# Density

**Density \((\rho)\)** is the amount of mass \((m)\) per unit volume \((V)\) of a substance.

\[\rho = \frac{m}{V}\]

Density is an intensive property, which means the density does not change as the amount of the substance present changes. This contrasts with mass (an extensive property), which appears quite often in physics, but becomes cumbersome to apply to fluids macroscopically.

The density of gold is 19,320 \(\frac{\text{kg}}{\text{m}^3}\). What is the density of a 12.5 \(\text{kg}\) sample of gold?

Since density does not depend on the amount of gold in the sample, the mass is meaningless, and the density is always

19,320 \(\frac{\text{kg}}{\text{m}^3}.\)

## Density and pressure

The density of a substance is generally given near the surface of the earth. At this location, the pressure is 1 \(\text{atm}\) or \(1.01\times 10^5 \text{Pa}.\) However, if the pressure changes, the density will change as well. Boyle's law says that the product of the pressure and the volume must remain constant if the amount of the substance present remains constant.

Boyle's lawIf the temperature and amount of an ideal gas remain constant, the product of the pressure and volume must also remain constant, or:

\[P_1V_1 = P_2V_2.\]

A rigid box is filled with air. If the pressure is doubled but the volume remains the same, does the density of the air increase or decrease?

By Boyle's law, \(PV\) must remain constant if the amount of the substance is unchanged. In this case, \(PV\) is doubled, which means the amount of air in the box must have changed. In order to increase the pressure, the amount of air must increase. Hence, the density of the air in the box

increases.

Tom's football is found to be inflated at a pressure below the pressure required by the rules. In order to bring the pressure up to the required level, he inflates the ball further. What happens to the density of the air inside the football?