TheInfoList

In
thermodynamics Thermodynamics is a branch of physics that deals with heat, Work (thermodynamics), work, and temperature, and their relation to energy, entropy, and the physical properties of matter and radiation. The behavior of these quantities is governed b ...

, the specific heat capacity (symbol ) of a substance is the
heat capacity Heat capacity or thermal capacity is a physical property A physical property is any property Property is a system of rights that gives people legal control of valuable things, and also refers to the valuable things themselves. Depending on t ...
of a sample of the substance divided by the
mass Mass is the quantity Quantity is a property that can exist as a multitude or magnitude, which illustrate discontinuity and continuity. Quantities can be compared in terms of "more", "less", or "equal", or by assigning a numerical value ...
of the sample. Specific heat is also sometimes referred to as massic heat capacity. Informally, it is the amount of
heat In thermodynamics Thermodynamics is a branch of physics that deals with heat, Work (thermodynamics), work, and temperature, and their relation to energy, entropy, and the physical properties of matter and radiation. The behavior of these ...

that must be added to one unit of mass of the substance in order to cause an increase of one unit in
temperature Temperature ( ) is a physical quantity that expresses hot and cold. It is the manifestation of thermal energy Thermal radiation in visible light can be seen on this hot metalwork. Thermal energy refers to several distinct physical concept ...

. The SI unit of specific heat capacity is
joule The joule ( ; symbol: J) is a SI derived unit, derived unit of energy in the International System of Units. It is equal to the energy transferred to (or work (physics), work done on) an object when a force of one Newton (unit), newton acts on th ...

per
kelvin The kelvin is the base unit of temperature Temperature is a physical quantity that expresses hot and cold. It is the manifestation of thermal energy, present in all matter, which is the source of the occurrence of heat, a flow of energy, ...

per
kilogram The kilogram (also kilogramme) is the base unit of mass Mass is the physical quantity, quantity of ''matter'' in a physical body. It is also a measure (mathematics), measure of the body's ''inertia'', the resistance to acceleration (change ...
, J⋅kg−1⋅K−1. For example, the heat required to raise the temperature of of water by is , so the specific heat capacity of water is . Specific heat capacity often varies with temperature, and is different for each
state of matter In physics Physics is the natural science that studies matter, its Elementary particle, fundamental constituents, its Motion (physics), motion and behavior through Spacetime, space and time, and the related entities of energy and force. " ...
. Liquid water has one of the highest specific heat capacities among common substances, about at 20 °C; but that of ice, just below 0 °C, is only . The specific heat capacities of
iron Iron () is a with Fe (from la, ) and 26. It is a that belongs to the and of the . It is, on , right in front of (32.1% and 30.1%, respectively), forming much of Earth's and . It is the fourth most common . In its metallic state, iron ...

,
granite Granite () is a coarse-grained (phaneritic A phanerite is an igneous rock Igneous rock (derived from the Latin Latin (, or , ) is a classical language belonging to the Italic languages, Italic branch of the Indo-European languages. Lat ...

, and
hydrogen Hydrogen is the chemical element with the Symbol (chemistry), symbol H and atomic number 1. Hydrogen is the lightest element. At standard temperature and pressure, standard conditions hydrogen is a gas of diatomic molecules having the che ...

