# Density

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Density
Common symbols
ρ
D
SI unit kg/m3

The density, or more precisely, the volumetric mass density, of a substance is its mass per unit volume. The symbol most often used for density is ρ (the lower case Greek letter rho), although the Latin letter D can also be used. Mathematically, density is defined as mass divided by volume: [1]

Mass is both a property of a physical body and a measure of its resistance to acceleration when a net force is applied. The object's mass also determines the strength of its gravitational attraction to other bodies.

Volume is the quantity of three-dimensional space enclosed by a closed surface, for example, the space that a substance or shape occupies or contains. Volume is often quantified numerically using the SI derived unit, the cubic metre. The volume of a container is generally understood to be the capacity of the container; i. e., the amount of fluid that the container could hold, rather than the amount of space the container itself displaces. Three dimensional mathematical shapes are also assigned volumes. Volumes of some simple shapes, such as regular, straight-edged, and circular shapes can be easily calculated using arithmetic formulas. Volumes of complicated shapes can be calculated with integral calculus if a formula exists for the shape's boundary. One-dimensional figures and two-dimensional shapes are assigned zero volume in the three-dimensional space.

## Contents

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

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 loosely defined as its weight per unit volume, [2] although this is scientifically inaccurate – this quantity is more specifically called specific weight.

In science and engineering, the weight of an object is related to the amount of force acting on the object, either due to gravity or to a reaction force that holds it in place.

The specific weight is the weight per unit volume of a material. The symbol of specific weight is γ.

For a pure substance the density has the same numerical value as its mass concentration. 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.

In chemistry, the mass concentrationρi is defined as the mass of a constituent mi divided by the volume of the mixture V:

In physics, buoyancy or upthrust, is an upward force exerted by a fluid that opposes the weight of an immersed object. In a column of fluid, pressure increases with depth as a result of the weight of the overlying fluid. Thus the pressure at the bottom of a column of fluid is greater than at the top of the column. Similarly, the pressure at the bottom of an object submerged in a fluid is greater than at the top of the object. The pressure difference results in a net upward force on the object. The magnitude of the force is proportional to the pressure difference, and is equivalent to the weight of the fluid that would otherwise occupy the volume of the object, i.e. the displaced fluid.

Osmium is a chemical element with symbol Os and atomic number 76. It is a hard, brittle, bluish-white transition metal in the platinum group that is found as a trace element in alloys, mostly in platinum ores. Osmium is the densest naturally occurring element, with an experimentally measured density of 22.59 g/cm3. Manufacturers use its alloys with platinum, iridium, and other platinum-group metals to make fountain pen nib tipping, electrical contacts, and in other applications that require extreme durability and hardness. The element's abundance in the Earth's crust is among the rarest.

To simplify comparisons of density across different systems of units, it is sometimes replaced by the dimensionless quantity "relative density" or "specific gravity", i.e. the ratio of the density of the material to that of a standard material, usually water. Thus a relative density less than one means that the substance floats in water.

Relative density, or specific gravity, is the ratio of the density of a substance to the density of a given reference material. Specific gravity usually means relative density with respect to water. The term "relative density" is often preferred in scientific usage. It is defined as a ratio of density of particular substance with that of water.

Specific gravity is the ratio of the density of a substance to the density of a reference substance; equivalently, it is the ratio of the mass of a substance to the mass of a reference substance for the same given volume. Apparent specific gravity is the ratio of the weight of a volume of the substance to the weight of an equal volume of the reference substance. The reference substance for liquids is nearly always water at its densest ; for gases it is air at room temperature. Nonetheless, the temperature and pressure must be specified for both the sample and the reference. Pressure is nearly always 1 atm (101.325 kPa).

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.

Convection is the heat transfer due to the bulk movement of molecules within fluids such as gases and liquids, including molten rock (rheid). Convection includes sub-mechanisms of advection, and diffusion.

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.

In thermodynamics, the specific volume of a substance is the ratio of the substance's volume to its mass. It is the reciprocal of density and an intrinsic property of matter as well. Specific volume is defined as the number of cubic meters occupied by one kilogram of a particular substance. The standard unit is the cubic meter per kilogram .

