Dimensional Analysis Tutorial

Using One Conversion Factor

Dimensional analysis is a systematic method for solving problems. In this method the units for all numbers are carried through the entire problem. Just as the numbers involved in the problem are multiplied and divided, the units are also multiplied together, divided into each other or cancelled out. Dimensional analysis is the best approach for solving many problems in chemistry (and other scientific fields). It helps ensure that you have set the problem up correctly and that the answers you get have the correct units.

Dimensional analysis uses conversion factors to change from one set of units to another. A conversion factor is a fraction whose numerator and denominator represent the same quantity but are expressed in different units.

Examples of conversion factors:

For the relationship 12 inches = 1 foot, two conversion factors can be written:

12 in
 or
1 ft
1 ft
12 in

For the relationship, 1000 mL = 1 L, two conversion factors can also be written:

1000 mL
 or
1 L
1 L
1000 mL

Notice that each relationship between two sets of units gives rise to two possible conversion factors. These conversion factors are the inverses of each other. One of your challenges when using dimensional analysis to solve problems will be to decide not only what relationship to use to write a conversion factor but also which set of units must go on the bottom (i.e. in the denominator) and which one must go on the top (i.e. in the numerator).

Problems Using One Conversion Factor

The best way to learn to use dimensional analysis is by actually working through some examples. In order to show you how it works we'll start off with an easy problem that most people can work without using dimensional analysis.

Example: A desk is 36 inches long. What is its length in feet?

  1. Write down the units that you are looking for and an equal sign:
    ft
    =

  2. Write down the length and the units you were given in the problem:
    ft = 36 in.

  3. Put a multiplication sign after the number (and units) you were given and draw a line
    ft
    =
    36 in.
    x
    ________

  4. Write a conversion factor above and below the line you've drawn in such a way that the inches cancel out.
    • put "12 in" on the bottom
    • put "1 ft" on the top

      ft

      =

      36 in

      x

      1 ft

       

       

       

       

      12 in

  5. Cancel out the "in" on the top and bottom. Notice that the only units left are "ft." Since these are the units you were looking for you are ready to do the math.

    ft

    =

    36 in

    x

    1 ft
    =
    		 3.0 ft

     

     

     

     

    12 in
    		 

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Example: A desk is 36 inches long. What is its length in cm?

  1. Write down the units that you are looking for and an equal sign:
    cm
    =

  2. Write down the length and the units you were given in the problem:
    cm = 36 in.

  3. Put a multiplication sign after the number (and units) you were given and draw a line
    cm
    =
    36 in.
    x
    ________

  4. Write a conversion factor above and below the line you've drawn in such a way that the inches cancel out. If you don't know the relationship between inches and cm (2.54 cm = 1 in), you can look in your text.
    • put "1 in" on the bottom
    • put "2.54 cm" on the top

      cm

      =

      36 in

      x

      2.54 cm

       

       

       

       

      1 in

  5. Cancel out the "in" on the top and bottom. Notice that the only units left are "cm." Since these are the units you were looking for you are ready to do the math.

    cm

    =

    36 in

    x

    2.54 cm
    =
    		 91 cm

     

     

     

     

    1in
    		 

Example: A flask holds 1.25 x 104 mL. What is its volume in L?

  1. Write down the units that you are looking for and an equal sign:
    L
    =

  2. Write down the volume and the units you were given in the problem:
    L = 1.25 x 104 mL

  3. Put a multiplication sign after the number (and units) you were given and draw a line
    L
    =
    1.25 x 104 mL
    x
    ________

  4. Write a conversion factor above and below the line you've drawn in such a way that the mL cancel out. If you don't know the relationship between mL and L (1000 mL = 1 L), you can look in your text. For your exam, YOU MUST KNOW THE METRIC TO METRIC CONVERSION FACTORS GIVEN IN YOUR NOTES.
    • put "1000 mL" on the bottom
    • put "1 L" on the top

      L

      =

      1.25 x 104 mL

      x

      1 L

       

       

       

       

      1000 mL

  5. Cancel out the "mL" on the top and bottom. Notice that the only units left are "L." Since these are the units you were looking for you are ready to do the math.

    cm

    =

    1.25 x 104 mL

    x

    1L
    =
    		 12.5 L

     

     

     

     

    1000 mL
    		 

 

Practice Problems:

  1. 12.5 kg = _______ g
  2. 1525 cm = _______ m
  3. 16 min = _______hr
  4. 27.25 mL = ______ cm3
  5. 25 qt = _____ L
  6. 135 km = ______ mi

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