Density

Density is the amount of mass per unit volume of a substance or the inverse of specific volume. Density is, therefore, a scalar quantity. The symbol used to represent density in equations is ρ with SI units of kilograms per cubic meter. This term is the reciprocal of the specific volume.

\[\rho=\dfrac{m}{V}\;\left[\dfrac{\textrm{kg}}{\textrm{m}^3}\right]\]

For gases, the density can vary over a wide range because the particles are free to move closer together when constrained by pressure or volume. This variation of density is referred to as compressibility. Like pressure and temperature, density is a state variable of gas, and the change in density during any process is governed by the laws of thermodynamics. For a static gas, the density is the same throughout the entire container.

Like pressure and temperature, density is a variable state of gas, and the change in density during any process is governed by the laws of thermodynamics. For a static gas, the density is the same throughout the entire container. It can be shown by the kinetic theory that the density is inversely proportional to the size of the container in which a fixed mass of gas is confined. In this case of a fixed mass, the density decreases as the volume increases.

The density of a substance is dependent on temperature and pressure; in gases, the density is directly proportional to the pressure and inversely proportional to the temperature, while in solids and liquids, being incompressible substances, the variation of their density when the pressure varies is negligible. The density of a liquid is usually close to that of a solid, and much higher than in a gas.

The term “density” can also be applied to other quantities that have a spatial distribution. For example, the ratio of the number of electrons in a given volume to the volume itself is called electron density. More generally, considering any set of objects, we talk about number density. The ratio of the total charge distributed in a volume to the volume itself is commonly referred to as charge density; light energy per unit volume is referred to as light energy density. Thus in general the volume mass of a quantity is expressed by the ratio between the quantity of the quantity contained in an assigned volume and the value of the latter. In an even broader sense the concept has been extended to other areas such as geography: “population density” measures the population of a territory per unit area.

Real density and apparent density

The definition of density given above refers to a massive amount of solid matter, i.e., without internal voids. It is also called real density since it takes into account only the volume of the solid fraction.

The apparent density of a body is calculated in a manner formally analogous to the absolute density, but takes into account the total volume occupied by the solid, i.e. its external dimensions, including empty spaces (solids with closed cavities, with open cavities or with spongy structure). The definition of bulk density is also valid for granular matter contained in vessels such as sand and grains or soil.

Density of a mixture

To calculate the density of a mixture consisting of several substances (components), the following relation can be used:

\[\rho=\dfrac{1}{V}\sum_i m_i=\sum_i\rho_i\]

where: mi is the mass of component i in the mixture; V is the volume of the mixture; ρi is the mass concentration of component i in the mixture.

Relative density (specific gravity)

If liquids having different densities are poured in a container, they do not mix but overlap in such a way that the liquid having higher density is placed on the bottom while the one having lower density is placed on the top. Credit image: Stephen Oliver. Getty Images.

In general, relative density means the ratio of the mass of the body under examination to that of a body taken as a reference, for given temperature and pressure. Relative density is often defined as the ratio of the density of the body under test to that of pure water at a temperature of 4 °C and a pressure of 1 bar, or equivalently as the ratio of the mass of the body under test to that of an equal volume of pure water (distilled or deionized) at a temperature of 4 °C and a pressure of 1 bar.

\[RD=\dfrac{\rho_{sample}}{\rho_{reference}}\]

Relative density can be determined in several ways. Solid bodies that have a density greater than that of water are weighed first in air and then in water under fully immersed conditions. Relative density is obtained by dividing the weight in air by the decrease in weight of the immersed body (see Archimedes’ principle). To determine the relative density of fluids, special instruments called densimeters are used. If very accurate measurements are required, we proceed by determining the mass of a known volume of liquid or gas under controlled temperature conditions.

