Density · aiwiki.page

Density is the mass of a sample or body divided by its volume, usually denoted by the Greek letter $\rho$ (rho). It describes how much mass occupies a given amount of space, rather than how large or heavy an object is in total. In physics and chemistry, the unqualified term generally means mass density. Other uses, such as probability density, specify a different quantity distributed over space or another domain. (goldbook.iupac.org)

Definition and units

The average density of a body is

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

where $m$ is its total mass and $V$ its volume. For a homogeneous material, this average also represents the density throughout the sample. For a heterogeneous body, density may vary with position, so a single average can conceal differences between its constituent regions. (openstax.org)

The coherent unit in the International System of Units (SI) is the kilogram per cubic metre, written $\mathrm{kg\,m^{-3}}$ . Its physical dimension is mass divided by length cubed, $ML^{-3}$ . Common laboratory units include grams per cubic centimetre and grams per millilitre:

1\ \mathrm{g\,cm^{-3}} =1\ \mathrm{g\,mL^{-1}} =1000\ \mathrm{kg\,m^{-3}}.

These conversions follow from the definitions of the units. (bipm.org)

For a homogeneous sample, the defining equation can be rearranged as $m=\rho V$ or $V=m/\rho$ . Thus, doubling the volume of a sample while retaining the same density doubles its mass; it does not double its density. Density is therefore an intensive property: under unchanged conditions, it does not depend on how much of a uniform material is present. (goldbook.iupac.org)

Local density and continuous descriptions

When density is not uniform, it is represented by a field $\rho(\mathbf r,t)$ , depending on position $\mathbf r$ and, if necessary, time $t$ . In a continuous description,

dm=\rho(\mathbf r,t)\,dV, \qquad m=\int_V \rho(\mathbf r,t)\,dV.

The volume integral adds the mass contributions of all parts of the body. Average density is consequently a volume-weighted average of local density, not necessarily the arithmetic average of densities measured at selected points. (openstax.org)

In fluid mechanics, this field participates in the local expression of conservation of mass:

\frac{\partial\rho}{\partial t} +\nabla\cdot(\rho\mathbf u)=0,

where $\mathbf u$ is fluid velocity. This continuity equation states that mass accumulating in a region must be accounted for by net mass flow into it, provided there are no mass sources or sinks. A density field therefore describes both the distribution of matter and its evolution during flow. (ocw.mit.edu)

Dependence on physical conditions

Density values must be associated with specified conditions, especially temperature, pressure, composition, and physical phase. A material's name alone is not always enough to determine its density: expansion changes volume, compression reduces it, and changes in composition can alter both mass and volume. (openstax.org)

For an ideal gas, combining $pV=nRT$ with $m=nM$ gives

\rho=\frac{pM}{RT},

where $p$ is absolute pressure, $M$ is molar mass, $R$ is the gas constant, and $T$ is absolute temperature. At fixed composition and temperature, ideal-gas density is proportional to pressure; at fixed pressure, it is inversely proportional to absolute temperature. Gases with different molar masses have different densities at the same pressure and temperature. This relationship is conditional on the ideal-gas approximation, rather than a universal equation for all materials. (openstax.org)

Water illustrates why temperature and phase must be stated. At approximately atmospheric pressure, liquid freshwater reaches its maximum density near $4^\circ\mathrm C$ , with a value close to $1000\ \mathrm{kg\,m^{-3}}$ . Cooling it toward freezing from that temperature lowers its density. Ordinary ice is less dense than liquid water because its crystal arrangement occupies more space for a given mass, allowing ice to float. Dissolved substances also affect water density. (usgs.gov)

Related quantities and definitions

Relative density, also called specific gravity, is the ratio of a material's density to that of a reference substance:

d=\frac{\rho}{\rho_{\mathrm{ref}}}.

It is dimensionless. For liquids and solids, water is a common reference, but the temperatures and pressures of both the sample and reference must be specified when precision matters. A relative density is not itself a density value with units. (media.iupac.org)

For granular and porous materials, the volume used in the calculation is especially important. Bulk density includes void space within the chosen bulk volume. In soil science, dry bulk density is dry solid mass divided by the total sample volume, including pores. Particle density instead relates solid mass to the volume of the solid particles, excluding the spaces between them. These quantities answer different questions and need not have the same value. (nrcs.usda.gov)

