{"id":6602,"date":"2018-02-20T17:59:20","date_gmt":"2018-02-20T17:59:20","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/suny-albany-chemistry\/?post_type=chapter&p=6602"},"modified":"2018-02-22T22:05:13","modified_gmt":"2018-02-22T22:05:13","slug":"dimensional-analysis","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/suny-albany-chemistry\/chapter\/dimensional-analysis\/","title":{"raw":"1.7 Dimensional Analysis","rendered":"1.7 Dimensional Analysis"},"content":{"raw":"[embed]https:\/\/youtu.be\/QLmRvuF5oxw[\/embed]\r\n\r\nDimensional Analysis<\/strong>\r\n\r\nDimensional analysis involves using units as a guide for problem solving.\u00a0 Dimensional analysis can be used to solve many types of problems, including chemistry problems.\u00a0 In this early stage of the course, the dimensional analysis problems may not have anything to do with chemistry, but the same approach can be used to solve chemistry problems.\r\n\r\nIn dimensional analysis we start with an equality, an equation expressing the relationship between two different units or quantities, and make it into a conversion factor.\u00a0 For example if one wants to know how many seconds are in 1.3 minutes, we need an equality between seconds and minutes.\u00a0 Equation 1 shows the equality that one minute (min) is equal to 60 seconds (s).\r\n\r\nEquation 1. An equality equation between minute and second.\u00a0 <\/strong>\r\n\r\nIn this example we use minutes and seconds as the equality because we want to find out how many seconds are in 1.3 minutes, but there are countless equalities that can be expressed as has been done in Equation 1.\r\n\r\nOnce an equality is known, the next step is to turn the equality into a conversion factor as shown in Calculation 1.\u00a0 Notice that in the conversion factor, 60 seconds is in the numerator and 1 minute is in the denominator.\u00a0 The number in the denominator<\/strong> (on the bottom) is the unit that is being canceled out and the numerator<\/strong>, the number on top, is the new unit that is desired.\u00a0 To mathematically solve the expression we first multiply by the number in the numerator (60 seconds) and then divide by the number in the denominator (1 minute).\u00a0 If there were multiple conversion factors present, this process would be repeated.\r\n\r\nCalculation 1. Converting 1.3 minutes to seconds using dimensional analysis.\u00a0 <\/strong>\r\n\r\nIt is worth pointing out that since 1 minute and 60 seconds are equal as shown in Equation 1, placing them in a conversion factor with 60 seconds in the numerator and 1 minute as the denominator is placing two equal values in the fraction.\u00a0 The mathematical result of dividing two equal things is 1.\u00a0 Therefore multiplying a value, in this case 1.3 minutes, by a conversion factor with two equal values in a fraction is mathematically valid because it is the same as multiplying the number by 1.\u00a0 Multiplying any value by 1 gives the same value.\u00a0 Said more simply, all conversion factors must be made from two equal values expressed as a fraction.\r\n\r\nIn the example above only one conversion was needed because there is a well-known equality between seconds and minutes.\u00a0 Now let\u2019s consider the case that we wish to know how many minutes are in 3.4 days.\u00a0 It\u2019s possible that one might know how many minutes are in a day, but for this example, let\u2019s assume that we only know the more common equalities shown in Equation 2.\u00a0 These equalities are 24 hours (h) is one day and 60 minutes is one hour.\r\n\r\nEquation 2. Equalities between day, hour and minute\u00a0 <\/strong>\r\n\r\nIn this example there is no direct equality between minutes and days (although one can be known or calculated).\u00a0 If there is no direct equality then multiple conversion factors are needed.\u00a0 In this case 3.4 days can be converted to hours and then hours can be converted to minutes.\u00a0 This plan is outlined in Figure 1 below.\u00a0 Note that although a plan is not required to solve the problem, it can be useful when you start doing dimensional analysis problems.\r\n\r\nFigure 1.\u00a0 Plan to convert days to minutes. <\/strong>\r\n\r\nIn Calculation 2, dimensional analysis is used to do the conversion planned in Figure 1. The answer should be reported based on the number of significant figures in 3.4 days (2 significant figures) because the equality between days and hours and minutes and hours are exact numbers.\u00a0\u00a0 When converting between two systems of measurement, like centimeters to inches, the conversion is a measured number.\u00a0 When converting within a single system of measurement, like centimeters to meters, the conversions are exact.\u00a0 In this case, the time units are in a single system of measurement so the conversion factors are exact numbers and should be ignored when determining significant figures.\r\n\r\n\u00a0<\/strong>\r\n\r\nCalculation 2. Converting 3.4 days to minutes using dimensional analysis.\u00a0 <\/strong>\r\n\r\nTo solve the equation in Calculation 2, first multiply 3.4 by 24, then divide by 1, then multiply by 60 and finally divide by 1.\u00a0 Said another way, we multiply by the top then divide by the bottom of each conversion factor, one at a time, for as many conversion factors as are needed.\u00a0 Another way to obtain the result from Calculation 2 is to multiply all numbers on the numerator first and then divide each number on the denominator. You may practice doing this same math two different ways to make sure the result come out the same. You may quickly realize that it is not necessary to multiply and divide by 1 on the calculator as you get the same number.\r\n\r\nSome of the equalities, used to make conversion factors, mentioned above are well known, such as 1 hour equals 60 minutes.\u00a0 However, sometimes equalities are not as obvious.\u00a0 Below a series of less obvious conversion factors is presented.\u00a0 If apples cost $1.99 per pound, an equality between dollars and pounds of apples exists as shown in Equation 3.\r\n\r\nEquation 3: An equality between dollars and pounds.<\/strong>\r\n\r\nIf 85% of students love chemistry, than equality exists between students and students who love chemistry as shown in Equation 4.\u00a0 Notice that 100 students is chosen for one part of the equality because then the percent number of students, in this case 85 students who love chemistry, can be used for the other part of the equality.\u00a0 Also notice that sometimes equalities are between two groups of similar things, in this case students overall and students who love chemistry, so it is important to properly label our equalities and conversion factors.\r\n\r\nEquation 4: An equality between a total students and students who love chemistry<\/strong>\r\n\r\nDensity can also be used as an equality.\u00a0 For example, aluminum has a density of 2.70 grams per centimeter cubed.\u00a0 In Equation 5 the density of aluminum is expressed as an equality.\r\n\r\nEquation 5: The density of aluminum expressed as an equality<\/strong>\r\n\r\nAll of the equalities (and there could be countless other examples) listed in Equations 3 to 5 can be used to make conversion factors.\u00a0 In dimensional analysis, equalities are used to convert from one set of units to another.\u00a0 The conversion factors always place the unit that you want to covert from in the denominator and the unit that you want to convert to in the numerator.\u00a0 In this way the unit we have is cancelled out and the new unit remains.\u00a0 As demonstrated earlier in this section, sometimes more than one conversion is necessary to find the desired unit.\u00a0 This methodical approach can be used to convert units both in chemistry and in general.\u00a0 Practicing dimensional analysis, as a method, will help you in future chemistry problems like stoichiometry.","rendered":"