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The volumetric heat capacity of a material is the
heat capacity Heat capacity or thermal capacity is a physical property of matter, defined as the amount of heat to be supplied to an object to produce a unit change in its temperature. The SI unit of heat capacity is joule per kelvin (J/K). Heat cap ...
of a sample of the substance divided by the volume of the sample. It is the amount of
energy In physics, energy (from Ancient Greek: ἐνέργεια, ''enérgeia'', “activity”) is the quantitative property that is transferred to a body or to a physical system, recognizable in the performance of work and in the form of ...
that must be added, in the form of
heat In thermodynamics, heat is defined as the form of energy crossing the boundary of a thermodynamic system by virtue of a temperature difference across the boundary. A thermodynamic system does not ''contain'' heat. Nevertheless, the term is ...
, to one unit of
volume Volume is a measure of occupied three-dimensional space. It is often quantified numerically using SI derived units (such as the cubic metre and litre) or by various imperial or US customary units (such as the gallon, quart, cubic inch). ...
of the material in order to cause an increase of one unit in its
temperature Temperature is a physical quantity that expresses quantitatively the perceptions of hotness and coldness. Temperature is measured with a thermometer. Thermometers are calibrated in various temperature scales that historically have relied o ...
. The SI unit of volumetric heat capacity is joule per kelvin per
cubic meter The cubic metre (in Commonwealth English and international spelling as used by the International Bureau of Weights and Measures) or cubic meter (in American English) is the unit of volume in the International System of Units (SI). Its symbol is m ...
, J⋅K−1⋅m−3. The volumetric heat capacity can also be expressed as the
specific heat capacity In thermodynamics, the specific heat capacity (symbol ) of a substance is the heat capacity of a sample of the substance divided by the mass of the sample, also sometimes referred to as massic heat capacity. Informally, it is the amount of heat t ...
(heat capacity per unit of mass, in J⋅K−1kg−1) times the
density Density (volumetric mass density or specific mass) is the substance's mass per unit of 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. Mathematicall ...
of the substance (in kg/ L, or g/ mL). This quantity may be convenient for materials that are commonly measured by volume rather than mass, as is often the case in
engineering Engineering is the use of scientific principles to design and build machines, structures, and other items, including bridges, tunnels, roads, vehicles, and buildings. The discipline of engineering encompasses a broad range of more speciali ...
and other technical disciplines. The volumetric heat capacity often varies with temperature, and is different for each
state of matter In physics, a state of matter is one of the distinct forms in which matter can exist. Four states of matter are observable in everyday life: solid, liquid, gas, and plasma. Many intermediate states are known to exist, such as liquid crystal, ...
. While the substance is undergoing a
phase transition In chemistry, thermodynamics, and other related fields, a phase transition (or phase change) is the physical process of transition between one state of a medium and another. Commonly the term is used to refer to changes among the basic states ...
, such as melting or boiling, its volumetric heat capacity is technically
infinite Infinite may refer to: Mathematics * Infinite set, a set that is not a finite set *Infinity, an abstract concept describing something without any limit Music *Infinite (group), a South Korean boy band *''Infinite'' (EP), debut EP of American m ...
, because the heat goes into changing its state rather than raising its temperature. The volumetric heat capacity of a substance, especially a gas, may be significantly higher when it is allowed to expand as it is heated (volumetric heat capacity ''at constant pressure'') than when is heated in a closed vessel that prevents expansion (volumetric heat capacity ''at constant volume''). If the amount of substance is taken to be the number of
moles Moles can refer to: * Moles de Xert, a mountain range in the Baix Maestrat comarca, Valencian Community, Spain *The Moles (Australian band) *The Moles, alter ego of Scottish band Simon Dupree and the Big Sound People * Abraham Moles, French engin ...
in the sample (as is sometimes done in chemistry), one gets the molar heat capacity (whose SI unit is joule per kelvin per mole, J⋅K−1⋅mol−1).


Definition

The volumetric heat capacity is defined as :s(T) = \frac = \frac \lim_\frac where V(T) is the volume of the sample at temperature T, and \Delta Q(T) is the amount of heat energy needed to raise the temperature of the sample from T to T + \Delta T. This parameter is an intensive property of the substance. Since both the heat capacity of an object and its volume may vary with temperature, in unrelated ways, the volumetric heat capacity is usually a function of temperature too. It is equal to the specific heat c(T) of the substance times its
density Density (volumetric mass density or specific mass) is the substance's mass per unit of 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. Mathematicall ...
(mass per volume) \rho(T), both measured at the temperature T. Its SI unit is joule per kelvin per cubic meter (J⋅K−1⋅m−3). This quantity is used almost exclusively for liquids and solids, since for gases it may be confused with the "specific heat capacity at constant volume", which generally has very different values. International standards now recommend that "specific heat capacity" always refer to capacity per unit of mass. Therefore, the word "volumetric" should always be used for this quantity.


