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1.6 Dimensional Analysis

Solving problems in chemistry requires careful manipulation of numbers and their associated units, a method known as dimensional analysis.

For example: What is the volume of a 5.25-gram sample of a liquid having a density of 1.23 g/mL? The density of the liquid can be used as a conversion factor. For the liquid in the example, 1.23 grams are equal to 1.00 milliliter (mL). When the numerator and denominator of a fraction are equal, the fraction has a value of 1, meaning that we can multiply by it for the purpose of changing units. The density conversion factor can be expressed in either of the following two ways.

The one we choose to multiply by depends on what units we want in our result. In this case we want an answer in units of milliliters. So we choose the fraction on the right and multiply it by the mass given in the problem.

Note that if we had chosen the other version of the density conversion factor, we would have ended up with a different number and also nonsensical units.

This illustrates the importance of carrying units throughout a calculation. One way to check your work is to cancel units carefully to make sure that you arrive at an answer with the appropriate units. When you end up with units that don't seem to have any reasonable physical meaning to the problem, such as grams squared per milliliter, you will realize that you must have made some sort of mistake. Go back and check your work.


How many cubic millimeters are there in a cubic centimeter?

1 10-3


Which sample has the greater volume?

16.2 g of a liquid with density 1.045 g/cm3 or 52.0 g of a solid with density 3.354 g/cm3

The solid
The liquid
They both have the same volume