Density of a substance: essence and applications

Density of a substance — a scalar physical quantity defined as the ratio of a body's mass to the volume it occupies, or as the derivative of mass with respect to volume:

.

These expressions are not equivalent, and the choice depends on exactly which density is being considered. The following are distinguished:

• average density of a body — the ratio of the body's mass to its volume . In the homogeneous case it is simply called the density of the body;
• density of a substance — the density of a homogeneous body made of that substance;
• density of a body at a point — the limit of the ratio of the mass of a small part of the body () containing that point to the volume of that small part () as the volume tends to zero , or, in short, . Since at the atomic level any body is inhomogeneous, the limiting process must stop at a volume corresponding to the physical model being used.

For a point mass the density is infinite. Mathematically it can be defined either as a measure or as a Radon–Nikodym derivative with respect to some reference measure.

Density is usually denoted by the Greek letter [rho] (the origin of the notation remains to be clarified); sometimes the Latin letters D and d are used (from Latin densitas, «density»). Based on the definition of density, its dimension is kg/m³ in SI and g/cm³ in the CGS system.

The concept of «density» in physics may have a broader interpretation. There are surface density (the ratio of mass to area) and linear density (the ratio of mass to length), applied respectively to flat (two-dimensional) and elongated (one-dimensional) objects. In addition, one speaks not only of the density of mass, but also of the density of other quantities, for example, energy or electric charge. In such cases, clarifying words are added to the term «density», say «linear charge density». «By default», density is understood to mean the above-mentioned (three-dimensional, kg/m3) mass density.

Here density plays a decisive role — the greater it is, the more the body will sink. Density and hardness are different concepts. Water and ice are a good example of this. Water has a higher density than ice, which is why ice floats on its surface rather than sinking.

A vivid example of this is the simple «Density Tower» experiment. It can be carried out at home.

fig. Density Tower

Formula for finding density

Density (the density of a homogeneous body or the average density of an inhomogeneous one) is found by the formula:

where M — the mass of the body, V — its volume; the formula is simply a mathematical statement of the definition of the term «density» given above.

When calculating the density of gases at standard conditions, this formula can also be written as:

where — the molar mass of the gas, — the molar volume (at standard conditions approximately equal to 22.4 L/mol).

The density of a body at a point is written as

then the mass of an inhomogeneous body (a body whose density depends on coordinates) is calculated as

The case of granular and porous bodies : Bulk density

In the case of granular and porous bodies, a distinction is made between

• true density, determined without accounting for voids;
• bulk density, calculated as the ratio of the mass of the substance to the entire volume it occupies.

True density is obtained from the bulk (apparent) density using the value of the porosity coefficient — the fraction of void volume in the occupied volume.

Dependence of density on temperature

As a rule, when temperature decreases, density increases, although there are substances whose density in a certain temperature range behaves differently, for example, water, bronze and cast iron. Thus, the density of water has a maximum value at 4 °C and decreases both as the temperature rises and as it falls relative to this value.

When the state of aggregation changes, the density of a substance changes abruptly: density increases during the transition from the gaseous state to the liquid state and during solidification of a liquid. Water, silicon, bismuth and some other substances are exceptions to this rule, since their density decreases upon solidification.

Range of densities in nature

For various natural objects, density varies over a very wide range.

• The lowest density is that of the intergalactic medium (2·10⁻³¹—5·10⁻³¹ kg/m³, not counting dark matter) .
• The density of the interstellar medium is approximately 10⁻²³—10⁻²¹ kg/m³.
• The average density of red giants within their photospheres is much lower than that of the Sun — because their radius is hundreds of times larger for a comparable mass.
• The density of gaseous hydrogen (the lightest gas) at standard conditions is 0.0899 kg/m³.
• The density of dry air at standard conditions is 1.293 kg/m³.
• One of the heaviest gases, tungsten hexafluoride, is about 10 times heavier than air (12.9 kg/m³ at +20 °C)
• Liquid hydrogen at atmospheric pressure and a temperature of −253 °C has a density of 70 kg/m³.
• The density of liquid helium at atmospheric pressure is 130 kg/m³.
• The average density of the human body ranges from 940—990 kg/m³ at full inhalation, to 1010—1070 kg/m³ at full exhalation.
• The density of fresh water at 4 °C is 1000 kg/m³.
• The average density of the Sun within the photosphere is about 1410 kg/m³, roughly 1.4 times higher than the density of water.
• Granite has a density of 2600 kg/m³.
• The average density of the Earth is 5520 kg/m³.
• The density of iron is 7874 kg/m³.
• The density of metallic uranium is 19100 kg/m³.
• The density of gold is 19320 kg/m³.
• The density of neptunium — the densest actinide — is 20200 kg/m³.
• The densest substances at standard conditions are the platinum-group metals of the sixth period (osmium, iridium, platinum), as well as rhenium. They have a density of 21000—22700 kg/m³.
• The density of atomic nuclei is approximately 2·10¹⁷ kg/m³.
• Theoretically, the upper bound of density according to current physical concepts is the Planck density, 5.1⋅10⁹⁶ kg/m³.