gas are about 449 J⋅kg−1⋅K−1, 790 J⋅kg−1⋅K−1, and 14300 J⋅kg−1⋅K−1, respectively. While the substance is undergoing a
phase transition In , , and many other related fields, phase transitions (or phase changes) are the of transition between a state of a medium, identified by some parameters, and another one, with different values of the parameters. Commonly the term is used to ...
, such as melting or boiling, its specific 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 (band), a South Korean boy band *''Infinite'' (EP), debut EP of American musi ...
, because the heat goes into changing its state rather than raising its temperature. The specific heat capacity of a substance, especially a gas, may be significantly higher when it is allowed to expand as it is heated (specific heat capacity ''at constant pressure'') than when it is heated in a closed vessel that prevents expansion (specific heat capacity ''at constant volume''). These two values are usually denoted by $c_p$ and $c_V$, respectively; their quotient $\gamma = c_p/c_V$is the
heat capacity ratio In thermal physics Example of a thermal column between the ground and a cumulus A thermal column (or thermal) is a column of rising air in the lower altitudes of Earth's atmosphere File:Atmosphere gas proportions.svg, Composition of Ear ...
. The term ''specific heat'' may also refer to the ratio between the specific heat capacities of a substance at a given temperature and of a reference substance at a reference temperature, such as water at 15 °C;(2001): ''Columbia Encyclopedia'', 6th ed.; as quoted b
Encyclopedia.com
Columbia University Press. Accessed on 2019-04-11.
much in the fashion of
specific gravity Relative density, or specific gravity, is the ratio In mathematics, a ratio indicates how many times one number contains another. For example, if there are eight oranges and six lemons in a bowl of fruit, then the ratio of oranges to lemon ...
. Specific heat capacity is also related to other intensive measures of heat capacity with other denominators. If the amount of substance is measured as a number of moles, one gets the
molar heat capacity The molar heat capacity of a is the amount of that must be added, in the form of , to one of the substance in order to cause an increase of one unit in its . Alternatively, it is the of a sample of the substance divided by the of the sample; ...
instead, whose SI unit is joule per kelvin per mole, J⋅mol−1⋅K−1. If the amount is taken to be the
volume Volume is a scalar quantity expressing the amount Quantity or amount is a property that can exist as a multitude Multitude is a term for a group of people who cannot be classed under any other distinct category, except for their shared fact ...

of the sample (as is sometimes done in engineering), one gets the
volumetric heat capacity The volumetric heat capacity of a material is the heat capacity Heat capacity or thermal capacity is a physical property A physical property is any property Property (''latin: Res Privata'') in the Abstract and concrete, abstract is what be ...
, whose SI unit is joule per kelvin per
cubic meter The cubic metre (in Commonwealth English The use of the English language English is a West Germanic languages, West Germanic language first spoken in History of Anglo-Saxon England, early medieval England, which has eventually become t ...
, J⋅m−3⋅K−1. One of the first scientists to use the concept was
Joseph Black Joseph Black (16 April 1728 – 6 December 1799) was a Scottish physicist and chemist, known for his discoveries of magnesium, latent heat, specific heat, and carbon dioxide. He was Professor of Anatomy and Chemistry at the University of Glasgo ...
, an 18th-century medical doctor and professor of medicine at
Glasgow University , image_name = University_of_Glasgow_Coat_of_Arms.jpg , image_size = 150px , latin_name = Universitas Glasguensis , motto = la, Via, Veritas, Vita ''Via et veritas et vita'' (, ) is a Latin language, Latin phrase meaning "the way and the t ...
. He measured the specific heat capacities of many substances, using the term ''capacity for heat''.

# Definition

The specific heat capacity of a substance, usually denoted by $c$ or , is the heat capacity $C$ of a sample of the substance, divided by the mass $M$ of the sample: :$c = \frac = \frac \cdot \frac$ where $\mathrm Q$ the amount of heat needed to uniformly raise the temperature of the sample by a small increment $\mathrm T$. Like the heat capacity of an object, the specific heat capacity of a substance may vary, sometimes substantially, depending on the starting temperature $T$ of the sample and the
pressure Pressure (symbol: ''p'' or ''P'') is the force In physics, a force is an influence that can change the motion (physics), motion of an Physical object, object. A force can cause an object with mass to change its velocity (e.g. moving fr ...

