Density

Units

From the equation for density (ρ = m/V), mass density has any unit that is 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. 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.

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 g/cm³ or 1000 kg/m³, making any of these SI units numerically convenient to use as most solids and liquids have densities between 0.1 and 20 g/cm³.

• gram per cubic centimetre (g/cm³)
• kilogram per cubic decimetre (kg/dm³)
• megagram per cubic metre (Mg/m³)

Units employing the metric units litre and tonne (metric ton), which are not part of the SI, include:

• gram per millilitre (g/mL)
• kilogram per litre (kg/L)
• tonne per cubic metre (t/m³)

In US customary units density can be stated in:

• Avoirdupois ounce per cubic inch (1 g/cm³ ≈ 0.578036672 oz/cu in)
• Avoirdupois ounce per fluid ounce (1 g/cm³ ≈ 1.04317556 oz/US fl oz = 1.04317556 lb/US fl pint)
• Avoirdupois pound per cubic inch (1 g/cm³ ≈ 0.036127292 lb/cu in)
• pound per cubic foot (1 g/cm³ ≈ 62.427961 lb/cu ft)
• pound per cubic yard (1 g/cm³ ≈ 1685.5549 lb/cu yd)
• pound per US liquid gallon (1 g/cm³ ≈ 8.34540445 lb/US gal)
• pound per US bushel (1 g/cm³ ≈ 77.6888513 lb/bu)
• slug per cubic foot

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/cm³ ≈ 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.

The density of a crystalline material and be calculated from its formula mass (in daltons) and the volume of its unit cell, the density can be calculated. One dalton per cubic ångström is equal to a density of 1.66053906892(52) g/cm³, based on the 2022 CODATA recommended value of the dalton.

Measurement

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 weighing (a buoyancy method for liquids and solids), immersed body method (a buoyancy method for liquids), pycnometer (liquids and solids), air comparison pycnometer (solids), oscillating U-tube (liquids), as well as pour and tap (solids). 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: ${\displaystyle \rho (\overrightarrow{r})=dm/dV}$, where ${\displaystyle dV}$ is an elementary volume at position ${\displaystyle \overrightarrow{r}}$. The mass of the body then can be expressed as $${\displaystyle m={\int }_V\rho (\overrightarrow{r})dV.}$$

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 space 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 the material volumetric mass density. To determine the material 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 $${\displaystyle \rho =\frac{{\rho }_{T_0}}{1+\alpha \cdot \mathrm{\Delta }T},}$$ 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 ${\displaystyle {\rho }_i}$ in a solution sums to density of the solution, $${\displaystyle \rho =\sum _i{\rho }_i.}$$

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: $${\displaystyle \rho =\sum _i{\rho }_i\frac{V_i}{V}=\sum _i{\rho }_i{\phi }_i=\sum _i{\rho }_i\frac{V_i}{\sum _i{V}_i+\sum _i{V^E}_i},}$$ 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: $${\displaystyle {\overline{V^E}}_i=RT\frac{\mathrm{\partial }\ln {\gamma }_i}{\mathrm{\partial }P}.}$$

List of densities

Various materials

Others

Water

Notes:

Air

Molar volumes of liquid and solid phase of elements

Generalization: volumic quantities

The qualifier volumic is recommended in the International System of Quantities (ISO 80000-1) to denote the quotient of any physical quantity by volume. The expressions "per unit volume" or "volume ... density" (or simply "density") are also often used, with resulting units involving reciprocal cubic metre (m⁻³), for example:

• mass density or volumic mass
• charge density or volumic electric charge, the electric charge per volume
• number density or volumic number, the number of entities per unit volume
• activity density or volumic activity
• energy density, potential energy per unit volume
• force density, force per unit volume
• power density, power per unit volume