History

Dulong and Petit predicted in 1818 that the product of solid substance density and specific heat capacity (ρcp) would be constant for all solids. This amounted to a prediction that volumetric heat capacity in solids would be constant. In 1819 they found that volumetric heat capacities were not quite constant, but that the most constant quantity was the heat capacity of solids adjusted by the presumed weight of the atoms of the substance, as defined by Dalton (the Dulong–Petit law). This quantity was proportional to the heat capacity per
atomic weight Relative atomic mass (symbol: ''A''; sometimes abbreviated RAM or r.a.m.), also known by the deprecated synonym atomic weight, is a dimensionless physical quantity defined as the ratio of the average mass of atoms of a chemical element in a giv ...
(or per
molar mass In chemistry, the molar mass of a chemical compound is defined as the mass of a sample of that compound divided by the amount of substance which is the number of moles in that sample, measured in moles. The molar mass is a bulk, not molecular, ...
), which suggested that it is the heat capacity ''per atom'' (not per unit of volume) which is closest to being a constant in solids. Eventually it became clear that heat capacities per particle for all substances in all states are the same, to within a factor of two, so long as temperatures are not in the cryogenic range.


Typical values

The volumetric heat capacity of solid materials at room temperatures and above varies widely, from about 1.2 MJ⋅K−1⋅m−3 (for example
bismuth Bismuth is a chemical element with the symbol Bi and atomic number 83. It is a post-transition metal and one of the pnictogens, with chemical properties resembling its lighter group 15 siblings arsenic and antimony. Elemental bismuth occurs ...
) to 3.4 MJ⋅K−1⋅m−3 (for example ironBased o
NIST data
and density.
). This is mostly due to differences in the physical size of atoms. Atoms vary greatly in density, with the heaviest often being more dense, and thus are closer to taking up the same average volume in solids than their mass alone would predict. If all atoms ''were'' the same size, molar and volumetric heat capacity would be proportional and differ by only a single constant reflecting ratios of the atomic molar volume of materials (their atomic density). An additional factor for all types of specific heat capacities (including molar specific heats) then further reflects degrees of freedom available to the atoms composing the substance, at various temperatures. For most liquids, the volumetric heat capacity is narrower, for example
octane Octane is a hydrocarbon and an alkane with the chemical formula , and the condensed structural formula . Octane has many structural isomers that differ by the amount and location of branching in the carbon chain. One of these isomers, 2,2,4-t ...
at 1.64 MJ⋅K−1⋅m−3 or
ethanol Ethanol (abbr. EtOH; also called ethyl alcohol, grain alcohol, drinking alcohol, or simply alcohol) is an organic compound. It is an alcohol with the chemical formula . Its formula can be also written as or (an ethyl group linked to a ...
at 1.9. This reflects the modest loss of degrees of freedom for particles in liquids as compared with solids. However,
water Water (chemical formula ) is an inorganic, transparent, tasteless, odorless, and nearly colorless chemical substance, which is the main constituent of Earth's hydrosphere and the fluids of all known living organisms (in which it acts as ...
has a very high volumetric heat capacity, at 4.18 MJ⋅K−1⋅m−3, and
ammonia Ammonia is an inorganic compound of nitrogen and hydrogen with the formula . A stable binary hydride, and the simplest pnictogen hydride, ammonia is a colourless gas with a distinct pungent smell. Biologically, it is a common nitrogenous ...
is also fairly high: 3.3 MJ⋅K−1⋅m−3. For gases at room temperature, the range of volumetric heat capacities per atom (not per molecule) only varies between different gases by a small factor less than two, because every
ideal gas An ideal gas is a theoretical gas composed of many randomly moving point particles that are not subject to interparticle interactions. The ideal gas concept is useful because it obeys the ideal gas law, a simplified equation of state, and is a ...
has the same molar volume. Thus, each gas molecule occupies the same mean volume in all ideal gases, regardless of the type of gas (see kinetic theory). This fact gives each gas molecule the same effective "volume" in all ideal gases (although this volume/molecule in gases is far larger than molecules occupy on average in solids or liquids). Thus, in the limit of ideal gas behavior (which many gases approximate except at low temperatures and/or extremes of pressure) this property reduces differences in gas volumetric heat capacity to simple differences in the heat capacities of individual molecules. As noted, these differ by a factor depending on the degrees of freedom available to particles within the molecules.