$p$ applied to it. Therefore, it should be considered a function $c\left(p,T\right)$ of those two variables. These parameters are usually specified when giving the specific heat capacity of a substance. For example, "Water (liquid): $c_p$ = 4187 J⋅kg−1⋅K−1 (15 °C)" When not specified, published values of the specific heat capacity $c$ generally are valid for some
standard conditions for temperature and pressure Standard temperature and pressure (STP) are standard Standard may refer to: Flags * Colours, standards and guidons * Standard (flag), a type of flag used for personal identification Norm, convention or requirement * Standard (metrology), ...
. However, the dependency of $c$ on starting temperature and pressure can often be ignored in practical contexts, e.g. when working in narrow ranges of those variables. In those contexts one usually omits the qualifier $\left(p,T\right)$, and approximates the specific heat capacity by a constant $c$ suitable for those ranges. Specific heat capacity is an
intensive propertyIn grammar, an intensive word form is one which denotes stronger, more forceful, or more concentrated action relative to the root on which the intensive is built. Intensives are usually lexical formations, but there may be a regular process for formi ...
of a substance, an intrinsic characteristic that does not depend on the size or shape of the amount in consideration. (The qualifier "specific" in front of an extensive property often indicates an intensive property derived from it.)

## Variations

The injection of heat energy into a substance, besides raising its temperature, usually causes an increase in its volume and/or its pressure, depending on how the sample is confined. The choice made about the latter affects the measured specific heat capacity, even for the same starting pressure $p$ and starting temperature $T$. Two particular choices are widely used: * If the pressure is kept constant (for instance, at the ambient atmospheric pressure), and the sample is allowed to expand, the expansion generates
work Work may refer to: * Work (human activity), intentional activity people perform to support themselves, others, or the community ** Manual labour, physical work done by humans ** House work, housework, or homemaking * Work (physics), the product of ...
as the force from the pressure displaces the enclosure or the surrounding fluid. That work must come from the heat energy provided. The specific heat capacity thus obtained is said to be measured at constant pressure (or isobaric), and is often denoted etc. * On the other hand, if the expansion is prevented — for example by a sufficiently rigid enclosure, or by increasing the external pressure to counteract the internal one — no work is generated, and the heat energy that would have gone into it must instead contribute to the internal energy of the sample, including raising its temperature by an extra amount. The specific heat capacity obtained this way is said to be measured at constant volume (or isochoric) and denoted etc. The value of $c_$ is usually less than the value of $c_p$. This difference is particularly notable in gases where values under constant pressure are typically 30% to 66.7% greater than those at constant volume. Hence the
heat capacity ratio In thermal physics Example of a thermal column between the ground and a cumulus A thermal column (or thermal) is a column of rising air in the lower altitudes of Earth's atmosphere File:Atmosphere gas proportions.svg, Composition of Ear ...
of gases is typically between 1.3 and 1.67.Lange's Handbook of Chemistry, 10th ed. page 1524

## Applicability

The specific heat capacity can be defined and measured for gases, liquids, and solids of fairly general composition and molecular structure. These include gas mixtures, solutions and alloys, or heterogenous materials such as milk, sand, granite, and concrete, if considered at a sufficiently large scale. The specific heat capacity can be defined also for materials that change state or composition as the temperature and pressure change, as long as the changes are reversible and gradual. Thus, for example, the concepts are definable for a gas or liquid that dissociates as the temperature increases, as long as the products of the dissociation promptly and completely recombine when it drops. The specific heat capacity is not meaningful if the substance undergoes irreversible chemical changes, or if there is a
phase changePhase change may refer to: * Phase transition, the transformation from one thermodynamic state to another. * Phase-change memory, a type of random-access memory. * Phase change (waves), concerning the physics of waves. {{disambiguation ...
, such as melting or boiling, at a sharp temperature within the range of temperatures spanned by the measurement.

# Measurement

The specific heat capacity of a substance is typically determined according to the definition; namely, by measuring the heat capacity of a sample of the substance, usually with a
calorimeter A calorimeter is an object used for calorimetry, or the process of measuring the heat of chemical reaction A chemical reaction is a process that leads to the chemical transformation of one set of chemical substances to another. Classicall ...

