View source for Volumetric heat capacity

{{short description|Thermal quality}}
{{for|volumetric heats of particular substances|Table of specific heat capacities}}

The '''volumetric heat capacity''' of a material is the [[heat capacity]] of a sample of the substance divided by the volume of the sample.  It is the amount of [[energy]] that must be added, in the form of [[heat]], to one unit of [[volume]] of the material in order to cause an increase of one unit in its [[temperature]].  The [[International System of Units|SI]] unit of volumetric heat capacity is joule per kelvin per [[cubic meter]], J⋅K<sup>−1</sup>⋅m<sup>−3</sup>.

The volumetric heat capacity can also be expressed as the [[specific heat capacity]] (heat capacity per unit of mass, in J⋅K<sup>−1</sup>⋅[[kilogram|kg]]<sup>−1</sup>) times the [[density]] of the substance (in kg/[[litre|L]], or [[gram|g]]/[[millilitre|mL]]).<ref>[https://www.wbdg.org/ccb/ARMYCOE/COETM/ARCHIVES/tm_5_852_6.pdf ''U.S. Army Corps of Engineers Technical Manual: Arctic and Subarctic Construction: Calculation Methods for Determination of Depths of Freeze and Thaw in Soils'', TM 5-852-6/AFR 88-19, Volume 6, 1988, Equation 2-1]</ref>

This quantity may be convenient for materials that are commonly measured by volume rather than mass, as is often the case in [[engineering]] and other technical disciplines. The volumetric heat capacity often varies with temperature, and is different for each [[state of matter]].<!-- REVISE EXAMPLES IF APPROPRIATE. Liquid water has one of the highest specific heats among common substances, about 4182&nbsp;J⋅K<sup>−1</sup>⋅kg<sup>−1</sup> at 20&nbsp;°C; but that of ice just below 0&nbsp;°C is only 2093&nbsp;J⋅K<sup>−1</sup>⋅kg<sup>−1</sup>.  The specific heats of [[iron]], [[granite]], [[oak|oak wood]], and [[hydrogen]] gas are about 449, 790, 2400, and 14300&nbsp;J⋅K<sup>−1</sup>⋅kg<sup>−1</sup>, respectively.--> While the substance is undergoing a [[phase transition]], such as melting or boiling, its volumetric heat capacity is technically [[infinity (mathematics)|infinite]], 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'').<!-- REVISE: These two values are usually denoted by <math>c_{\mathrm{p}}</math> and <math>c_{\mathrm{v}}</math>, respectively; their quotient <math>c_{\mathrm{p}}/c_{\mathrm{v}}</math> is the [[heat capacity ratio]].-->

If the amount of substance is taken to be the number of [[mole (unit)|moles]] 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<sup>−1</sup>⋅mol<sup>−1</sup>).

==Definition==
The volumetric heat capacity is defined as
:<math>s(T) = \frac{C(T)}{V(T)} = \frac{1}{V(T)} \lim_{\Delta T \to 0}\frac{\Delta Q(T)}{\Delta T}</math>

where <math>V(T)</math> is the volume of the sample at temperature <math>T</math>, and <math>\Delta Q(T)</math> is the amount of heat energy needed to raise the temperature of the sample from <math>T</math> to <math>T + \Delta T</math>.  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 <math>c(T)</math> of the substance times its [[density]] (mass per volume) <math>\rho(T)</math>, both measured at the temperature <math>T</math>.  Its SI unit is joule per kelvin per cubic meter (J⋅K<sup>−1</sup>⋅m<sup>−3</sup>).

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.<ref>{{SIbrochure8th}}</ref> Therefore, the word "volumetric" should always be used for this quantity.

==History==
[[Pierre Louis Dulong|Dulong]] and [[Alexis Thérèse Petit|Petit]] predicted in 1818 {{Citation needed|date=February 2010}} that the product of solid substance density and specific heat capacity (ρc<sub>p</sub>) 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]] (or per [[molar mass]]), 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.<!-- TO MERGE:
For very cold temperatures, heat capacities fall drastically and eventually approach zero as temperature approaches zero.
-->

==Typical values==
The volumetric heat capacity of solid materials at room temperatures and above varies widely, from about 1.2&nbsp;MJ⋅K<sup>−1</sup>⋅m<sup>−3</sup> (for example [[bismuth]]<ref>Based on values in [http://hyperphysics.phy-astr.gsu.edu/hbase/tables/sphtt.html#c1 this table] and density.</ref>) to 3.4&nbsp;MJ⋅K<sup>−1</sup>⋅m<sup>−3</sup> (for example iron<ref>Based on [http://webbook.nist.gov/cgi/cbook.cgi?ID=C7439896&Mask=2&Type=JANAFS&Table=on#JANAFS NIST data] and density.</ref>). 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]] at 1.64&nbsp;MJ⋅K<sup>−1</sup>⋅m<sup>−3</sup> or [[ethanol]] at 1.9. This reflects the modest loss of degrees of freedom for particles in liquids as compared with solids.

