# Densities: Wikis

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The density of a material is defined as its mass per unit volume. The symbol of density is ρ (the Greek letter rho).

## Formula

Mathematically:

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

where:

ρ (rho) is the density,
m is the mass,
V is the volume.

Different materials usually have different densities, so density is an important concept regarding buoyancy, metal purity and packaging.

In some cases density is expressed as the dimensionless quantities specific gravity (SG) or relative density (RD), in which case it is expressed in multiples of the density of some other standard material, usually water or air/gas.

## History

In a well-known tale, Archimedes was given the task of determining whether King Hiero's goldsmith was embezzling gold during the manufacture of a wreath dedicated to the gods and replacing it with another, cheaper alloy.[1]

Archimedes knew that the irregularly shaped wreath could be crushed into a cube whose volume could be calculated easily and compared with the mass; but the king did not approve of this.

Baffled, Archimedes took a relaxing immersion bath and observed from the rise of the warm water upon entering that he could calculate the volume of the gold crown through the displacement of the water. Allegedly, upon this discovery, he went running naked through the streets shouting, "Eureka! Eureka!" (Εύρηκα! Greek "I found it"). As a result, the term "eureka" entered common parlance and is used today to indicate a moment of enlightenment.

The story first appeared in written form in Vitruvius' books of architecture, two centuries after it supposedly took place.[2] Some scholars have doubted the accuracy of this tale, saying among other things that the method would have required precise measurements that would have been difficult to make at the time.[3][4]

## Measurement of density

For a homogeneous object, the mass divided by the volume gives the density. The mass is normally measured with an appropriate scale or balance; the volume may be measured directly (from the geometry of the object) or by the displacement of a fluid. Hydrostatic weighing is a method that combines these two.

If the body is not homogeneous, then the density is a function of the position: $\rho(\vec{r})=dm/dv$, where dv is an elementary volume at position $\vec{r}$. The mass of the body then can be expressed as

$m = \int_V \rho(\vec{r}) dV.$

The density of a solid material can be ambiguous, depending on exactly how its volume is defined, and this may cause confusion in measurement. A common example is sand: if it is gently poured into a container, the density will be low; if the same sand is compacted into the same container, it will occupy less volume and consequently exhibit a greater density. This is because sand, like all powders and granular solids, contains a lot of air space in between individual grains. The density of the material including the air spaces is the bulk density, which differs significantly from the density of an individual grain of sand with no air included.

## Formal definition

Density is defined as mass per unit volume. A concise statement of what this means may be obtained by considering a small box in a Cartesian coordinate system, with dimensions Δx, Δy, Δz. If the mass is represented by a net mass function, then the density at some point (x,y,z) is:

\begin{align} \rho(x,y,z) & = \lim_{Volume \to 0}\frac{\mbox{mass of box}}{\mbox{volume of box}} \ & = \lim_{\Delta x, \Delta y, \Delta z \to 0}\left(\frac{ m(x + \Delta x, y + \Delta y, z + \Delta z) - m(x, y, z)}{\Delta x \Delta y \Delta z}\right) \ & = \frac{d m}{d V}.\ \end{align}\,

For a homogeneous substance, this derivative is equal to the net mass divided by the net volume. For a nonhomogeneous substance, m is a nonconstant function of position: m = m(x,y,z).

## Common units

The SI unit for density is:

Densities using the following metric units all have exactly the same numerical value, one thousandth of the value in (kg/m³). Liquid water has a density of about 1 kg/dm³, making any of these SI units numerically convenient to use as most solids and liquids have densities between 0.1 and 20 kg/dm³.

• kilograms per cubic decimeter (kg/dm³)
• grams per cubic centimeter (g/cc, gm/cc or g/cm³)
• megagrams per cubic meter (Mg/m³)

Liters and metric tons are not part of the SI, but are acceptable for use with it. Since 1 L = 1 dm³, we also have these of the same size:

In U.S. customary units density can be stated in:

In principle there are Imperial units different from the above as the Imperial gallon and bushel differ from the U.S. units, but in practice they are no longer used, though found in older documents. The density of precious metals could conceivably be based on Troy ounces and pounds, a possible cause of confusion.