, and dividing by the sample's mass . Several techniques can be applied for estimating the heat capacity of a substance as for example fast differential scanning calorimetry. The specific heat capacities of gases can be measured at constant volume, by enclosing the sample in a rigid container. On the other hand, measuring the specific heat capacity at constant volume can be prohibitively difficult for liquids and solids, since one often would need impractical pressures in order to prevent the expansion that would be caused by even small increases in temperature. Instead, the common practice is to measure the specific heat capacity at constant pressure (allowing the material to expand or contract as it wishes), determine separately the
coefficient of thermal expansion Thermal expansion is the tendency of matter to change its shape A shape or figure is the form of an object or its external boundary, outline, or external surface File:Water droplet lying on a damask.jpg, Water droplet lying on a damask ...
and the
compressibility In thermodynamics Thermodynamics is a branch of physics that deals with heat, Work (thermodynamics), work, and temperature, and their relation to energy, entropy, and the physical properties of matter and radiation. The behavior of these quan ...
of the material, and compute the specific heat capacity at constant volume from these data according to the laws of thermodynamics.

# Units

## International system

The SI unit for specific heat capacity is joule per kelvin per kilogram ((J/K)/kg, J/(kg⋅K), J⋅K−1⋅kg−1, etc.). Since an increment of temperature of one
degree Celsius The degree Celsius is a unit of temperature on the Celsius scale, a temperature scale Scale of temperature is a methodology of calibrating the physical quantity temperature in metrology. Empirical scales measure temperature in relation to conven ...
is the same as an increment of one kelvin, that is the same as joule per degree Celsius per kilogram (J⋅kg−1⋅°C−1). Sometimes the
gram The gram (alternative spelling: gramme; SI unit symbol: g) is a metric system The metric system is a that succeeded the decimalised system based on the introduced in France in the 1790s. The historical development of these systems culm ...
is used instead of kilogram for the unit of mass: 1 J⋅g−1⋅K−1 = 0.001 J⋅kg−1⋅K−1. The specific heat capacity of a substance (per unit of mass) has
dimension In physics Physics is the that studies , its , its and behavior through , and the related entities of and . "Physical science is that department of knowledge which relates to the order of nature, or, in other words, to the regular s ...
L2⋅Θ−1⋅T−2, or (L/T)2/Θ. Therefore, the SI unit J⋅kg−1⋅K−1 is equivalent to
metre The metre ( Commonwealth spelling) or meter (American spelling Despite the various English dialects spoken from country to country and within different regions of the same country, there are only slight regional variations in English ...
squared per
second The second (symbol: s, also abbreviated: sec) is the base unit of time Time is the continued sequence of existence and event (philosophy), events that occurs in an apparently irreversible process, irreversible succession from the past, th ...
squared per
kelvin The kelvin is the base unit of temperature Temperature is a physical quantity that expresses hot and cold. It is the manifestation of thermal energy, present in all matter, which is the source of the occurrence of heat, a flow of energy, ...

(m2⋅K−1⋅s−2).

## Imperial engineering units

Professionals in
construction Construction is a general term meaning the and to form , , or ,"Construction" def. 1.a. 1.b. and 1.c. ''Oxford English Dictionary'' Second Edition on CD-ROM (v. 4.0) Oxford University Press 2009 and comes from ''constructio'' (from ''com-' ...

,
civil engineering Civil engineering is a professional engineering Regulation and licensure in engineering is established by various jurisdictions of the world to encourage public welfare, safety, well-being and other interests of the general public and to defin ...
,
chemical engineering Chemical engineering is a certain type of which deals with the study of operation and design of chemical plants as well as methods of improving production. Chemical engineers develop economical commercial processes to convert raw material into u ...
, and other technical disciplines, especially in the
United States The United States of America (U.S.A. or USA), commonly known as the United States (U.S. or US) or America, is a country Continental United States, primarily located in North America. It consists of 50 U.S. state, states, a Washington, D.C., ...

