows... 

A measured quantity consists of a number and a unit. Conversion factors are used to express a quantity in different units and are constructed as a ratb of equivalent quantities. The problem-solving approach used in this text usually has four parts (1) devise a plan for the solution, (2) put the plan into effect in the calculatbns, (3) check to see if the answer makes sense, and (4) practice with similar problems. 

We will use these ideas as we consider unit conversions in this chapter. Then we will have much more to say about problem solving in Chapter 3, where we will start to consider more complex problems. 

The units in the answer, as well as the value of the answer, are incorrect. The unit cm /in. is not correct, and, based on our knowledge that centimeters are smaller than inches, we know that 44.7 cm cannot be equivalent to 114 in. In solving problems, always check if the final units are correct, and consider whether or not the magnitude of the answer makes sense. In this case, our mistake was in how we used the conversion factor. We must invert it. 

We can diagram conversions using a solution map. A solution map is a visual outline that shows the strategic route required to solve a problem. For unit conversion, the solution map focuses on units and how to convert from one unit to another. The solution map for converting from inches to centimeters is ... 

In Examples 2.8 and 2.9, you will find this problem-solving procedure applied to unit conversion problems. The procedure is summarized in the left column, and two examples of applying the procedure are shown in the middle and right columns. This three-column format is used in selected examples throughout this text. It allows you to see how a particular procedure can be applied to two different problems. Work through one problem first (from top to bottom) and then examine how the same procedure is applied to the other problem. Recognizing the commonalities and differences between problems is a key part of problem solving. 

When solving multistep xmit conversion problems, we follow the preceding procedure, but we add more steps to the solution map. Each step in the solution map should have a conversion factor with the units of the previous step in the denominator and the units of the following step in the numerator. For example, suppose we want to convert 194 cm to feet. The solution map begins with cm, and we use the relationship 2.54 cm = 1 in to convert to in. We then use the relationship 12 in. = 1 ft to convert to ft. 

Solve unit conversion problems using the problemsolving strategies outlined in Section 2.6. 

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. 

---