## 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; similar behaviour is observed in silicon at low temperatures.

The effect of pressure and temperature on the densities of liquids and solids is small. The compressibility for a typical 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. The density of an ideal gas is

$\rho = \frac {MP}{RT} \,$

where R is the universal gas constant, P is the pressure, M is the molar mass, and T is the absolute temperature. This means that the density of an ideal gas can be doubled by doubling the pressure, or by halving the absolute temperature.

Osmium is the densest known substance at standard conditions for temperature and pressure.

## Density of water

Temp (°C) Density (kg/m3)
100 958.4
80 971.8
60 983.2
40 992.2
30 995.6502
25 997.0479
22 997.7735
20 998.2071
15 999.1026
10 999.7026
4 999.9720
0 999.8395
−10 998.117
−20 993.547
−30 983.854
The density of water in kilograms per cubic meter (SI unit)
at various temperatures in degrees Celsius.
The values below 0 °C refer to supercooled water.

## Density of air

T in °C ρ in kg/m3 (at 1 atm)
–25 1.423
–20 1.395
–15 1.368
–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
35 1.146

## Density of solutions

The density of a solution is the sum of mass (massic) concentrations of the components of that solution.
Mass (massic) concentration of a given component ρi in a solution can be called partial density of that component.

$\rho = \sum_i \rho_i \,$

## Density of composite material

ASTM specification D792-00[5] describes the steps to measure the density of a composite material.

$\rho = \frac{W_a}{W_a + W_w - W_b} \left (0.9975 \right ) \,$

where:

ρ is the density of the composite material, in g/cm3

and

Wa is the weight of the specimen when hung in the air
Ww is the weight of the partly immersed wire holding the specimen
Wb is the weight of the specimen when immersed fully in distilled water, along with the partly immersed wire holding the specimen
0.9975 is the density in g/cm3 of the distilled water at 23 °C.

## Densities of various materials

Material ρ in kg/m3 Notes
Interstellar medium 10−25 − 10−15 Assuming 90% H, 10% He; variable T
Earth's atmosphere 1.2 At sea level
Aerogel 1 − 2
Styrofoam 30 − 120 From
Cork 220 − 260 From
Water 1000 At STP
Plastics 850 − 1400 For polypropylene and PETE/PVC
Glycerol[6][7] 1261
The Earth 5515.3 Mean density
Iron 7874 Near room temperature
Copper 8920 − 8960 Near room temperature
The Inner Core of the Earth ~13000 As listed in Earth
Uranium 19100 Near room temperature
Tungsten 19250 Near room temperature
Gold 19300 Near room temperature
Platinum 21450 Near room temperature
Iridium 22500 Near room temperature
Osmium 22610 Near room temperature
The core of the Sun ~150000
White dwarf star 1 × 109[8]
Atomic nuclei 2.3 × 1017 [9] Does not depend strongly on size of nucleus
Neutron star 8.4 × 1016 — 1 × 1018
Black hole 4 × 1017 Mean density inside the Schwarzschild radius of an earth-mass black hole (theoretical)

## References

1. ^ Archimedes, A Gold Thief and Buoyancy - by Larry "Harris" Taylor, Ph.D.
2. ^ Vitruvius on Architecture, Book IX, paragraphs 9-12, translated into English and in the original Latin.
3. ^ The first Eureka moment, Science 305: 1219, August 2004.
4. ^ Fact or Fiction?: Archimedes Coined the Term "Eureka!" in the Bath, Scientific American, December 2006.
5. ^ (2004). Test Methods for Density and Specific Gravity (Relative Density) of Plastics by Displacement. ASTM Standard D792-00. Vol 81.01. American Society for Testing and Materials. West Conshohocken. PA.
6. ^ glycerol composition at physics.nist.gov
7. ^ Glycerol density at answers.com
8. ^ Extreme Stars: White Dwarfs & Neutron Stars, Jennifer Johnson, lecture notes, Astronomy 162, Ohio State University. Accessed on line May 3, 2007.
9. ^ Nuclear Size and Density, HyperPhysics, Georgia State University. Accessed on line June 26, 2009.