Density: The Compactness of Matter
What Exactly is Density?
Imagine you have two boxes that are exactly the same size. One is filled with feathers and the other is filled with books. The box with books is much heavier. This is because the books pack more mass into the same amount of space than the feathers do. This simple idea is the core of density. Density is a measure of how much mass is contained in a given volume. It tells us how tightly matter is packed together.
For example, the density of water is approximately 1 g/cm³. This means that a cube of water measuring 1 cm on each side has a mass of 1 gram. A substance like gold, with a density of about 19.3 g/cm³, has much more mass packed into the same volume—that same 1 cm³ cube would have a mass of over 19 grams!
Calculating Density Step-by-Step
To find the density of any object, you need to perform two measurements: find its mass and find its volume.
Step 1: Measure Mass. Mass is the amount of matter in an object. It is typically measured using a balance or a scale and is expressed in grams (g) or kilograms (kg).
Step 2: Measure Volume. Volume is the amount of space an object occupies. For a regular-shaped object, like a cube or a rectangular prism, you can measure its length, width, and height and multiply them: $ V = l \times w \times h $. The unit will be cubic centimeters (cm³) if you used centimeters.
For an irregular object, like a rock, you can use the water displacement method. Fill a graduated cylinder with water, note the initial volume, submerge the object, and note the new volume. The difference is the object's volume: $ V_{object} = V_{final} - V_{initial} $.
Step 3: Apply the Formula. Divide the mass by the volume. $ \rho = \frac{m}{V} $.
Example: A metal cube has a mass of 144 g. Each side is 2 cm long. What is its density?
- Mass, $ m = 144 g $
- Volume, $ V = 2 cm \times 2 cm \times 2 cm = 8 cm³ $
- Density, $ \rho = \frac{144 g}{8 cm³} = 18 g/cm³ $
Density in Action: Sinking and Floating
The most common and visible application of density is in determining whether an object will sink or float in a fluid (a liquid or a gas). This principle was first described by the ancient Greek mathematician Archimedes.
The Rule: An object will float if its density is less than the density of the fluid it is placed in. It will sink if its density is greater.
- A wooden log (density ~0.7 g/cm³) floats on water (density = 1.0 g/cm³).
- A steel nail (density ~7.8 g/cm³) sinks in water.
- A helium balloon (density of helium is very low) rises in air (density of air is ~0.0013 g/cm³).
This is also why a giant, heavy aircraft carrier can float. It is not made of solid steel. It is mostly hollow, filled with air. Its average density (the total mass of the ship divided by the total volume it occupies, including the air inside) is less than the density of water.
| Substance | Density (g/cm³) |
|---|---|
| Helium Gas | ~0.0002 |
| Cork | 0.24 |
| Pine Wood | 0.5 |
| Water (at 4°C) | 1.00 |
| Aluminum | 2.7 |
| Iron / Steel | 7.8 |
| Lead | 11.3 |
| Mercury | 13.6 |
| Gold | 19.3 |
Density and States of Matter
Density changes with the state of matter (solid, liquid, gas). Generally, for the same substance, the solid state is the most dense and the gaseous state is the least dense. However, a famous exception is water. Ice is less dense than liquid water, which is why ice cubes float in your glass. When water freezes, its molecules form a crystalline structure that holds them farther apart than in liquid water, increasing the volume and thus decreasing the density.
Real-World Applications of Density
Density is not just a topic in a science textbook; it has countless practical applications.
Shipbuilding and Submarines: Engineers design ships to have an average density less than water. Submarines, however, can control their density. They have ballast tanks that can be filled with water (to increase density and sink) or emptied and filled with air (to decrease density and rise).
Hot Air Balloons: A hot air balloon rises because heating the air inside the balloon causes it to expand. The mass of air inside remains roughly the same, but its volume increases. According to the density formula $ \rho = m/V $, if volume increases and mass stays constant, density decreases. The less dense hot air inside the balloon is buoyant in the cooler, denser air outside.
Geology and Mining: The density of rocks and minerals helps geologists identify them. For example, gold is much denser than "fool's gold" (iron pyrite). A simple density test can distinguish between them.
Food and Cooking: Density is used to measure the sugar content in syrups and soft drinks. A hydrometer, which measures the density of a liquid, can tell a brewer if fermentation is complete.
Common Mistakes and Important Questions
Q: Is density the same as weight?
A: No. Weight is the force of gravity pulling on an object's mass. It can change depending on location (e.g., you weigh less on the Moon). Density, however, is a property of the material itself and does not change with location, as long as the mass and volume remain constant. A gold bar has the same density on Earth, the Moon, or in outer space.
Q: If I cut an object in half, does its density change?
A: No. When you cut a homogeneous object (the same material throughout) in half, you halve both its mass and its volume. Since density is mass divided by volume, the ratio remains the same. Half a brick has half the mass and half the volume of a whole brick, so its density is unchanged.
Q: What is relative density or specific gravity?
A: Relative density, also called specific gravity[1], is a dimensionless[2] number. It is the ratio of the density of a substance to the density of a reference substance, usually water for liquids and solids. For example, the relative density of mercury is 13.6, meaning it is 13.6 times denser than water. If a substance has a relative density less than 1, it will float in water.
Footnote
[1] Specific Gravity: A ratio of the density of a substance to the density of a reference substance (typically water at 4°C). It is a dimensionless value.
[2] Dimensionless: A quantity that has no physical units associated with it. It is a pure number.