Volumetric heat capacity of gases

Large complex gas molecules may have high heat capacities per mole (of molecules), but their heat capacities per mole of atoms are very similar to those of liquids and solids, again differing by less than a factor of two per mole of atoms. This factor of two represents vibrational degrees of freedom available in solids vs. gas molecules of various complexities. In monatomic gases (like argon) at room temperature and constant volume, volumetric heat capacities are all very close to 0.5 kJ⋅K−1⋅m−3, which is the same as the theoretical value of RT per kelvin per mole of gas molecules (where R is the
gas constant The molar gas constant (also known as the gas constant, universal gas constant, or ideal gas constant) is denoted by the symbol or . It is the molar equivalent to the Boltzmann constant, expressed in units of energy per temperature increment per ...
and T is temperature). As noted, the much lower values for gas heat capacity in terms of volume as compared with solids (although more comparable per mole, see below) results mostly from the fact that gases under standard conditions consist of mostly empty space (about 99.9% of volume), which is not filled by the atomic volumes of the atoms in the gas. Since the molar volume of gases is very roughly 1000 times that of solids and liquids, this results in a factor of about 1000 loss in volumetric heat capacity for gases, as compared with liquids and solids. Monatomic gas heat capacities per atom (not per molecule) are decreased by a factor of 2 with regard to solids, due to loss of half of the potential
degrees of freedom Degrees of freedom (often abbreviated df or DOF) refers to the number of independent variables or parameters of a thermodynamic system. In various scientific fields, the word "freedom" is used to describe the limits to which physical movement or ...
per atom for storing energy in a monatomic gas, as compared with regard to an ideal solid. There is some difference in the heat capacity of monatomic vs. polyatomic gasses, and also gas heat capacity is temperature-dependent in many ranges for polyatomic gases; these factors act to modestly (up to the discussed factor of 2) increase heat capacity per atom in polyatomic gases, as compared with monatomic gases. Volumetric heat capacities in polyatomic gases vary widely, however, since they are dependent largely on the number of atoms per molecule in the gas, which in turn determines the total number of atoms per volume in the gas. The volumetric heat capacity is defined as having SI units of J/( m3K). It can also be described in Imperial units of BTU/( ft3°F).


Volumetric heat capacity of solids

Since the
bulk density Bulk density, also called apparent density or volumetric density, is a property of powders, granules, and other "divided" solids, especially used in reference to mineral components ( soil, gravel), chemical substances, ( pharmaceutical) ingredi ...
of a solid chemical element is strongly related to its molar mass (usually about 3''R'' per mole, as noted above), there exists noticeable inverse correlation between a solid's density and its specific heat capacity on a per-mass basis. This is due to a very approximate tendency of atoms of most elements to be about the same size, despite much wider variations in density and atomic weight. These two factors (constancy of atomic volume and constancy of mole-specific heat capacity) result in a good correlation between the ''volume'' of any given solid chemical element and its total heat capacity. Another way of stating this, is that the volume-specific heat capacity (volumetric heat capacity) of solid elements is roughly a constant. The molar volume of solid elements is very roughly constant, and (even more reliably) so also is the molar heat capacity for most solid substances. These two factors determine the volumetric heat capacity, which as a bulk property may be striking in consistency. For example, the element uranium is a metal which has a density almost 36 times that of the metal lithium, but uranium's volumetric heat capacity is only about 20% larger than lithium's. Since the volume-specific corollary of the Dulong–Petit specific heat capacity relationship requires that atoms of all elements take up (on average) the same volume in solids, there are many departures from it, with most of these due to variations in atomic size. For instance,
arsenic Arsenic is a chemical element with the symbol As and atomic number 33. Arsenic occurs in many minerals, usually in combination with sulfur and metals, but also as a pure elemental crystal. Arsenic is a metalloid. It has various allotropes, b ...
, which is only 14.5% less dense than
antimony Antimony is a chemical element with the symbol Sb (from la, stibium) and atomic number 51. A lustrous gray metalloid, it is found in nature mainly as the sulfide mineral stibnite (Sb2S3). Antimony compounds have been known since ancient ti ...
, has nearly 59% more specific heat capacity on a mass basis. In other words; even though an ingot of arsenic is only about 17% larger than an antimony one of the same mass, it absorbs about 59% more heat for a given temperature rise. The heat capacity ratios of the two substances closely follows the ratios of their molar volumes (the ratios of numbers of atoms in the same volume of each substance); the departure from the correlation to simple volumes in this case is due to lighter arsenic atoms being significantly more closely packed than antimony atoms, instead of similar size. In other words, similar-sized atoms would cause a mole of arsenic to be 63% larger than a mole of antimony, with a correspondingly lower density, allowing its volume to more closely mirror its heat capacity behavior.