However, [[water]] has a very high volumetric heat capacity, at 4.18&nbsp;MJ⋅K<sup>−1</sup>⋅m<sup>−3</sup>, and [[ammonia]] is also fairly high: 3.3&nbsp;MJ⋅K<sup>−1</sup>⋅m<sup>−3</sup>.

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]] 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 of gases|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&nbsp;kJ⋅K<sup>−1</sup>⋅m<sup>−3</sup>, which is the same as the theoretical value of {{sfrac|3|2}}RT per kelvin per mole of gas molecules (where '''R''' is the [[gas constant]] 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 (mechanics)|degrees of freedom]] 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 [[joule|J]]/([[cubic metre|m<sup>3</sup>]]⋅[[Kelvin|K]]). It can also be described in Imperial units of [[British thermal unit|BTU]]/([[cubic foot|ft<sup>3</sup>]]⋅[[Fahrenheit|°F]]).

==Volumetric heat capacity of solids ==
Since the [[bulk density]] 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]], which is only 14.5% less dense than [[antimony]], 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.

==Volumetric heat capacity of liquids ==
The volumetric heat capacity of liquids could be measured from the thermal conductivity and thermal diffusivity correlation. The volumetric heat capacity of liquids could be directly obtained during thermal conductivity analysis using thermal conductivity analyzers that use techniques like the transient plane source method.<ref>{{cite journal |last1=Sreekumar |first1=Sreehari. |last2=Ganguly |first2=Abhijit. |last3=Khalil |first3=Sameh. |last4=Chakrabarti |first4=Supriya. |last5=Hewitt |first5=Neil. |last6=Mondol |first6=Jayanta. |last7=Shah |first7=Nikhilkumar. |year=2023 |title=Thermo-optical characterization of novel MXene/Carbon-dot hybrid nanofluid for heat transfer applications |journal=Journal of Cleaner Production |volume=434 |issue=29 |page=140395 |doi=10.1016/j.jclepro.2023.140395|doi-access=free }}</ref>

==Thermal inertia==
'''Thermal inertia''' is a term commonly used to describe the observed delays in a body's temperature response during [[heat transfer]]s.   The phenomenon exists because of a body's ability to both store and transport heat relative to its environment.  Since the configuration of system components and mix of heat transfer mechanisms (e.g. conduction, convection, radiation, phase change) varies substantially between instances, there is no generally applicable mathematical definition for thermal inertia.<ref>{{cite book |last1=Sala-Lizarraga |first1=Jose |last2=Picallo-Perez |first2=Ana |title=Exergy Analysis and Thermoeconomics of Buildings |pages=272–273 |publisher=Elsevier |isbn=9780128176115 |year=2019 |doi=10.1016/B978-0-12-817611-5.00004-7|s2cid=210737476 }}</ref>  The phenomenon occurs in conjunction with a material's or a transport medium's [[heat transfer coefficient|heat transfer properties]].  A larger thermal storage capacity typically yields a more sluggish temperature response.

[[File:Thermal wave animation.gif|thumb|upright=2.0|The thermal wave induced by a stepped heat source illustrates the phenomenon of thermal inertia.]]
A system containing one or more components with large volumetric heat capacity indicates that dynamic, or transient, effects must be considered when [[model (abstract)|modelling]] system 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.  Response times for complex systems can be evaluated with detailed numerical simulation, or a [[thermal time constant]] estimated from a [[lumped system analysis]].<ref>{{cite journal |last1=Keshavarz |first1=P. |last2=Taheri |first2=M. |title=An improved lumped analysis for transient heat conduction by using the polynomial approximation method |journal=Heat and Mass Transfer |volume=43 |year=2007 |pages=1151–1156 |doi=10.1007/s00231-006-0200-0}}</ref><ref>{{cite book |title=Physically-based Modelling and Simulation of Climate and Climatic Change |editor=Michael E. Schlesinger |agency=NATO |author=Gerald R. North |chapter=Lessons from energy balance models |edition=NATO Advanced Study Institute on Physical-Based Modelling |publisher=Springer |year=1988 |isbn=978-90-277-2789-3}}</ref>{{rp|627}}  A higher value of volumetric heat capacity generally means a longer time for the system to reach [[Thermodynamic equilibrium|equilibrium]].