Specific gravity SG usually means relative density with respect to water. It is a dimensionless quantity.

\[SG=\dfrac{\rho_{sample}}{\rho_{water}}\]

The choice of the reference substance is arbitrary: it is also possible to define the specific gravity with reference to other substances, but specifying the conditions of temperature and pressure since the density of the material varies with temperature and pressure. This physical quantity is of fundamental importance in the context of fluid statics and in particular in the formulation of Archimedes’ law, which explains the floating of bodies in a fluid.

Density in electromagnetism

Density refers to both stationary and moving electrical charges. For electric charges distributed on elements of lines, surfaces, or volumes, the quantities are defined: electric charge density, or volumic electric charge, measured in the International System (SI) in coulombs per cubic meter (C/m3); electric charge surface density, or areic electric charge, measured in coulombs per square meter (C/m2); linear charge density, or lineal electric charge, measured in coulombs per meter (C/m). For moving electric charges, the quantities surface density and linear current density are defined.

The surface density of electric current, or areic electric current, is the ratio of the intensity of current flowing perpendicularly through the cross-section of a conductor to the area of that cross-section; it is measured in the International System (SI) in amperes per meter (A/m). The surface density of the current is constant over the entire crossed surface, if the current is continuous; it grows from the center to the periphery, if the current is alternating. Introducing this quantity, Ohm’s law is written in vector form E=ρJ, where E is the electric field strength, ρ the resistivity of the conductor, and J the vector of electric current density; J is related to the number N of charge carriers q per unit volume and their average velocity v by the relation J=Nqv.

The linear density of electric current, or lineal electric current, is a vector analogous to the areic electric current, but definable in the case that we must consider currents that cross surfaces of infinite area, which is necessary when the current density is not constant throughout the section of the conductor. It is measured in amperes per meter (A/m).

Density in astronomy

The term density appears in astronomy in many different contexts.

  • In its most generic use the density is the mass per unit volume of an object or region and might have units like kg/m3 or Mo/pc3.
  • The Number density is the number of a particular object or species per unit volume and might be used when describing the number of electrons per cubic centimetre in a plasma or the number of stars in the core of a globular cluster.
  • Density profile is the density as a function of some variable like distance from the centre of a star.
  • The mean or average density is the total mass of a region divided by the total volume of that region.

Since the behaviour of matter is often a function of density it is a very important quantity to determine when trying to explain the underlying physics of an object.

Density Functional Theory [DFT]

Theory that relates all the properties of a quantum system (for example of an atom, a molecule or their aggregates) with its electronic density. This theory offers an alternative method of quantum computation to the traditional one, in which instead the properties of the system (total energy, molecular geometry, frequencies of vibration of the nuclei, polarizability, ionization energy, etc.) are derived from its state function (or wave function), which is therefore the first objective of the calculation.

The density functional theory shifts the central problem of the quantum treatment of a microscopic system from the evaluation of its state function to that of its electronic density. The latter, often denoted by the symbol ρ(), is a function that provides point by point the probability of finding any of the N electrons of the system under consideration in a small volume centered at the point , with an arbitrary spin (α or β), while the other N-1 electrons have arbitrary spin and position.

Although the first attempts to follow this approach go back almost to the beginnings of quantum theory (Thomas and Fermi’s model for a gas of electrons, 1927), modern density functional theory arose shortly before the mid-1960s, with the rigorous demonstration, due to W. Kohn and P. Hoehenberg, that the electron density uniquely determines the Hamiltonian operator of a system and, therefore, its total energy in the fundamental state (the lowest energy state) and all other properties. In other words, the energy of the system is a functional of the electronic density (hence the name of the theory). Since the exact form of the functional is not known, however, it is necessary to devise approximate forms of it that allow the properties of interest to be derived concretely.

The density functional theory has gained wide popularity in the field of theoretical chemistry and physics research with application purposes, especially since the nineties of the twentieth century, alongside the traditional chemistry and quantum physics. Calculations based on the density functional theory, in fact, allow to obtain with considerable accuracy some molecular quantities (in particular, the distances and angles of bonds in molecules, and the frequencies of vibration of nuclear motions) with a lower computational cost (in terms of calculation time, and therefore also economic costs).

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