, may use
English Engineering units Some fields of engineering in the United States use a system of measurement of physical quantities known as the English Engineering Units. Despite its name, the system is based on United States customary units of measure; it is not used in England ...
including the
Imperial Imperial is that which relates to an empire, emperor, or imperialism. Imperial or The Imperial may also refer to: Places United States * Imperial, California * Imperial, Missouri * Imperial, Nebraska * Imperial, Pennsylvania * Imperial, Texas * ...
pound Pound or Pounds may refer to: Units * Pound (currency) A pound is any of various units of currency A currency, "in circulation", from la, currens, -entis, literally meaning "running" or "traversing" in the most specific sense is money Im ...
(lb = 0.45359237 kg) as the unit of mass, the
degree Fahrenheit The Fahrenheit scale ( or ) is a temperature scale based on one proposed in 1724 by the physicist A physicist is a scientist A scientist is a person who conducts Scientific method, scientific research to advance knowledge in an Branches ...

or Rankine (°F = 5/9 K, about 0.555556 K) as the unit of temperature increment, and the
British thermal unit The British thermal unit (BTU or Btu) is a unit of heat In thermodynamics Thermodynamics is a branch of physics that deals with heat, Work (thermodynamics), work, and temperature, and their relation to energy, entropy, and the physica ...
(BTU ≈ 1055.06 J), Published under the auspices of the ''Verein Deutscher Ingenieure'' (VDI). as the unit of heat. In those contexts, the unit of specific heat capacity is BTU⋅°F−1⋅lb−1 = 4177.6 J⋅kg−1⋅K−1. The BTU was originally defined so that the average specific heat capacity of water would be 1 BTU⋅°F−1⋅lb−1.

## Calories

In chemistry, heat amounts were often measured in
calorie The calorie is a unit of energy Unit may refer to: Arts and entertainment * UNIT Unit may refer to: Arts and entertainment * UNIT, a fictional military organization in the science fiction television series ''Doctor Who'' * Unit of action, a di ...
s. Confusingly, two units with that name, denoted "cal" or "Cal", have been commonly used to measure amounts of heat: * the "small calorie" (or "gram-calorie", "cal") is 4.184 J, exactly. It was originally defined so that the specific heat capacity of liquid water would be 1 cal⋅°C−1⋅g−1. *The "grand calorie" (also "kilocalorie", "kilogram-calorie", or "food calorie"; "kcal" or "Cal") is 1000 small calories, that is, 4184 J, exactly. It was originally defined so that the specific heat capacity of water would be 1 Cal⋅°C−1⋅kg−1. While these units are still used in some contexts (such as kilogram calorie in
nutrition Nutrition is the biochemical Biochemistry or biological chemistry, is the study of chemical processes within and relating to living organisms. A sub-discipline of both chemistry and biology, biochemistry may be divided into three fields: st ...
), their use is now deprecated in technical and scientific fields. When heat is measured in these units, the unit of specific heat capacity is usually : 1 cal⋅°C−1⋅g−1 ("small calorie") = 1 Cal⋅°C−1⋅kg−1 = 1 kcal⋅°C−1⋅kg−1 ("large calorie") = 4184 J⋅kg−1⋅K−1. In either unit, the specific heat capacity of water is approximately 1. The combinations cal⋅°C−1⋅kg−1 = 4.184 J⋅kg−1⋅K−1 and kcal⋅°C−1⋅g−1 = 4,184,000 J⋅kg−1⋅K−1 do not seem to be widely used.

# Physical basis of specific heat capacity

The temperature of a sample of a substance reflects the average
kinetic energy In physics Physics is the that studies , its , its and behavior through , and the related entities of and . "Physical science is that department of knowledge which relates to the order of nature, or, in other words, to the regular ...
of its constituent particles (atoms or molecules) relative to its center of mass. However, not all energy provided to a sample of a substance will go into raising its temperature, exemplified via the
equipartition theorem In , the equipartition theorem relates the of a system to its average . The equipartition theorem is also known as the law of equipartition, equipartition of energy, or simply equipartition. The original idea of equipartition was that, in , e ...
.