Analogies of thermal inertia to the inertial behaviors observed in other disciplines of engineering and physics can sometimes be used with caution.<ref name=veto>{{cite journal |url=https://www.hou.usra.edu/meetings/lpsc2015/pdf/2914.pdf |last1=Veto |first1=M.S. |last2=Christensen |first2=P.R. |title=Mathematical Theory of Thermal Inertia Revisited |journal=46th Lunar and Planetary Science Conference |year=2015}}</ref>   In building design, thermal inertia is also known as the thermal flywheel effect, and a [[thermal mass]] can produce a delay between [[diurnal cycle|diurnal]] heat flow and temperature which is similar to the delay between current and voltage in an AC-driven [[RC circuit]].  Thermal inertia is less directly comparable to the mass-and-velocity term used in mechanics, where inertia restricts the acceleration 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.

===Thermal effusivity===
{{main|Thermal effusivity}}
For a semi-infinite rigid body where heat transfer is dominated by the diffusive process of conduction only,  the thermal inertia response at a surface can be approximated from the material's ''[[thermal effusivity]]'' (''e'').  It is defined as the square root of the product of the material's bulk [[thermal conductivity]] and volumetric heat capacity, where the latter is the product of [[density]] and [[specific heat capacity]]:<ref name=dante>{{cite book |last=Dante |first=Roberto C. |title=Handbook of Friction Materials and Their Applications |year=2016 |publisher=Elsevier |doi=10.1016/B978-0-08-100619-1.00009-2 |pages=123–134}}</ref><ref name=carjae>{{cite book |last1=Carslaw |first1=H.S. |last2=Jaeger |first2=J.C. |title=Conduction of Heat in Solids |publisher=Clarendon Press, Oxford |year=1959 |isbn=978-0-19-853368-9 |url=https://books.google.com/books?id=ySRRAAAAMAAJ}}</ref>
: <math>e = \sqrt{k \rho c}</math>
* <math>k</math> is thermal conductivity, with unit W⋅m<sup>−1</sup>⋅K<sup>−1</sup>
* <math>\rho</math> is density, with unit kg⋅m<sup>−3</sup>
* <math>c</math> is specific heat capacity, with unit J⋅kg<sup>−1</sup>⋅K<sup>−1</sup>
<br>
Thermal effusivity has units of a [[heat transfer coefficient]] multiplied by square root of time:
* [[SI]] units of W⋅m<sup>−2</sup>⋅K<sup>−1</sup>⋅s<sup>1/2</sup> or J⋅m<sup>−2</sup>⋅K<sup>−1</sup>⋅s<sup>−1/2</sup>. 
* Non-SI units of kieffers: Cal⋅cm<sup>−2</sup>⋅K<sup>−1</sup>⋅s<sup>−1/2</sup>, are also used informally in older references.{{efn-lr|Coined by the planetary [[geophysicist]] Hugh H. Kieffer.}}
When a constant flow of heat is abruptly imposed upon a surface, ''e'' performs nearly the same role in limiting the surfaces initial ''dynamic'' "thermal inertia" response ( ''U<sub>dyn</sub> ≈ e ⋅ t <sup>−1/2</sup>'' ) as the rigid body's usual heat transfer coefficient (''U'') plays in determining the final ''static'' surface temperature.<ref>{{cite journal |last1=van der Maas |first1=J. |last2=Maldonado |first2=E. |title=A New Thermal Inertia Model Based on Effusivity |journal=International Journal of Solar Energy |url=https://www.researchgate.net/profile/Claude-Alain-Roulet/post/What_is_Thermal_Mass_in_Passive_Solar_Building/attachment/62388d0119eb97629d3df771/AS%3A1136077033218052%401647873280997/download/ThermalInertia.pdf |volume=19 |issue=1-3 |year=1997 |pages=131-160 |doi=10.1080/01425919708914334 |doi-access=}}</ref><ref>{{cite journal |last=Bunn |first=J.P. |title=The thermal response of a homogeneous slab to a constant heat flux |journal=Building and Environment |volume=18 |issue=1-2 |pages=61-64 |year=1983 |doi=10.1016/0360-1323(83)90019-7 |doi-access=}}</ref>

== 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]], 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]] (the latter meaning either mass-specific or mole-specific heat capacity).

==See also==
* [[Heat capacity]]
* [[Specific heat capacity]]
* [[Temperature]]
* [[Thermal effusivity]]
* [[Thermodynamic equations]]

==References==
{{Notelist-lr}}

{{Reflist}}

{{DEFAULTSORT:Volumetric Heat Capacity}}
[[Category:Thermodynamic properties]]
[[Category:Physical quantities]]
[[Category:Volume]]
[[Category:Heat transfer]]