CH 104: Lesson 1a

Unit Conversions

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A common problem in chemistry (and in everyday life) is to convert one value of units to another.  For example, at home we may need to convert units such as inches to feet, cups to quarts, pounds to ounces, and square feet to acres.  Here is a uniform method for making conversions.

I.  A Single Factor Example--Familiar Units

Our question:  How many feet are in 3.250 inches?

Step 1:  Evaluate the question:   We are starting with inches and need to find feet.

Step 2:  Find the conversion value between these units.  In our example, we use the conversion value of:

            1 foot = 12 inches  (exactly)

Step 3:  Begin setting up the math.  We start with 3.250 inches and divide it by 1 (this makes things easier to see):

       3.250 inches

            1

Step 4:    Set up the conversion by putting the units that you want on the top and the units that you have on the bottom.  This way the original units cancel and we are left with the units that we want:

       3.250 inches      feet

            1                 inch

Step 5:  Now place the values from Step 2 in their respective places as shown below.  Note that it is very important to associate the 12 with the inches and the 1 with the foot.  Otherwise, you would have things backwards by saying 1 inch = 12 feet!   This is a very common type of error.

   3.250 inches           1.000 foot     

            1               12.000 inches

Step 6:  Multiply the tops together and divide by the bottoms:

  3.250 inches  *    1.000 foot        =  3.250 inches*feet    =    0.270833 feet  =  0.2708 feet (with 4 significant figures)

            1      *  12.000 inches          12.000 inches

            Note how the original units cancel out.   Anytime that your units cancel out to give you the units that you need, you should have more confidence in your answer.  In chemistry, if your units do NOT work out properly, you almost certainly made a mistake somewhere.  This is a valuable lesson to use throughout your time learning (and using) chemistry.

Step 7:  Check your units and significant figures.

    This was a multiplication and division problem where we rely on the figure containing the LEAST number of significant digits.  In our case, this was the 3.250 inches which has 4 significant figures.  Note that the conversion between inches and feet is exact.  We implied that 1.00000 foot= 12.000 inches (or more significant digits if we needed them).  We showed feet as our final units.      BE VERY CAREFUL WITH BOTH SIGNIFICANT FIGURES AND UNITS AS GRADE POINTS ARE DEDUCTED THROUGHOUT ALL CHEMISTRY COURSES IF THEY ARE INCORRECTLY SHOWN.

II.  A Single Factor Example--Metric Units

Our question:  How many grams are in 5.65 kilograms?

Step 1:  Evaluate the question:   We are starting with kg and need to find grams.

Step 2:  Find the conversion value between these units.  In our example, we use the conversion value of:

            1 kg = 1000. g  (exactly)

Step 3:  Begin setting up the math.  We start with 5.65 kg and divide it by 1 (this makes things easier to see):

        5.65 kg

            1

Step 4:    Set up the conversion by putting the units that you want on the top and the units that you have on the bottom.  This way the original units cancel and we are left with the units that we want:

         5.65 kg      grams

            1              kg

Step 5:  Now place the values from Step 2 in their respective places as shown below.  Note that it is very important to associate the 1000 with the grams and the 1 with the kg  Otherwise, you would have things backwards by saying 1 gram = 1000 kilograms!

       5.65 kg      1000 grams

            1                1 kg

Step 6:  Multiply the tops together and divide by the bottoms:

    5.65 kg  * 1000. grams           =    5,650 grams    =  5.65 x 10³ grams

        1           *     1 kg

            Again, note that the original units cancel out.  

Step 7:  Check your units and significant figures.

    This was a multiplication and division problem where we rely on the figure containing the LEAST number of significant digits.  In our case, this was the 5.65 kg which has 3 significant figures.  Note that the metric conversions are exact.  We implied that 1.00000 kg = 1000.00 grams (or more significant digits if we needed them).  The answer needs to have 3 significant figures which we show by leaving the trailing decimal point off the 5,650 or (even better), by showing it in scientific notation (5.65 x 10³).  We also showed grams as our final units.     

III.  A multi-factor example. 

Sometimes, we need to use multiple conversion factors to go from our starting units to our desired units.  Here is an example of the conversion of weight units that are used by pharmacists:

Our question:  What is the weight in milligrams of a 5.00 grain aspirin?

Step 1:  Evaluate the question:   We are starting with grains and need to find milligrams.

Step 2:  Find the conversion factors that we can use to convert between these units.  In our example, we can use:

            1.000 pound = 453.6 grams

            1.000 pound = 7000. grains

            1 gram = 1000. mg  (exactly)

Step 3:  Begin setting up the math.  We start with 5.00 grains. and divide it by 1 (this makes things easier to see):

        5.00 grains

            1

Step 4:    Set up the conversion by putting the units that you want on the top and the units that you have on the bottom.  Unfortunately, this is not too useful by itself as we do not have a direct conversion factor here.  We will deal with this issue in Step 5 below.

         5.00 grains           mg    

            1                    grains

Step 5:  As we do not have the direct conversion factor, we look to see what we can use (from step 2):  Note that we arrange these so the units will cancel to give us the units that we want (everything should cancel except for mg):

    5.00 grains         pound      grams        mg   

            1                grains      pound     grams

Step 6:  Now place the values from Step 2 in their respective places as shown below.  This is very similar to what we did in the first example, but now we have more values to place in their respective places:

   (5.00 grains)  *  (1.000 pound) *  (453.6 grams)  * (1000 mg)

      ( 1 ) *           ( 7000. grains) *  (1.000 pound) *   ( 1 grams)

Step 6:  Multiply the tops together and divide by the bottoms:

   (5.00 grains)  *  (1.000 pound) *  (453.6 grams)  * (1000 mg)   =  2268000  mg  =   324 mg

      ( 1 )        *    ( 7000. grains) *  (1.000 pound) *   ( 1 grams)            7000

            Again, note how the original units cancel out.   Thus, we have more confidence in our answer. 

Step 7:  Check your units and significant figures.

    In this problem, the least number of significant figures was in our 5.00 grains (3 significant figures).  Thus our answer needs to be shown with 3 significant figures.  Our units (milligrams) are exactly what we wanted. 

Note that a standard aspirin tablet is now 325 mg instead of 324 mg.  The pharmaceutical companies used a slightly different dosage when they converted from English to Metric units.

P. L. Hanrahan, July 3, 2005

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