Density

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The mass density or density of a material is its mass per unit volume. The symbol most often used for density is ρ (the lower case Greek letter rho). Mathematically, density is defined as mass divided by volume:[1]

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

where ρ is the density, m is the mass, and V is the volume. In some cases (for instance, in the United States oil and gas industry), density is also defined as its weight per unit volume,[2] although this quantity is more properly called specific weight.

Different materials usually have different densities, and density may be relevant to buoyancy, purity and packaging. Osmium and iridium are the densest known elements at standard conditions for temperature and pressure but certain chemical compounds may be denser.

Less dense fluids float on more dense fluids if they do not mix. This concept can be extended, with some care, to less dense solids floating on more dense fluids. If the average density (including any air below the waterline) of an object is less than water it will float in water and if it is more than water it will sink in water.

Density is sometimes expressed by the dimensionless quantity "specific gravity" or "relative density", i.e. the ratio of the density of the material to that of a standard material, usually water. Thus a specific gravity less than one means that the substance floats in water.

The density of a material varies with temperature and pressure. This variation is typically small for solids and liquids but much greater for gases. Increasing the pressure on an object decreases the volume of the object and thus increases its density. Increasing the temperature of a substance (with a few exceptions) decreases its density by increasing its volume. In most materials, heating the bottom of a fluid results in convection of the heat from the bottom to the top, due to the decrease in the density of the heated fluid. This causes it to rise relative to more dense unheated material.

The reciprocal of the density of a substance is occasionally called its specific volume, a term sometimes used in thermodynamics. Density is an intensive property in that increasing the amount of a substance does not increase its density; rather it increases its mass.

History

In a well-known but probably apocryphal tale, Archimedes was given the task of determining whether King Hiero's goldsmith was embezzling gold during the manufacture of a golden wreath dedicated to the gods and replacing it with another, cheaper alloy.[3] 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 is said to have taken an immersion bath and observed from the rise of the water upon entering that he could calculate the volume of the gold wreath through the displacement of the water. Upon this discovery, he leapt from his bath and ran naked through the streets shouting, "Eureka! Eureka!" (Εύρηκα! Greek "I have 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.[4] 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.[5][6]

From the equation for density (ρ = m / V), mass density has units of mass divided by volume. As there are many units of mass and volume covering many different magnitudes there are a large number of units for mass density in use. The SI unit of kilogram per cubic metre (kg/m3) and the cgs unit of gram per cubic centimetre (g/cm3) are probably the most commonly used units for density. (The cubic centimeter can be alternately called a millilitre or a cc.) 1,000 kg/m3 equals one g/cm3. In industry, other larger or smaller units of mass and or volume are often more practical and US customary units may be used. See below for a list of some of the most common units of density.

Measurement of density

The density at all points of a homogeneous object equals its total mass divided by its total volume. The mass is normally measured with a scale or balance; the volume may be measured directly (from the geometry of the object) or by the displacement of a fluid. To determine the density of a liquid or a gas, a hydrometer or dasymeter may be used, respectively. Similarly, hydrostatic weighing uses the displacement of water due to a submerged object to determine the density of the object.

If the body is not homogeneous, then its density varies between different regions of the object. In that case the density around any given location is determined by calculating the density of a small volume around that location. In the limit of an infinitesimal volume the density of an inhomogeneous object at a point becomes: ρ(r) = dm/dV, where dV is an elementary volume at position r. The mass of the body then can be expressed as

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

The density of granular 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 then compacted, 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.

Changes of density

In general, density can be changed by changing either the pressure or the temperature. Increasing the pressure always increases the density of a material. Increasing the temperature generally decreases the density, but there are notable exceptions to this generalization. For example, the density of water increases between its melting point at 0 °C and 4 °C; similar behavior 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. This roughly translates into needing around ten thousand times atmospheric pressure to reduce the volume of a substance by one percent. (Although the pressures needed may be around a thousand times smaller for sandy soil and some clays.) A one percent expansion of volume typically requires a temperature increase on the order of thousands of degrees Celsius.

In contrast, the density of gases is strongly affected by pressure. The density of an ideal gas is

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

where M is the molar mass, P is the pressure, R is the universal gas constant, 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.

In the case of volumic thermal expansion at constant pressure and small intervals of temperature the temperature dependence of density is :

$\rho = \frac {{\rho_{T_0}}}{{(1 + \alpha \cdot \Delta T)}}$

where $\rho_{T_0}$ is the density at a reference temperature, $\alpha$ is the thermal expansion coefficient of the material at temperatures close to $T_0$.

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 each given component ρi in a solution sums to density of the solution.

$\rho = \sum_i \varrho_i \,$

Expressed as a function of the densities of pure components of the mixture and their volume participation, it reads:

$\rho = \sum_i \rho_i \frac{V_i}{V}.\,$

provided that there is no interaction between the components.

Densities

Water

Density of water at 1 atm pressure:

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 values below 0 °C refer to supercooled water.