Thermal inertia

Thermal inertia (or thermal effusivity) is a term commonly used for modelling
heat transfer Heat transfer is a discipline of thermal engineering that concerns the generation, use, conversion, and exchange of thermal energy ( heat) between physical systems. Heat transfer is classified into various mechanisms, such as thermal conducti ...
s. It is a bulk material property related to
thermal conductivity The thermal conductivity of a material is a measure of its ability to conduct heat. It is commonly denoted by k, \lambda, or \kappa. Heat transfer occurs at a lower rate in materials of low thermal conductivity than in materials of high thermal ...
and volumetric heat capacity. For example, "this material has a high thermal inertia", or "thermal inertia plays an important role in this system", mean that dynamic effects need to be considered when modelling their behavior. Steady-state calculations, many of which produce valid estimates of equilibrium heat flows and temperatures without an accounting for thermal inertia, nevertheless yield no information on the pace of changes between equilibrium states. The term is a scientific analogy, and is not directly related to the mass-and-velocity term used in
mechanics Mechanics (from Ancient Greek: μηχανική, ''mēkhanikḗ'', "of machines") is the area of mathematics and physics concerned with the relationships between force, matter, and motion among physical objects. Forces applied to objec ...
, where
inertia Inertia is the idea that an object will continue its current motion until some force causes its speed or direction to change. The term is properly understood as shorthand for "the principle of inertia" as described by Newton in his first law ...
is that which limits the
acceleration In mechanics, acceleration is the rate of change of the velocity of an object with respect to time. Accelerations are vector quantities (in that they have magnitude and direction). The orientation of an object's acceleration is given by ...
of an object. In a similar way, thermal inertia is a measure of the thermal mass and the velocity of the thermal wave which controls the surface temperature of a material. In
heat transfer Heat transfer is a discipline of thermal engineering that concerns the generation, use, conversion, and exchange of thermal energy ( heat) between physical systems. Heat transfer is classified into various mechanisms, such as thermal conducti ...
, a higher value of the volumetric heat capacity means a longer time for the system to reach equilibrium. The thermal inertia of a material is defined as the square root of the product of the material's bulk
thermal conductivity The thermal conductivity of a material is a measure of its ability to conduct heat. It is commonly denoted by k, \lambda, or \kappa. Heat transfer occurs at a lower rate in materials of low thermal conductivity than in materials of high thermal ...
and volumetric heat capacity, where the latter is the product of
density Density (volumetric mass density or specific mass) is the substance's mass per unit of 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. Mathematicall ...
and
specific heat capacity In thermodynamics, the specific heat capacity (symbol ) of a substance is the heat capacity of a sample of the substance divided by the mass of the sample, also sometimes referred to as massic heat capacity. Informally, it is the amount of heat t ...
: : I = \sqrt * k is thermal conductivity, with unit W⋅m−1⋅K−1 * \rho is density, with unit kg⋅m−3 * c is specific heat capacity, with unit J⋅kg−1⋅K−1 * I has SI units of thermal inertia of J⋅m−2⋅K−1⋅s. Non-SI units of kieffers: Cal⋅cm−2⋅K−1⋅s−1/2, are also used informally in older references.http://scienceworld.wolfram.com/physics/ThermalInertia.html ''Eric Weisstein's World of Science – Thermal Inertia''


Constant volume and constant pressure

For gases it is necessary to distinguish between volumetric heat capacity at constant volume and volumetric heat capacity at constant
pressure Pressure (symbol: ''p'' or ''P'') is the force applied perpendicular to the surface of an object per unit area over which that force is distributed. Gauge pressure (also spelled ''gage'' pressure)The preferred spelling varies by country a ...
, which is always larger due to the pressure–volume work done as a gas expands during heating at constant pressure (thus absorbing heat which is converted to work). The distinctions between constant-volume and constant-pressure heat capacities are also made in various types of
specific heat capacity In thermodynamics, the specific heat capacity (symbol ) of a substance is the heat capacity of a sample of the substance divided by the mass of the sample, also sometimes referred to as massic heat capacity. Informally, it is the amount of heat t ...
(the latter meaning either mass-specific or mole-specific heat capacity).


See also

*
Heat capacity Heat capacity or thermal capacity is a physical property of matter, defined as the amount of heat to be supplied to an object to produce a unit change in its temperature. The SI unit of heat capacity is joule per kelvin (J/K). Heat cap ...
*
Specific heat capacity In thermodynamics, the specific heat capacity (symbol ) of a substance is the heat capacity of a sample of the substance divided by the mass of the sample, also sometimes referred to as massic heat capacity. Informally, it is the amount of heat t ...
*
Temperature Temperature is a physical quantity that expresses quantitatively the perceptions of hotness and coldness. Temperature is measured with a thermometer. Thermometers are calibrated in various temperature scales that historically have relied o ...
* Thermal effusivity * Thermodynamic equations


References

{{DEFAULTSORT:Volumetric Heat Capacity Thermodynamic properties Physical quantities Volume Heat transfer