Thermodynamics is the branch of physics that has to do with heat and temperature and their relation to energy and work. The behavior of these quantities is governed by the four laws of thermodynamics, irrespective of the composition or specific properties of the material or system in question. The laws of thermodynamics are explained in terms of microscopic constituents by statistical mechanics. Thermodynamics applies to a wide variety of topics in science and engineering, especially physical chemistry, chemical engineering and mechanical engineering.

## 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. One g/cm3 is equal to one thousand kg/m3. One cubic centimetre (abbreviation cc) is equal to one millilitre. 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

A number of techniques as well as standards exist for the measurement of density of materials. Such techniques include the use of a hydrometer (a buoyancy method for liquids), Hydrostatic balance (a buoyancy method for liquids and solids), immersed body method (a buoyancy method for liquids), pycnometer (liquids and solids), air comparison pycnometer (solids), oscillating densitometer (liquids), as well as pour and tap (solids). [7] However, each individual method or technique measures different types of density (e.g. bulk density, skeletal density, etc.), and therefore it is necessary to have an understanding of the type of density being measured as well as the type of material in question.

### Homogeneous materials

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, a dasymeter or a Coriolis flow meter may be used, respectively. Similarly, hydrostatic weighing uses the displacement of water due to a submerged object to determine the density of the object.

### Heterogeneous materials

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: ${\displaystyle \rho ({\vec {r}})=dm/dV}$, where ${\displaystyle dV}$ is an elementary volume at position ${\displaystyle r}$. The mass of the body then can be expressed as

${\displaystyle m=\int _{V}\rho ({\vec {r}})\,dV.}$

### Non-compact materials

In practice, bulk materials such as sugar, sand, or snow contain voids. Many materials exist in nature as flakes, pellets, or granules.

Voids are regions which contain something other than the considered material. Commonly the void is air, but it could also be vacuum, liquid, solid, or a different gas or gaseous mixture.

The bulk volume of a material—inclusive of the void fraction—is often obtained by a simple measurement (e.g. with a calibrated measuring cup) or geometrically from known dimensions.

Mass divided by bulk volume determines bulk density. This is not the same thing as volumetric mass density.

To determine volumetric mass density, one must first discount the volume of the void fraction. Sometimes this can be determined by geometrical reasoning. For the close-packing of equal spheres the non-void fraction can be at most about 74%. It can also be determined empirically. Some bulk materials, however, such as sand, have a variable void fraction which depends on how the material is agitated or poured. It might be loose or compact, with more or less air space depending on handling.

In practice, the void fraction is not necessarily air, or even gaseous. In the case of sand, it could be water, which can be advantageous for measurement as the void fraction for sand saturated in water—once any air bubbles are thoroughly driven out—is potentially more consistent than dry sand measured with an air void.

In the case of non-compact materials, one must also take care in determining the mass of the material sample. If the material is under pressure (commonly ambient air pressure at the earth's surface) the determination of mass from a measured sample weight might need to account for buoyancy effects due to the density of the void constituent, depending on how the measurement was conducted. In the case of dry sand, sand is so much denser than air that the buoyancy effect is commonly neglected (less than one part in one thousand).

Mass change upon displacing one void material with another while maintaining constant volume can be used to estimate the void fraction, if the difference in density of the two voids materials is reliably known.

## 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

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

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 :

${\displaystyle \rho ={\frac {\rho _{T_{0}}}{1+\alpha \cdot \Delta T}}}$

where ${\displaystyle \rho _{T_{0}}}$ is the density at a reference temperature, ${\displaystyle \alpha }$ is the thermal expansion coefficient of the material at temperatures close to ${\displaystyle T_{0}}$.

## 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.

${\displaystyle \rho =\sum _{i}\varrho _{i}\,}$

Expressed as a function of the densities of pure components of the mixture and their volume participation, it allows the determination of excess molar volumes:

${\displaystyle \rho =\sum _{i}\rho _{i}{\frac {V_{i}}{V}}\,=\sum _{i}\rho _{i}\varphi _{i}=\sum _{i}\rho _{i}{\frac {V_{i}}{\sum _{i}V_{i}+\sum _{i}{V^{E}}_{i}}}}$

provided that there is no interaction between the components.

Knowing the relation between excess volumes and activity coefficients of the components, one can determine the activity coefficients.

