# Density

## Overview

In physics, density is mass m per unit volume V—how heavy something is compared to its size. A small, heavy object, such as a rock or a lump of lead, is denser than a lighter object of the same size or a larger object of the same weight, such as pieces of cork or foam.

For the common case of a homogeneous substance, density is expressed as:

$\rho = \frac {m}{V}$

where, in SI Units:

ρ (rho) is the density of the substance, measured in kg·m–3
m is the mass of the substance, measured in kg
V is the volume of the substance, measured in m3

## History

In a famous problem, Archimedes was given the task of determining if King Hiero's goldsmith was embezzling gold during the manufacture of the king's crown and replacing it with another, cheaper alloy.

Archimedes knew that the crown could be smashed into a cube or sphere, where the volume could be calculated more easily when compared with the weight; the king did not approve of this.

Baffled, Archimedes went to take a bath and observed from the rise of the water upon entering that he could calculate the volume of the crown through the displacement of the water. Allegedly, upon this discovery Archimedes went running though the streets naked shouting, "Eureka! Eureka!" Greek for "I've found it!"

Following a test of the crown based upon this new discovery, the goldsmith was executed.

## Measurement of density

For a homogeneous object, the formula mass/volume may be used. The mass is normally measured with an appropriate scale; the volume may be measured directly (from the geometry of the object) or by the displacement of a liquid. A very common instrument for the direct measurement of the density of a liquid is the hydrometer. A less common device for measuring fluid density is a pycnometer, a similar device for measuring the absolute density of a solid is a gas pycnometer.

The density of a solid material can be ambiguous, depending on exactly how it is defined, and this may cause confusion in measurement. A common example is sand: if gently filled into a container, the density will be small; when the same sand is compacted into the same container, it will occupy less volume and consequently carry a greater density. This is because "sand" contains a lot of air space in between individual grains; this overall density is called the bulk density, which differs significantly from the density of an individual grain of sand.

## Common units

In U.S. customary units or Imperial units, the units of density include:

ounces per cubic inch (oz/in3)
pounds per cubic inch (lb/in3)
pounds per cubic foot (lb/ft3)
pounds per cubic yard (lb/yd3)
pounds per gallon (for U.S. or imperial gallons) (lb/gal)
pounds per U.S. bushel (lb/bu)
slugs per cubic foot.

## Changes of density

In general density can be changed by changing either the pressure or the temperature. Increasing the pressure will always increase the density of a material. Increasing the temperature generally decreases the density, but there are notable exceptions to this generalisation. For example, the density of water increases between its melting point at 0 °C and 4 °C and similar behaviour is observed in silicon at low temperatures.

The effect of pressure and temperature on the densities of liquids and solids is small so that a typical compressibility for a liquid or solid is 10–6 bar–1 (1 bar=0.1 MPa) and a typical thermal expansivity is 10–5 K–1.

In contrast, the density of gases is strongly affected by pressure. Boyle's law says that the density of an ideal gas is given by

$\rho = \frac {mP}{RT}$

where $R$ is the universal gas constant, $P$ is the pressure, $m$ the molar mass, and $T$ the absolute temperature.

This means that a gas at 300 K and 1 bar will have its density doubled by increasing the pressure to 2 bar or by reducing the temperature to 150 K.

## Density of water

Temperature Density (at 1 atm)
°C °F kg/m³
0.0 32.0 999.8425
4.0 39.2 999.9750
15.0 59.0 999.1026
20.0 68.0 998.2071
25.0 77.0 997.0479
37.0 98.6 993.3316
50.0 122.0 988.04
100.0 212.0 958.3665

## Density of air

T in °C ρ in kg/m³ (at 1 atm)
–10 1.342
–5 1.316
0 1.293
5 1.269
10 1.247
15 1.225
20 1.204
25 1.184
30 1.164

## Books

• Fundamentals of Aerodynamics Second Edition, McGraw-Hill, John D. Anderson, Jr.
• Fundamentals of Fluid Mechanics Wiley, B.R. Munson, D.F. Young & T.H. Okishi
• Introduction to Fluid Mechanics Fourth Edition, Wiley, SI Version, R.W. Fox & A.T. McDonald
• Thermodynamics: An Engineering Approach Second Edition, McGraw-Hill, International Edition, Y.A. Cengel & M.A. Boles