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Jump to one of the following dimensional analysis parts:   Choose One Introduction to Dimensional Analysis Part 1- The Need for Units Part 2- Setting up Conversions Part 3- Basic Conversion Problems Part 4- Complex Conversion Problems Part 5- Problem Dissection I Part 6- Problem Dissection II Part 7- Density Problems Part 8- Quizzes

Let's take a look at the different types of density problems you might see.

Density is a physical property and demonstrates the relationship between mass and volume of a material. You might remember that water is commonly assigned a density of 1 g/mL (it's an approximation since density tends to be temperature dependent). If you take one mL of water and put it on a balance, you'll see that the mass is about 1 g.

Obtaining the mass is simple enough. You whip the material up on the balance and read the display. Measuring the volume, however, ranges in difficulty. If it's a liquid, you can use a graduated cylinder. If it's a solid then you have a couple of possibilities. If it's a geometric solid like a cube or a cylinder then you can take length measurements and, with the appropriate formula, calculate the volume. If it's an irregular solid, however, you need to use a different method. Let's look at each of the previous cases in turn before moving on to manipulating the density formula.

The formula for density is:

 or 

If you're given an unknown liquid with a mass of 3.444 g and a volume of 3.82 mL, what's the density?

Volumes for liquids are generally measured in mL, solids in cm³, and gases in L. However, remember that 1 mL is defined to be equivalent to 1 cm³ so it's a snap to convert between the two.

Let's try a solid. What if you've got a rectangular solid? How do you get volume? (length)x(width)x(height) allows you to calculate volume. Therefore, what's the density of a 8.490 g solid with dimensions 0.98 cm, 2.11 cm, and 1.99 cm? Remember how to calculate volume-

And now let's put it all together-

Make special note of how the units were used when calculating the volume in the density formula.

It's not always quite so simple to calculate the volume. If you have what's called an "irregular" solid, then it's too complicated to meaure the volume. Think about trying to determine your volume. The way to get it is to use the Archimedes Principle. Imagine if you filled the bathtub with water up to the very brim. What would happen if you then got into the tub? Water would overflow and the amount would be equal to your own volume. That's how we get the volume of irregular solids. You fill a container with water to a specific volume and then measure the change when you drop your object into the water.

Let's try it. A solid has a mass of 5.019 g and is placed into a graduated cylinder with an initial volume of 49.8 mL. The volume of the water rises to 71.3 mL. What is the density of the solid?

Let's tackle the volume, first-

And now let's put it all together-

The conversion from mL to cm³ was done at the end simply to show you how to do it by unit analysis.

Not all density problems want you to actually calculate density. The formula involves density, mass, and volume. If you are given any 2, you can get the third. So, just how do we go about solving for mass or for volume if we are given the other 2? Let's walk through the formula manipulation this one time to demonstrate it.

1. Given density and volume, solve for mass. The cycle button will cycle you through the process until you've solved for mass-

2. Given density and mass, solve for volume. The cycle button will cycle you through the process until you've solved for volume-

Whether you are solving for mass, density, or volume, you must keep careful watch over the units. The common type of mistake in rearranging the density formula is not properly checking the units at the end. If you are looking for mass, your units must be in g and not 1/g. The same goes for volume.

Let's try one more problem before turning you loose on the quizzes.

A material has a density of 0.103 g/mL and a mass of 6.142 g. What's the volume of the substance in cm³?

Whoops! What happened to the units? I'm not referring to the mL instead of cm³ (since it's a one-to-one ratio, it's simple to convert between them... I trust you!). This is where unit analysis comes into play along with algebra and fractions. Don't make the mistake of assuming your units are correct. Understand how to check them in situations like these! Here, let's check-

One more time, just for emphasis-

"Treat units just like numbers. They can be multiplied and divided."