If $\phi$ is total porosity, the definitions give

\rho_{\mathrm{bulk}}=(1-\phi)\rho_{\mathrm{particle}},

for dry solid mass and consistently defined solid and pore volumes. Consequently, bulk density can increase through compaction even when the density of the constituent solid material remains unchanged. For porous samples, the measurement method and its treatment of accessible or inaccessible pores are part of the meaning of the reported result. (nrcs.usda.gov)

Measurement

Density measurements determine mass and volume directly or infer their ratio from a calibrated physical response. For an irregular solid, hydrostatic weighing compares its apparent weight in air with its apparent weight while submerged in a liquid of known density. The difference reflects the displaced liquid and allows the solid's volume and density to be calculated. (openstax.org)

Common liquid-density methods include:

Pycnometry: a vessel of calibrated volume is filled with liquid, and the liquid's mass is determined by weighing.
Known-volume sampling: a calibrated pipette delivers a measured volume whose mass is determined.
Oscillating-tube measurement: an instrument infers density from the oscillation characteristics of a tube containing the sample.

These methods differ in apparatus and calibration, but each ultimately establishes the relationship between sample mass and occupied volume. (nist.gov)

A hydrometer floats in a liquid and indicates density or relative density from its immersion depth. It sinks farther in a less dense liquid and less far in a denser one. Temperature affects both the liquid and the instrument, so readings must correspond to the instrument's calibration conditions or be corrected appropriately. (usgs.gov)

Precision work belongs to metrology and requires attention to calibration and measurement uncertainty. Air buoyancy can affect weighing, while temperature changes can affect both sample volume and the dimensions of volumetric apparatus. A density result is therefore more informative when accompanied by its conditions, method, and uncertainty than when presented as an isolated number. (nvlpubs.nist.gov)

Buoyancy and practical applications

Density is central to buoyancy. Archimedes' principle states that the upward buoyant force equals the weight of displaced fluid. In a uniform fluid,

F_{\mathrm b}=\rho_{\mathrm f}gV_{\mathrm{displaced}},

where $\rho_{\mathrm f}$ is fluid density and $g$ is gravitational acceleration. When buoyancy and weight are the relevant vertical forces, a freely immersed object denser than the fluid tends to sink, while one less dense tends to rise. For a freely floating body at equilibrium,

\frac{V_{\mathrm{submerged}}}{V_{\mathrm{body}}} =\frac{\bar\rho_{\mathrm{body}}}{\rho_{\mathrm f}}.

The relevant object density is its overall average, which can include enclosed low-density regions; it is not necessarily the density of its structural material. (openstax.org)

Density also determines how pressure changes with depth. For a stationary fluid of uniform density in a uniform gravitational field,

p=p_0+\rho gh,

where $p_0$ is pressure at the reference surface and $h$ is depth below it. The formula shows why equal depths in different fluids can produce different pressure increases. (openstax.org)

In analytical chemistry, liquid density supports conversion between mass-based and volume-based composition measurements. Hydrometers can also indicate changes in liquid composition, such as sugar content during fermentation. In soil studies, bulk density provides information about pore space and compaction, while water-density differences help explain the vertical arrangement of water in lakes. These applications require interpreting density together with composition, temperature, and the relevant volume definition. (nist.gov)

Probability density

In probability and statistics, “density” has a related mathematical meaning but is not mass per physical volume. A probability density function $f(x)$ distributes probability over possible values of a random variable. For a variable admitting such a density,

P(a\leq X\leq b)=\int_a^b f(x)\,dx, \qquad \int_{-\infty}^{\infty}f(x)\,dx=1.

The probability is obtained from the integral over an interval, not from the density's value at a single point. The shared idea is that a total quantity—mass in the physical case, probability in the statistical case—is recovered by integrating a density over the appropriate domain. (itl.nist.gov)

References

SI Brochure — English versionbipm.org
1 Fluids, Density, and Pressure — University Physics Volume 1openstax.org
09(F14) Chapter 6: Fluid Mechanicsocw.mit.edu
3 Stoichiometry of Gaseous Substances, Mixtures, and Reactions — Chemistry: Atoms First 2eopenstax.org
Water Density — U.S. Geological Surveyusgs.gov
Soil Survey Laboratorynrcs.usda.gov
Soil Tech Note 16A — Compacted Zone in Soilnrcs.usda.gov
Determination of Liquid Density — NISTnist.gov
NIST Calibration Services for Hydrometersnvlpubs.nist.gov
The electronic balance and some gravimetric applications (the density of solids and liquids, pycnometry and mass)nvlpubs.nist.gov