## Monatomic gases

Quantum mechanics Quantum mechanics is a fundamental theory A theory is a reason, rational type of abstraction, abstract thinking about a phenomenon, or the results of such thinking. The process of contemplative and rational thinking is often associated with ...
predicts that, at room temperature and ordinary pressures, an isolated atom in a gas cannot store any significant amount of energy except in the form of kinetic energy. Thus, heat capacity per mole is the same for all monatomic gases (such as the noble gases). More precisely, $c_ = 3R/2 \approx \mathrm$ and $c_ = 5R/2 \approx \mathrm$, where $R \approx \mathrm$ is the ideal gas unit (which is the product of Boltzmann conversion constant from
kelvin The kelvin is the base unit of temperature Temperature is a physical quantity that expresses hot and cold. It is the manifestation of thermal energy, present in all matter, which is the source of the occurrence of heat, a flow of energy, ...

microscopic energy unit to the macroscopic energy unit
joule The joule ( ; symbol: J) is a SI derived unit, derived unit of energy in the International System of Units. It is equal to the energy transferred to (or work (physics), work done on) an object when a force of one Newton (unit), newton acts on th ...

, and the
Avogadro number The Avogadro constant (''N''A or ''L'') is the proportionality factor that relates the number of constituent particles (usually molecule File:Pentacene on Ni(111) STM.jpg, A scanning tunneling microscopy image of pentacene molecules, which ...
). Therefore, the specific heat capacity (per unit of mass, not per mole) of a monatomic gas will be inversely proportional to its (adimensional)
atomic weight Relative atomic mass (symbol: ''A'') or atomic weight is a dimensionless physical quantity A physical quantity is a physical property of a material or system that can be Quantification (science), quantified by measurement. A physical quantity ca ...
$A$. That is, approximately, :$c_V \approx \mathrm/A \quad\quad\quad c_p \approx \mathrm/A$ For the noble gases, from helium to xenon, these computed values are

## Polyatomic gases

On the other hand, a polyatomic gas molecule (consisting of two or more atoms bound together) can store heat energy in other forms besides its kinetic energy. These forms include rotation of the molecule, and vibration of the atoms relative to its center of mass. These extra
degrees of freedom Degrees of Freedom (often abbreviated df or DOF) refers to the number of independent variables or parameters of a system. In various scientific fields, the word "freedom" is used to describe the limits to which physical movement or other physical ...
or "modes" contribute to the specific heat capacity of the substance. Namely, when heat energy is injected into a gas with polyatomic molecules, only part of it will go into increasing their kinetic energy, and hence the temperature; the rest will go to into those other degrees of freedom. In order to achieve the same increase in temperature, more heat energy will have to be provided to a mol of that substance than to a mol of a monatomic gas. Therefore, the specific heat capacity of a polyatomic gas depends not only on its molecular mass, but also on the number of degrees of freedom that the molecules have. Quantum mechanics further says that each rotational or vibrational mode can only take or lose energy in certain discrete amount (quanta). Depending on the temperature, the average heat energy per molecule may be too small compared to the quanta needed to activate some of those degrees of freedom. Those modes are said to be "frozen out". In that case, the specific heat capacity of the substance is going to increase with temperature, sometimes in a step-like fashion, as more modes become unfrozen and start absorbing part of the input heat energy. For example, the molar heat capacity of
nitrogen Nitrogen is the chemical element upright=1.0, 500px, The chemical elements ordered by link=Periodic table In chemistry Chemistry is the science, scientific study of the properties and behavior of matter. It is a natural science ...

