Dimensional Analysis

Dimensional analysis involves using units as a guide for problem solving.  Dimensional analysis can be used to solve many types of problems, including chemistry problems.  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.

In 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.  For example if one wants to know how many seconds are in 1.3 minutes, we need an equality between seconds and minutes.  Equation 1 shows the equality that one minute (min) is equal to 60 seconds (s).

Equation 1. An equality equation between minute and second. 

In 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.

Once an equality is known, the next step is to turn the equality into a conversion factor as shown in Calculation 1.  Notice that in the conversion factor, 60 seconds is in the numerator and 1 minute is in the denominator.  The number in the denominator (on the bottom) is the unit that is being canceled out and the numerator, the number on top, is the new unit that is desired.  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).  If there were multiple conversion factors present, this process would be repeated.

Calculation 1. Converting 1.3 minutes to seconds using dimensional analysis. 

It 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.  The mathematical result of dividing two equal things is 1.  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.  Multiplying any value by 1 gives the same value.  Said more simply, all conversion factors must be made from two equal values expressed as a fraction.

In the example above only one conversion was needed because there is a well-known equality between seconds and minutes.  Now let’s consider the case that we wish to know how many minutes are in 3.4 days.  It’s possible that one might know how many minutes are in a day, but for this example, let’s assume that we only know the more common equalities shown in Equation 2.  These equalities are 24 hours (h) is one day and 60 minutes is one hour.

Equation 2. Equalities between day, hour and minute 

In this example there is no direct equality between minutes and days (although one can be known or calculated).  If there is no direct equality then multiple conversion factors are needed.  In this case 3.4 days can be converted to hours and then hours can be converted to minutes.  This plan is outlined in Figure 1 below.  Note that although a plan is not required to solve the problem, it can be useful when you start doing dimensional analysis problems.

Figure 1.  Plan to convert days to minutes.

In 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.   When converting between two systems of measurement, like centimeters to inches, the conversion is a measured number.  When converting within a single system of measurement, like centimeters to meters, the conversions are exact.  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.

Calculation 2. Converting 3.4 days to minutes using dimensional analysis. 

To 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.  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.  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.

Some of the equalities, used to make conversion factors, mentioned above are well known, such as 1 hour equals 60 minutes.  However, sometimes equalities are not as obvious.  Below a series of less obvious conversion factors is presented.  If apples cost $1.99 per pound, an equality between dollars and pounds of apples exists as shown in Equation 3.

Equation 3: An equality between dollars and pounds.

If 85% of students love chemistry, than equality exists between students and students who love chemistry as shown in Equation 4.  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.  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.

Equation 4: An equality between a total students and students who love chemistry

Density can also be used as an equality.  For example, aluminum has a density of 2.70 grams per centimeter cubed.  In Equation 5 the density of aluminum is expressed as an equality.

Equation 5: The density of aluminum expressed as an equality

All of the equalities (and there could be countless other examples) listed in Equations 3 to 5 can be used to make conversion factors.  In dimensional analysis, equalities are used to convert from one set of units to another.  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.  In this way the unit we have is cancelled out and the new unit remains.  As demonstrated earlier in this section, sometimes more than one conversion is necessary to find the desired unit.  This methodical approach can be used to convert units both in chemistry and in general.  Practicing dimensional analysis, as a method, will help you in future chemistry problems like stoichiometry.