Dimensional analysis conversion factors

If you know the percent yield, you can use it as a dimensional analysis conversion factor. Percent yield is grams actual yield per 100 grams theoretical yield. For example, assume that a manufacturer of magnesium hydroxide knows from experience that the percent yield is 81.3% from the production process. This can be used as either of 81.3 g (act) 100 g (theo)... 

This time you can use percentage (grams solute/100 grams solution) as a dimensional-analysis conversion factor to find the grams of Na2C03- Calculate that quantity first. 

Multiplication by unity (by one) does not change the value of an expression. If we represent one in a useful way, we can do many conversions by just multiplying by one. This method of performing calculations is known as dimensional analysis, the factor-label method, or the unit factor method. Regardless of the name chosen, it is a powerful mathematical tool that is almost foolproof. 

Use dimensional analysis to obtain correct conversion factors. Rearrange 1 mL = 10 L to give 1 = (lO L/mL), with mL in the denominator so it cancels mL in the numerator. 

It should be noted that a dimensional analysis of this problem results in one more dimensionless group than for the Newtonian fluid, because there is one more fluid rheological property (e.g., m and n for the power law fluid, versus fi for the Newtonian fluid). However, the parameter n is itself dimensionless and thus constitutes the additional dimensionless group, even though it is integrated into the Reynolds number as it has been defined. Note also that because n is an empirical parameter and can take on any value, the units in expressions for power law fluids can be complex. Thus, the calculations are simplified if a scientific system of dimensional units is used (e.g., SI or cgs), which avoids the necessity of introducing the conversion factor gc. In fact, the evaluation of most dimensionless groups is usually simplified by the use of such units. 

Chemistry is full of calculations. Our basic goal is to help you develop the knowledge and strategies you need to solve these problems. In this chapter, you will review the Metric system and basic problem solving techniques, such as the Unit Conversion Method. Your textbook or instructor may call this problem solving method by a different name, such as the Factor-Label Method and Dimensional Analysis. Check with your instructor or textbook as to for which SI (Metric) prefixes and SI-English relationships will you be responsible. Finally, be familiar with the operation of your calculator. (A scientific calculator will be the best for chemistry purposes.) Be sure that you can correctly enter a number in scientific notation. It would also help if you set your calculator to display in scientific notation. Refer to your calculator s manual for information about your specific brand and model. Chemistry is not a spectator sport, so you will need to Practice, Practice, Practice. 

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. 

Dimensional analysis, sometimes called the factor label (unit conversion) method, is a method for setting up mathematical problems. Mathematical operations are conducted with the units associated with the numbers, and these units are cancelled until only the unit of the desired answer is left. This results in a setup for the problem. Then the mathematical operations can efficiently be conducted and the final answer calculated and rounded off to the correct number of significant figures. For example, to determine the number of centimeters in 2.3 miles ... 

The simplest way to carry out calculations that involve different units is to use the dimensional-analysis method. In this method, a quantity described in one unit is converted into an equivalent quantity with a different unit by using a conversion factor to express the relationship between units ... 

The dimensional-analysis method gives the right answer only if the equation is set up so that the unwanted units cancel. If the equation is set up in any other way, the units won t cancel properly, and you won t get the right answer. Thus, if you were to multiply your height in inches by the incorrect conversion factor inches per meter, you would end up with an incorrect answer expressed in meaningless units ... 

The known information is the speed in kilometers per hour the unknown is the speed in miles per hour. Find the appropriate conversion factor, and use the dimensional-analysis method to set up an equation so the km units cancel. 

Dimensional analysis is a structured way to convert units. It involves a conversion factor that allows the units to be cancelled out when multiplied or divided. 

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... 

Using Dimensional Analysis and Conversion Factors in Problem Solving... 

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 ... 

Cross-multiplying and dividing gives 24 = lx and x equals 24 eggs upon solving. To use dimensional analysis, you would start with the unit and value given, and then multiply it by the conversion factor. The conversion factor has a numerator and a denominator. Place the unit that you want to cancel out in the denominator and the unit you are converting to in the numerator ... 