at constant volume is $c_ = \mathrm$ (at 15 °C, 1 atm), which is $2.49 R$.Thornton, Steven T. and Rex, Andrew (1993) ''Modern Physics for Scientists and Engineers'', Saunders College Publishing That is the value expected from theory if each molecule had 5 degrees of freedom. These turn out to be three degrees of the molecule's velocity vector, plus two degrees from its rotation about an axis through the center of mass and perpendicular to the line of the two atoms. Because of those two extra degrees of freedom, the specific heat capacity $c_V$ of (736 J⋅K−1⋅kg−1) is greater than that of an hypothetical monoatomic gas with the same molecular mass 28 (445 J⋅K−1⋅kg−1), by a factor of . This value for the specific heat capacity of nitrogen is practically constant from below −150 °C to about 300 °C. In that temperature range, the two additional degrees of freedom that correspond to vibrations of the atoms, stretching and compressing the bond, are still "frozen out". At about that temperature, those modes begin to "un-freeze", and as a result $c_V$ starts to increase rapidly at first, then slower as it tends to another constant value. It is 35.5 J⋅K−1⋅mol−1 at 1500 °C, 36.9 at 2500 °C, and 37.5 at 3500 °C.Chase, M.W. Jr. (1998)
NIST-JANAF Themochemical Tables, Fourth Edition
', In ''Journal of Physical and Chemical Reference Data'', Monograph 9, pages 1–1951.
The last value corresponds almost exactly to the predicted value for 7 degrees of freedom per molecule.

# Derivations of heat capacity

## Relation between specific heat capacities

Starting from the
fundamental Thermodynamic Relation In thermodynamics, the fundamental thermodynamic relation are four fundamental equations which demonstrate how four important thermodynamic quantities depend on variables that can be controlled and measured experimentally. Thus, they are essentiall ...
one can show, :$c_p - c_v = \frac$ where, *$\alpha$ is the
coefficient of thermal expansion Thermal expansion is the tendency of matter to change its shape A shape or figure is the form of an object or its external boundary, outline, or external surface File:Water droplet lying on a damask.jpg, Water droplet lying on a damask ...
, *$\beta_T$ is the
isothermal In thermodynamics, an isothermal process is a type of thermodynamic process in which the temperature ''T'' of a Thermodynamic system, system remains constant: Δ''T'' = 0. This typically occurs when a system is in contact with an outside the ...

compressibility In thermodynamics Thermodynamics is a branch of physics that deals with heat, Work (thermodynamics), work, and temperature, and their relation to energy, entropy, and the physical properties of matter and radiation. The behavior of these quan ...
, and *$\rho$ is
density The density (more precisely, the volumetric mass density; also known as specific mass), of a substance is its per unit . The symbol most often used for density is ''ρ'' (the lower case Greek letter ), although the Latin letter ''D'' can also ...

. A derivation is discussed in the article Relations between specific heats. For an
ideal gas An ideal gas is a theoretical gas Gas is one of the four fundamental states of matter In physics Physics is the natural science that studies matter, its Elementary particle, fundamental constituents, its Motion (physics), motion ...
, if $\rho$ is expressed as molar density in the above equation, this equation reduces simply to 's relation, :$C_ - C_ = R \!$ where $C_$ and $C_$ are
intensive propertyIn grammar, an intensive word form is one which denotes stronger, more forceful, or more concentrated action relative to the root on which the intensive is built. Intensives are usually lexical formations, but there may be a regular process for formi ...
heat capacities expressed on a per mole basis at constant pressure and constant volume, respectively.

