Factor-Label Method

A very useful technique for converting physical quantities to alternative sets of units is the factor-label method. The units themselves are regarded as algebraic quantities subject to the rules of arithmetic, particularly to cancellation. To illustrate, let us calculate the speed of light in miles/sec, given the metric value c = 2.9979 X 10 m/sec. First, write this as an equation 

Multiplying Eq. (3.22) hy 1 in the form of the last expression, and cancelling the units m from numerator and denominator, we find 

We continue by multiplying the result by successive factors of 1, expressed in appropriate forms, namely 

we can continue our multiplication and cancellation sequence beginning with Eq. (3.24)  

As another example, let us calculate the number of seconds in a year. 

The units of each measurement are as important as the numeric value, and must always bq stated with th number. Moreover, we will use the units to help us in our calculations (Sec. 2.2). 

The units of a measurement are an integral part of the measurement. In many ways, thi as algebraic quantities, like x and y in mathematical equations. You must always state the measurements and calculations. 

For example, by. titles to arrive af )d, may be used other does so in ount in dollars. Let us look at n 

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How many dimes are there in 9.40 dollars We know that 

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To use the factor-label method, start with the quantity given (not a rate or ratio). Multiply that quantity by a factor, or more than one factor, until an answer with the desired units is obtained. Compare this method with that of Example 2.23(b). 

Since it is a quantitative property, it is often more useful for identification than a qualitative property like color or smell. Moreover, density determines whether an object will float in a given liquid. If the object is less dense than the liquid, it will float if it is more dense, it will sink. It is also useful to discuss density here for practice with the factor-label method of solving problems, and as such, it is often emphasized on early quizzes and examinations. 

In doing numerical density problems, you may always use the equation d = m/V or the same equation rearranged into the forms V = m/d or m = dV. You are often given two of these quantities and asked for the third. You will use the equation d = m/V if you are given mass and volume, but if you are given density and either of the others, you probably should use the factor-label method. That way, you need not manipulate the equation and then substitute you can solve immediately. 

Use of the equation requires manipulation of the basic equation followed by substitution. The factor-label method gives the same result by merely relying on the units. It also allows combination of the solution with other conversions. 

Then the coefficients in the balanced chemical equation can be used as factors in the factor-label method to convert from moles of one chemical to moles of any other in the equation ... 

In the example given in the text above, with 4 mol of nitrogen, it was not necessary to use the factor-label method the numbers were easy enough to work with. However, when the numbers get even slightly complicated, it is useful to use the factor-label method. Note that any of the following factors could be used for this equation, but we used the one above because it is the one that changes moles of hydrogen to moles of ammonia. 

Ans. Section 2.4, factor-label method Sec. 4.4, calculation of formula weights Sec. 4.5. changing moles to grams and vice versa Sec. 4.5, Avogadro s number and/or Sec. 7.2, balancing chemical equations. 

Heat capacities may be used as factors in factor-label method solutions to problems. Be aware that there are two units in the denominator, mass (or moles) and temperature change. Thus, to get energy, one must multiply the heat capacity by both mass (or moles) and temperature change. 

Which of the equations given for AH is used here Both. You can see from the factor-label method solution that the atomic weight divided into the molar heat capacity is the specific heat capacity while the mass divided by atomic weight is the number of moles. Thus, we have cither moles times molar heat capacity times change in temperature or mass times specific heat times change in temperature. 

Next, use the factor-label method to solve the problem. 

This problem can be easily solved using the factor-label method ... 

The gram-equivalent weight (GEW) of a base is the mass of the base (in grams) that will provide 1 mole of hydroxide ions in a reaction or that will react with 1 mole of H+ ions. This problem can be done by using the factor-label method. 

E) Do this problem by using the factor-label method (m.t. stands for metric tons, 1 m.t. = 103 kg). 

C) This problem can be solved by using the factor-label method. 100. g solution - 10. g solute = 90 g solvent (H20)... 

I m a firm believer in the use of units, factor-label method, dimensional analysis, whatever you call it. I KNOW I ve screwed up if my units are (g 1-butanol) 2/mole 1-butanol. 

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. 

Dalton s law Dalton s law states that in a mixture of gases (A + B + C. . . ) the total pressure is simply the sum of the partial pressures (the pressures associated with each individual gas), decomposition reactions Decomposition reactions are reactions in which a compound breaks down into two or more simpler substances, diamagnetism Diamagnetism is the repulsion of a molecule from a magnetic field due to the presence of all electrons in pairs, dilute Dilute is a qualitative term that refers to a solution that has a relatively small amount of solute in comparison to the amount of solvent, dimensional analysis Dimensional analysis, sometimes called the factor label method, is a method for generating a correct setup for a mathematical problem. 

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