Air

Density vs. temperature

Density of air at 1 atm pressure:

T (°C) ρ (kg/m3)
−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

Various materials

Unless otherwise noted, all densities given are at standard conditions for temperature and pressure, that is, 273.15 K (0.00 °C) and 100 kPa (0.987 atm).

Material ρ (kg/m3) Notes
Air 1.2 At sea level
Aerographite 0.2 *[7][8]
Metallic microlattice 0.9 *
Aerogel 1.0 *
Styrofoam 75 Approx.[9]
liquid hydrogen 70 At ~ -255°C
Cork 240 Approx.[9]
Lithium 535
Wood 700 Seasoned, typical[10][11]
Potassium 860 [12]
Sodium 970
Ice 916.7 At temperature < 0°C
Water (fresh) 1,000
Water (salt) 1,030
Plastics 1,175 Approx.; for polypropylene and PETE/PVC
Tetrachloroethene 1,622
Magnesium 1,740
Beryllium 1,850
Glycerol 1,261 [13]
Silicon 2,330
Aluminium 2,700
Diiodomethane 3,325 liquid at room temperature
Diamond 3,500
Titanium 4,540
Selenium 4,800
Antimony 6,690
Zinc 7,000
Chromium 7,200
Manganese 7,325 Approx.
Tin 7,310
Iron 7,870
Niobium 8,570
Cobalt 8,900
Nickel 8,900
Copper 8,940
Bismuth 9,750
Molybdenum 10,220
Silver 10,500
Thorium 11,700
Rhodium 12,410
Mercury 13,546
Tantalum 16,600
Uranium 18,800
Tungsten 19,300
Gold 19,320
Plutonium 19,840
Platinum 21,450
Iridium 22,420
Osmium 22,570

*Air excluded when calculating density

Others

Entity ρ (kg/m3) Notes
Interstellar medium 1×10−19 Assuming 90% H, 10% He; variable T
The Earth 5,515 Mean density.[14]
The Inner Core of the Earth 13,000 Approx., as listed in Earth.[15]
The core of the Sun 33,000–160,000 Approx.[16]
Super-massive Black hole 9×105 Density of a 4.5-million-solar-mass black hole
Event horizon radius is 13.5 million km.
White dwarf star 2.1×109 Approx.[17]
Atomic nuclei 2.3×1017 Does not depend strongly on size of nucleus[18]
Neutron star 1×1018
Stellar-Mass Black hole 1×1018 Density of a 4-solar-mass black hole
Event horizon radius is 12 km.

Other common units

The SI unit for density is:

Litres and metric tons are not part of the SI, but are acceptable for use with it, leading to the following units:

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 decimetre (kg/dm3)
• grams per cubic centimetre (g/cc, gm/cc or g/cm3)
• 1 gram/cm3 = 1000 kg/m3
• megagrams (metric tons) per cubic metre (Mg/m3)

In US customary units density can be stated in:

Imperial units differing from the above (as the Imperial gallon and bushel differ from the US units) in practice are rarely 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.

References

1. ^
2. ^ "Density definition in Oil Gas Glossary". Oilgasglossary.com. Retrieved 2010-09-14.
3. ^ Archimedes, A Gold Thief and Buoyancy – by Larry "Harris" Taylor, Ph.D.
4. ^ Vitruvius on Architecture, Book IX, paragraphs 9–12, translated into English and in the original Latin.
5. ^ "EXHIBIT: The First Eureka Moment". Science 305 (5688): 1219e. 2004. doi:10.1126/science.305.5688.1219e.
6. ^ Fact or Fiction?: Archimedes Coined the Term "Eureka!" in the Bath, Scientific American, December 2006.
7. ^ New carbon nanotube struructure aerographite is lightest material champ. Phys.org (2012-07-13). Retrieved on 2012-07-14.
8. ^ Aerographit: Leichtestes Material der Welt entwickelt – SPIEGEL ONLINE. Spiegel.de (2012-07-11). Retrieved on 2012-07-14.
9. ^ a b "Re: which is more bouyant [sic] styrofoam or cork". Madsci.org. Retrieved 2010-09-14.
10. ^ "Wood Densities". www.engineeringtoolbox.com. Retrieved October 15, 2012.
11. ^ "Density of Wood". www.simetric.co.uk. Retrieved October 15, 2012.
12. ^ CRC Press Handbook of tables for Applied Engineering Science, 2nd Edition, 1976, Table 1-59
13. ^ glycerol composition at. Physics.nist.gov. Retrieved on 2012-07-14.
14. ^ Density of the Earth, wolframalpha.com
15. ^ Density of Earth's core, wolframalpha.com
16. ^ Density of the Sun's core, wolframalpha.com
17. ^ Extreme Stars: White Dwarfs & Neutron Stars, Jennifer Johnson, lecture notes, Astronomy 162, Ohio State University. Accessed: May 3, 2007.
18. ^ Nuclear Size and Density, HyperPhysics, Georgia State University. Accessed: June 26, 2009.