Dimensional Analysis in Chemistry



One of the first road blocks you might find when you start chemistry is how to do dimensional analysis problems.



Dimensional analysis problems do not lend themselves to memorization. The first factor of frustration comes when being able to solve a problem does not boil down to finding which equation to use.


Now you must learn to figure out each problem. However this is not as hard as it looks. 

The fundamental operation in all dimensional analysis problems is to put the given unit in the denominator and the unit to convert to into the numerator. The rest of this method merely extends this idea, Figure 1.

The strategy is to put the given unit and quantity to the left. The unit you want to have the quantity converted to has a ratio to the original unit. Arrange the ratio so the given unit appears in the denominator of the of the ratio. The unit you want appears in the numerator. As you carry out the computation, the units of the original quantity cancel out.



dimensional analysis basic operation
Figure 1 : Dimensional analysis is done with one basic operation.



First step : what kind of problem is this?


There are only two types of problems: 


one unit to another unit, or one ratio to another ratio, Figure 2.


This never changes. The problem follows one pattern or the other pattern. Once you know which pattern a problem follows, you can identify the both the specific question and where to begin the problem.



two kinds of problems
Figure 2: All dimensional analysis problems are unit conversions or ratio conversions


Unit to Unit


In the case of unit conversion problems, convert the known units to units of an unknown quantity with the pattern in Figure 3:


symbolic unit to unit conversion
Figure 3: An example of a two step equation to convert one unit to another


Instead of specific units, the units are shown as symbols in shapes. When you start with a square with dots, put the square with dots in the denominator with a different unit in the numerator: right pointed triangle. 


Ratio to Ratio


In contrast, ratio to ratio conversions follow the following kind of pattern, Figure 4.


ratio to ratio conversions
Figure 4 : dimensional analysis used to convert a given ratio to another ration


Like the previous example, symbols take the place of numerical values and units. Once again, find a ratio with the unit you want to change. Arrange the ratio so the unit you want to change appears in the denominator.


To change the bottom unit, find a ratio with the unit that  in the denominator. Place the unit to change in the numerator.


The initial ratio and final ratio used to make the conversion might not match. Insert any ratios needed until the units match.


When you break down dimensional analysis into smaller tasks, you will find it easier to tackle.


Step by Step


Approach each problem systematically. To solve problems, use a known process which gives proven results, Figure 5.


The steps break down into smaller self-contained steps. Once you find the question and given information, you know what kind of problem you have. Find the needed ratios in the problem statement or look the ratios up.



steps to solve conversion problems
Figure 5: Sequence of steps used to solve unit conversion problems


    1. Identify the Question

      The key to any question is to understand what question the problems asks. If the answer takes the form of a unit, then you have a unit to unit conversion. If you have a ratio as the final answer, then you know you will start the problem with a ratio.

    2. Find the Start

      Based on whether the final answer is a unit problem or ratio problem, you can spot the start of the conversion to see what ratio or unit remains to convert.

    3. Find the Ratios

      Once you know where the problems starts and ends, the rest of the ratios you need are in the problem statement or you are able to look up any needed ratios.

      You can find common ratios in your textbook or on the internet. These ratios include: miles to kilometers, psi to mm Hg, atm to mm Hg, inch to cm, or  gallons to liters.


    4. Arrange the First and Last Ratio

      Take the ratio which alters the unit in the denominator and arrange them so the numerator unit changes.

      Place the last ratio with the unit in the numerator so the desired unit is either canceled or ends up in the denominator.

    5. Add Middle Ratios

      Match any needed conversions until the units for the first conversion and last ratio match the units you need to carry out the first and last conversion. 

    6. Compute

      Calculate the numerical values associated with each unit. Check your work. If you have a correct answer, you can cancel any unit which appears in a ratio which is anywhere in both a numerator or denominator.