${\displaystyle {\overline {V^{E}}}_{i}=RT{\frac {\partial \ln \gamma _{i}}{\partial P}}}$

## Densities

### Various materials

Selected chemical elements are listed here. For the densities of all chemical elements, see List of chemical elements
Densities of various materials covering a range of values
Materialρ (kg/m3) [note 1] Notes
Hydrogen 0.0898
Helium 0.179
Aerographite 0.2 [note 2] [8] [9]
Metallic microlattice 0.9 [note 2]
Aerogel 1.0 [note 2]
Air 1.2At sea level
Tungsten hexafluoride 12.4One of the heaviest known gases at standard conditions
Liquid hydrogen 70At approx. −255 °C
Styrofoam 75Approx. [10]
Cork 240Approx. [10]
Pine 373 [11]
Lithium 535
Wood 700Seasoned, typical [12] [13]
Oak 710 [11]
Potassium 860 [14]
Ice 916.7At temperature < 0 °C
Cooking oil 910–930
Sodium 970
Water (fresh)1,000At 4 °C, the temperature of its maximum density
Water (salt)1,0303%
Liquid oxygen 1,141At approx. −219 °C
Nylon 1,150
Plastics 1,175Approx.; for polypropylene and PETE/PVC
Tetrachloroethene 1,622
Magnesium 1,740
Beryllium 1,850
Glycerol 1,261 [15]
Concrete 2,400 [16] [17]
Silicon 2,330
Aluminium 2,700
Diiodomethane 3,325Liquid at room temperature
Diamond 3,500
Titanium 4,540
Selenium 4,800
Antimony 6,690
Zinc 7,000
Chromium 7,200
Tin 7,310
Manganese 7,325Approx.
Iron 7,870
Niobium 8,570
Brass 8,600 [17]
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
Rhenium 21,020
Platinum 21,450
Iridium 22,420
Osmium 22,570
Notes:
1. 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).
2. Air contained in material excluded when calculating density

### Others

Entityρ (kg/m3)Notes
Interstellar medium 1×10−19Assuming 90% H, 10% He; variable T
The Earth 5,515Mean density. [18]
The inner core of the Earth13,000Approx., as listed in Earth. [19]
The core of the Sun 33,000–160,000Approx. [20]
Super-massive black hole 9×105Density of a 4.5-million-solar-mass black hole
Event horizon radius is 13.5 million km.
White dwarf star2.1×109Approx. [21]
Atomic nuclei 2.3×1017Does not depend strongly on size of nucleus [22]
Neutron star 1×1018
Stellar-mass black hole 1×1018Density of a 4-solar-mass black hole
Event horizon radius is 12 km.

### Water

Density of liquid water at 1 atm pressure
Temp. (°C) [note 1] Density (kg/m3)
−30983.854
−20993.547
−10998.117
0999.8395
4999.9720
10999.7026
15999.1026
20998.2071
22997.7735
25997.0479
30995.6502
40992.2
60983.2
80971.8
100958.4
Notes:
1. Values below 0 °C refer to supercooled water.

### Air

Density of air at 1 atm pressure
T (°C)ρ (kg/m3)
−251.423
−201.395
−151.368
−101.342
−51.316
01.293
51.269
101.247
151.225
201.204
251.184
301.164
351.146

## Common units

The SI unit for density is:

The litre 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.

• kilogram per cubic decimetre (kg/dm3)
• gram per cubic centimetre (g/cm3)
• 1 g/cm3 = 1000 kg/m3
• megagram (metric ton) 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 Imperial gallon was based on the concept that an Imperial fluid ounce of water would have a mass of one Avoirdupois ounce, and indeed 1 g/cc ≈ 1.00224129 ounces per Imperial fluid ounce = 10.0224129 pounds per Imperial gallon. The density of precious metals could conceivably be based on Troy ounces and pounds, a possible cause of confusion.

## Related Research Articles

In physics and thermodynamics, an equation of state is a thermodynamic equation relating state variables which describe the state of matter under a given set of physical conditions, such as pressure, volume, temperature (PVT), or internal energy. Equations of state are useful in describing the properties of fluids, mixtures of fluids, solids, and the interior of stars.

Pressure is the force applied perpendicular to the surface of an object per unit area over which that force is distributed. Gauge pressure is the pressure relative to the ambient pressure.

