Dimensional Analysis

What is dimensional analysis and how can we use it to help us solve problems in chemistry?

Dimensional Analysis for Party Planning…

If you have every planned a party, you have used dimensional analysis. The amount of beer and munchies you will need depends on the number of people you expect. For example, if you are planning a Friday night party and expect 30 people you might estimate you need to go out and buy 120 bottles of beer and 10 large pizza's. How did you arrive at these numbers? The following indicates the type of dimensional analysis solution to party problem:

Finally, in going to buy the beer, you perform another dimensional analysis: should you buy the beer in six-packs or in cases?

Realizing that carrying around 20 six packs is a real headache, you get 5 cases instead.

In this party problem, we have used dimensional analysis in two different ways:

• In the first application, we have used dimensional analysis to calculate how much beer we need based on knowing 1) how much beer we need for one person, and 2) how many people we expect. Likewise for the Pizza.
• In the second application we used dimensional analysis to convert units (i.e. from individual beers to the equivalent amount of six packs or cases)

First, let's take a look at dimensional analysis as it relates to converting units:

• If we ignore the numbers for a moment, and just look at the units (i.e. dimensions), we have:

beers * (six pack/beer)

• We can treat the dimensions in a similar fashion as other numerical analyses, i.e. any number divided by itself is 1. Therefore:

beers * (six pack/beer) = beers * (six pack/beer) = six pack

• So, the dimensions of the numerical answer will be "six packs".

How can we use dimensional analysis to be sure we have set up our equation correctly?

• Consider the following alternative way to set up the above unit conversion analysis:

• While it is correct that there are 6 beers in one six pack, the above equation yields a value of 720 with units of beers2/six pack.
• These rather bizarre units indicate that the equation has been setup incorrectly (and as a consequence you will have a ton of extra beer at the party).

Next, let’s take a look at dimensional analysis as it relates to calculations:

• In the above case it was relatively straightforward keeping track of units during the calculation.
• What if the calculation involves powers, etc? For example, the equation relating kinetic energy to mass and velocity is:

• An example of units of mass is kilograms (kg) and velocity might be in meters/second (m/s)
• What are the dimensions of Ekinetic?

• Since velocity is squared, the dimensions associated with the numerical value of the velocity are also squared
• We can double check this by knowing the the Joule (J) is a measure of energy, and has units of kg m2 s-2 (1J = 1 kg m2 s-2)

Another example

• Pressure (P) is a measure of the Force (F) per unit area (A):

• Force, in turn, is a measure of the acceleration (a) on a mass (m):

• Thus, pressure (P) can be written as:

• What are the units of pressure from this relationship? (Note: acceleration is the change in velocity per unit time)

• We can simplify this description of the units of Pressure by dividing numerator and denominator by m:

In fact, these are the units of a Pascal (Pa), a measure of pressure

In summary…

• Dimensional analysis is used in numerical calculations, and in converting units
• Dimensional analysis can help us identify whether an equation is set up correctly (i.e. the resulting units should be as expected)
• Units are treated similarly to the associated numerical values, i.e. if a variable in an equation is supposed to be squared, then the associated dimensions are squared, etc.

1998 Dr. Michael Blaber