3.11: Density

Introduction

After trees are cut, logging companies often move these materials down a river to a sawmill where they can be shaped into building materials or other products. The logs float on the water because they are less dense than the water they are in. Knowledge of density is important in the characterization and separation of materials. Information about density allows us to make predictions about the behavior of matter.

Density

A golf ball and a table tennis ball are about the same size. However, the golf ball is much heavier than the table tennis ball. Now imagine a similar size ball made out of lead. That would be very heavy indeed! What are we comparing? By comparing the mass of an object relative to its size, we are studying a property called density. Density is the ratio of the mass of an object to its volume.

$\text{Density} = \frac{\text{mass}}{\text{volume}}$

Density is an intensive property, meaning that it does not depend on the amount of material present in the sample. Water has a density of $$1.0 \: \text{g/mL}$$. That density is the same whether you have a small glass of water or a swimming pool full of water. Density is a property that is constant for the particular identity of the matter being studied.

The SI units of density are kilograms per cubic meter $$\left( \text{kg/m}^3 \right)$$, since the $$\text{kg}$$ and the $$\text{m}$$ are the SI units for mass and length respectively. In everyday usage in a laboratory, this unit is awkwardly large. Most solids and liquids have densities that are conveniently expressed in grams per cubic centimeter $$\left( \text{g/cm}^3 \right)$$. Since a cubic centimeter is equal to a milliliter, density units can also be expressed as $$\text{g/mL}$$. Gases are much less dense than solids and liquids, so their densities are often reported in $$\text{g/L}$$. Densities of some common substances at $$20^\text{o} \text{C}$$ are listed in the table below.

 Densities of Some Common Substances Liquids and Solids Density at $$20^\text{o} \text{C} \: \left( \text{g/mL} \right)$$ Gases Density at $$20^\text{o} \text{C} \: \left( \text{g/L} \right)$$ Ethanol 0.79 Hydrogen 0.084 Ice $$\left( 0^\text{o} \text{C} \right)$$ 0.917 Helium 0.166 Corn oil 0.922 Air 1.20 Water 0.998 Oxygen 1.33 Water $$\left( 4^\text{o} \text{C} \right)$$ 1.000 Carbon dioxide 1.83 Corn syrup 1.36 Radon 9.23 Aluminum 2.70 Copper 8.92 Lead 11.35 Mercury 13.6 Gold 19.3

Since most materials expand as temperature increases, the density of a substance is temperature dependent and usually decreases as temperature increases.

You known that ice floats in water and it can be seen from the table that ice is less dense. Alternatively, corn syrup, being denser, would sink if placed in water.

Example 3.11.1

An $$18.2 \: \text{g}$$ sample of zinc metal has a volume of $$2.55 \: \text{cm}^3$$.Calculate the density of zinc.

Solution:

Step 1: List the known quantities and plan the problem.

Known

• Mass $$= 18.2 \: \text{g}$$
• Volume $$= 2.55 \: \text{cm}^3$$

Unknown

• Density $$= ? \: \text{g/cm}^3$$

Use the equation for density, $$D = \frac{m}{V}$$, to solve the problem.

Step 2: Calculate

$D = \frac{m}{V} = \frac{18.2 \: \text{g}}{2.55 \: \text{cm}^3} = 7.14 \: \text{g/cm}^3$

If $$1 \: \text{cm}^3$$ of zinc has a mass of about 7 grams, then 2 and a half $$\text{cm}^3$$ will have a mass about 2 and a half times as great. Metals are expected to have a density greater than that of water and zinc's density falls within the range of the other metals listed above.

Since density values are known for many substances, density can be used to determine an unknown mass or an unknown volume. Dimensional analysis will be used to ensure that units cancel appropriately.

Example 3.11.2

1. What is he mass of $$2.49 \: \text{cm}^3$$ of aluminum?

2. What is the volume of $$50.0 \: \text{g}$$ of aluminum?

Solution:

Step 1: List the known quantities and plan the problem.

Known

• Density $$= 2.70 \: \text{g/cm}^3$$
• 1. Volume $$= 2.49 \: \text{cm}^3$$
• 2. Mass $$= 50.0 \: \text{g}$$

Unknown

• 1. Mass $$= ? \: \text{g}$$
• 2. Volume $$= ? \: \text{cm}^3$$

Use the equation for density, $$D = \frac{m}{V}$$, and dimensional analysis to solve each problem.

Step 2: Calculate

$1. \: \: 2.49 \: \text{cm}^3 \times \frac{2.70 \: \text{g}}{1 \: \text{cm}^3} = 6.72 \: \text{g}$

$2. \: \: 50.0 \: \text{g} \times \frac{1 \: \text{cm}^3}{2.70 \: \text{g}} = 18.5 \: \text{cm}^3$

In problem 1, the mass is equal to the density multiplied by the volume. In problem 2, the volume is equal to the mass divided by the density.

Because a mass of $$1 \: \text{cm}^3$$ of aluminum is $$2.70 \: \text{g}$$, the mass of about $$2.5 \: \text{cm}^3$$ should be about 2.5 times larger. The $$50 \: \text{g}$$ of aluminum is substantially more than its density, so that amount should occupy a relatively large volume.