The molar volume, symbol Vm, is the volume occupied by one mole of a substance at a given temperature and pressure. It is equal to the molar mass (M) divided by the mass density (ρ). It has the SI unit cubic metres per mole (m3/mol), although it is more practical to use the units cubic decimetres per mole (dm3/mol) for gases and cubic centimetres per mole (cm3/mol) for liquids and solids.

In thermodynamics and fluid mechanics, compressibility is a measure of the relative volume change of a fluid or solid as a response to a pressure change. In its simple form, the compressibility may be expressed as

Molality, also called molal concentration, is a measure of the concentration of a solute in a solution in terms of amount of substance in a specified amount of mass of the solvent. This contrasts with the definition of molarity which is based on a specified volume of solution.

Foam is an object formed by trapping pockets of gas in a liquid or solid. A bath sponge and the head on a glass of beer are examples of foams. In most foams, the volume of gas is large, with thin films of liquid or solid separating the regions of gas. Soap foams are also known as suds.

The density of air or atmospheric density, denoted ρ, is the mass per unit volume of Earth's atmosphere. Air density, like air pressure, decreases with increasing altitude. It also changes with variation in atmospheric pressure, temperature and humidity. At 101.325 kPa (abs) and 15°C, air has a density of approximately 1.225 kg/m³ according to ISA.

A slurry is a mixture of solids with specific gravity greater than 1 suspended in liquid, usually water. Most common use of slurry is as a mean of transportation of solids being the liquid a carrier that is pumped on a device like a centrifugal pump. The size o solid particles may vary from 1 micron up to hundred of milliliters. The solids may settle at certain transport velocity and the mixture can behave as Newtonian or non-Newtonian fluid. Depending on the mixture the slurry may be abrasive and corrosive.

Soil mechanics is a branch of soil physics and applied mechanics that describes the behavior of soils. It differs from fluid mechanics and solid mechanics in the sense that soils consist of a heterogeneous mixture of fluids and particles but soil may also contain organic solids and other matter. Along with rock mechanics, soil mechanics provides the theoretical basis for analysis in geotechnical engineering, a subdiscipline of civil engineering, and engineering geology, a subdiscipline of geology. Soil mechanics is used to analyze the deformations of and flow of fluids within natural and man-made structures that are supported on or made of soil, or structures that are buried in soils. Example applications are building and bridge foundations, retaining walls, dams, and buried pipeline systems. Principles of soil mechanics are also used in related disciplines such as engineering geology, geophysical engineering, coastal engineering, agricultural engineering, hydrology and soil physics.

A fluidised bed is a physical phenomenon occurring when a quantity of a solid particulate substance is placed under appropriate conditions to cause a solid/fluid mixture to behave as a fluid. This is usually achieved by the introduction of pressurized fluid through the particulate medium. This results in the medium then having many properties and characteristics of normal fluids, such as the ability to free-flow under gravity, or to be pumped using fluid type technologies.

A liquid is a nearly incompressible fluid that conforms to the shape of its container but retains a (nearly) constant volume independent of pressure. As such, it is one of the four fundamental states of matter, and is the only state with a definite volume but no fixed shape. A liquid is made up of tiny vibrating particles of matter, such as atoms, held together by intermolecular bonds. Water is, by far, the most common liquid on Earth. Like a gas, a liquid is able to flow and take the shape of a container. Most liquids resist compression, although others can be compressed. Unlike a gas, a liquid does not disperse to fill every space of a container, and maintains a fairly constant density. A distinctive property of the liquid state is surface tension, leading to wetting phenomena.

Porosity or void fraction is a measure of the void spaces in a material, and is a fraction of the volume of voids over the total volume, between 0 and 1, or as a percentage between 0% and 100%. Strictly speaking, some tests measure the "accessible void", the total amount of void space accessible from the surface. There are many ways to test porosity in a substance or part, such as industrial CT scanning. The term porosity is used in multiple fields including pharmaceutics, ceramics, metallurgy, materials, manufacturing, hydrology, earth sciences, soil mechanics and engineering.

In thermodynamics, the volume of a system is an important extensive parameter for describing its thermodynamic state. The specific volume, an intensive property, is the system's volume per unit of mass. Volume is a function of state and is interdependent with other thermodynamic properties such as pressure and temperature. For example, volume is related to the pressure and temperature of an ideal gas by the ideal gas law.

Slip ratio in gas–liquid (two-phase) flow, is defined as the ratio of the velocity of the gas phase to the velocity of the liquid phase.

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