Dimensional analysis is the method to convert a number from one unit to another using a conversion factor. Conversion factors establish a relationship of equivalence in measurements between two different units. Examples of conversion factors are tabulated in Table 3.1 (metric prefixes) and Table 3.3 (common conversion factors). 

The previous calculation is an example of the use of the factor label method, also called dimensional analysis, in which a quantity is multiplied by a factor equal or equivalent to 1. The units inclnded in the factor are the labels. In the previous example, 9 is equivalent to 1 hour (h), and the calculation changes the number of hours worked to the equivalent number of dollars. To use the factor label method, first put down the given quantity, then multiply by a conversion factor (a rate or ratio) that will change the units given to the units desired for the answer. The factor may be a known constant or a value given in the problem. 

It is often necessary to convert results from one system of units to another. The most common way of converting units is by the unit factor method, more commonly called dimensional analysis. To illustrate the use of this method, we will look at a simple unit conversion. 

Dimensional analysis often uses conversion factors. Suppose you want to know how many meters are in 48 km. You need a conversion factor that relates kilometers to meters. You know that 1 km is equal to 1000 m. Because you are going to multiply 48 km by the conversion factor, you want to set up the conversion factor so the kilometer units will cancel out. 

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

Note the conversion factor is set up so that the unit meters cancels and the answer is in centimeters as required. When setting up a unit conversion, use dimensional analysis to check that the units cancel to give an answer in the desired units. And, always check your answer to be certain the units make sense. 

Convert SI Units In science, quantities such as length, mass, and time sometimes are measured using different units. A process called dimensional analysis can be used to change one unit of measure to another. This process involves multiplying your starting quantity and units by one or more conversion factors. A conversion factor is a ratio equal to one and can be made from any two equal quantities with different units. If 1,000 mL equal 1 L then two ratios can be made. 

The dimensional analysis provided in Equations 1.33-1.38 applies here as well, and the electron charge, e, needed here is equal to 4.803 x 10 l0StatC. To use Equation 1.40, a conversion factor must be added, as shown in Equation 1.41. 

Unit ananysis has other names, including the factor-label method, the conversion factor method, and dimensional analysis. 

To convert from one unit to another, we must have a conversion factor or series of conversion factors that relate two imits. The proper use of these conversion factors is called the factor-label method. This method is also termed dimensional analysis. 

Note that the alt factor serves to equate the analyte and titrant. To avoid a mistake in setting up the factor, it is helpful to remember that when you calculate the amount of analyte, you must multiply the amount of titrant by the dt ratio a comes first). Conversely, if you are calculating the amount of titrant (e.g., molarity) from a known amount of analyte titrated, you must multiply the amount of analyte by the tia ratio t comes first). The best way, of course, to ascertain the correct ratio is to always do a dimensional analysis to obtain the correct units. 

The use of conversion factors in calculations is known by various names, such as the factor-label method or dimensional analysis (because units represent physical dimensions). We use this method in quantitative problems throughout the text. 

Changing from one unit to another via conversion factors (based on the equivalence statements between the units) is often called dimensional analysis. We will use this method throughout our study of chemistry. 

We can convert from one system of units to another by a method called dimensional analysis using conversion factors. 

Dimensional analysis the process of using conversion factors to change from one unit to another. 

Using conversion factors A conversion factor used in dimensional analysis must accomplish two things it must cancel one unit and introduce a new one. While working through a solution, all of the units except the desired unit must cancel. Suppose you want to know how many meters there are in 48 km. The relationship between kilometers and meters is 1 km = 1000 m. The conversion factors are as follows. 

Dimensional analysis uses conversion factors to solve problems. 

Figure 10.3 A key to using dimensional analysis is correctly identifying the mathematical relationship between the units you are converting. The relationship shown here, 12 roses = 1 dozen roses, can be used to write two conversion factors. 

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