Densities of astronomical objects

Average density of the celestial bodies of the Solar
system (in g/cm³)

• See the average densities of the celestial bodies of the Solar System in the inset.
• The interplanetary medium in the Solar System is fairly inhomogeneous and can change over time; its density in the vicinity of Earth is ~10⁻²¹÷10⁻²⁰ kg/m³.
• The density of the interstellar medium is ~10⁻²³÷10⁻²¹ kg/m³.
• The density of the intergalactic medium is 2×10⁻³⁴÷5×10⁻³⁴ kg/m³.
• The average density of red giants is many orders of magnitude lower because their radius is hundreds of times larger than that of the Sun.
• The density of white dwarfs is 10⁸÷10¹² kg/m³
• The density of neutron stars is on the order of 10¹⁷÷10¹⁸ kg/m³.
• The average density (by volume within the event horizon) of a black hole depends on its mass and is expressed by the formula:

The average density falls off inversely proportional to the square of the black hole's mass (ρ~M⁻²). Thus, if a black hole with a mass on the order of the Sun's has a density of about 10¹⁹ kg/m³, exceeding nuclear density (2×10¹⁷ kg/m³), then a supermassive black hole with a mass of 10⁹ solar masses (such black holes are assumed to exist in quasars) has an average density of about 20 kg/m³, which is significantly less than the density of water (1000 kg/m³).

Densities of some gases

To calculate the density of an arbitrary ideal gas under arbitrary conditions, one can use a formula derived from the ideal gas equation of state:

,

where:

• — pressure,
• — molar mass,
• — the universal gas constant, equal to approximately 8.314 J/(mol·K)
• — thermodynamic temperature.

Densities of some liquids

Density of some types of wood

Density of some metals

The density values of metals can vary within very wide limits: from the smallest value for lithium, which is lighter than water, to the largest value for osmium, which is heavier than gold and platinum.

Measuring density

The following are used to measure density:

• Pycnometer — a device for measuring true density
• Various types of hydrometers — devices for measuring the density of liquids.
• Kachinsky's density borer and Zaidelman's auger — devices for measuring soil density.
• Vibrating densitometer — a device for measuring the density of a liquid or gas under pressure.

Applications of density

determining a substance by mass and volume

1. Over time, oil stops being squeezed out of the rock formation by the weight of the overlying strata. Then the reservoir pressure maintenance (RPM) system comes into play. Injection wells are drilled, and water is pumped into them under high pressure. Naturally, the injected or formation water will sooner or later reach the production wells and rise to the surface together with the oil.

2. Cream makes up the greater part of the fat contained in milk. When milk settles, the light fats rise to the surface; in the past they could simply be skimmed off fresh milk after it had settled for some time. Under industrial conditions this product is obtained by separation.

3. In countries where malaria-carrying mosquitoes are common, their larvae are combated by pouring liquid oils over the places where they live. The larvae need air. But they cannot pierce even a thin layer of oil with their spiracles. As a result – a sharp decrease in their numbers

See also

• List of chemical elements with their density
• Specific weight
• Specific density
• Relative density
• Bulk density
• Condensation
• Consistency (Latin consistere — to consist) — the state of a substance, the degree of softness or firmness (hardness) of something — of semi-solid, semi-soft substances (oils, soap, paints, building mortars, etc.); for example, glycerin has a syrup-like consistency.
• Consistometer — a device for measuring, in conventional physical units, the consistency of various colloidal and gel-like substances, as well as suspensions and coarsely dispersed media, for example, pastes, liniments, gels, creams, ointments.
• Particle concentration
• Solution concentration
• Charge density
• Continuity equation