## Specific heat capacity

The specific heat capacity of a material on a per mass basis is :$c=,$ which in the absence of phase transitions is equivalent to :$c=E_ m= = ,$ where *$C$ is the heat capacity of a body made of the material in question, *$m$ is the mass of the body, *$V$ is the volume of the body, and *$\rho = \frac$ is the density of the material. For gases, and also for other materials under high pressures, there is need to distinguish between different boundary conditions for the processes under consideration (since values differ significantly between different conditions). Typical processes for which a heat capacity may be defined include (constant pressure, $dp = 0$) or (constant volume, $dV = 0$) processes. The corresponding specific heat capacities are expressed as :$c_p = \left\left(\frac\right\right)_p,$ :$c_V = \left\left(\frac\right\right)_V.$ A related parameter to $c$ is $CV^$, the
volumetric heat capacity The volumetric heat capacity of a material is the heat capacity Heat capacity or thermal capacity is a physical property A physical property is any property Property (''latin: Res Privata'') in the Abstract and concrete, abstract is what be ...
. In engineering practice, $c_V$ for solids or liquids often signifies a volumetric heat capacity, rather than a constant-volume one. In such cases, the mass-specific heat capacity is often explicitly written with the subscript $m$, as $c_m$. Of course, from the above relationships, for solids one writes :$c_m = \frac = \frac.$ For pure homogeneous
chemical compound A chemical compound is a chemical substance composed of many identical molecules (or molecular entity, molecular entities) composed of atoms from more than one chemical element, element held together by chemical bonds. A homonuclear molecule, m ...
s with established molecular or molar mass or a molar quantity is established, heat capacity as an
intensive propertyIn grammar, an intensive word form is one which denotes stronger, more forceful, or more concentrated action relative to the root on which the intensive is built. Intensives are usually lexical formations, but there may be a regular process for formi ...
can be expressed on a per
mole Mole (or Molé) may refer to: Animals * Mole (animal) or "true mole", mammals in the family Talpidae, found in Eurasia and North America * Golden moles, southern African mammals in the family Chrysochloridae, similar to but unrelated to Talpidae ...
basis instead of a per mass basis by the following equations analogous to the per mass equations: :$C_ = \left\left(\frac\right\right)_p = \text$ :$C_ = \left\left(\frac\right\right)_V = \text$ where ''n'' = number of moles in the body or
thermodynamic system A thermodynamic system is a body of matter In classical physics and general chemistry, matter is any substance that has mass and takes up space by having volume. All everyday objects that can be touched are ultimately composed of atoms, whic ...
. One may refer to such a ''per mole'' quantity as molar heat capacity to distinguish it from specific heat capacity on a per-mass basis.

## Polytropic heat capacity

The heat capacity is calculated at processes if all the thermodynamic properties (pressure, volume, temperature) change :$C_ = \left\left(\frac\right\right) = \text$ The most important polytropic processes run between the adiabatic and the isotherm functions, the polytropic index is between 1 and the adiabatic exponent (''γ'' or ''κ'')

## Dimensionless heat capacity

The
dimensionless In dimensional analysis In engineering and science, dimensional analysis is the analysis of the relationships between different physical quantities by identifying their base quantity, base quantities (such as length, mass, time, and electric curre ...
heat capacity of a material is :$C^*= =$ where *''C'' is the heat capacity of a body made of the material in question (J/K) *''n'' is the
amount of substance In chemistry Chemistry is the study of the properties and behavior of . It is a that covers the that make up matter to the composed of s, s and s: their composition, structure, properties, behavior and the changes they undergo during a ...
in the body ( mol) *''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 The Boltzmann constant ( or ) is the proportionality f ...
(J⋅K−1⋅mol−1) *''N'' is the number of molecules in the body. (dimensionless) *''k''B is the
Boltzmann constant The Boltzmann constant ( or ) is the proportionality factor In mathematics, two varying quantities are said to be in a Binary relation, relation of proportionality, Multiplication, multiplicatively connected to a Constant (mathematics), c ...
(J⋅K−1) Again, units shown for example. Read more about the quantities of dimension one at BIPM In the
Ideal gas An ideal gas is a theoretical gas Gas is one of the four fundamental states of matter In physics Physics is the natural science that studies matter, its Elementary particle, fundamental constituents, its Motion (physics), motion ...
article, dimensionless heat capacity $C^*$ is expressed as $\hat c$ .

## Heat capacity at absolute zero

From the definition of
entropy Entropy is a scientific concept as well as a measurable physical property that is most commonly associated with a state of disorder, randomness, or uncertainty. The term and the concept are used in diverse fields, from classical thermodynamic ...
:$TdS=\delta Q$ the absolute entropy can be calculated by integrating from zero kelvins temperature to the final temperature ''Tf'' :$S\left(T_f\right)=\int_^ \frac =\int_0^ \frac\frac =\int_0^ C\left(T\right)\,\frac.$ The heat capacity must be zero at zero temperature in order for the above integral not to yield an infinite absolute entropy, thus violating the
third law of thermodynamics The third law of thermodynamics states as follows, regarding the properties of closed systems in thermodynamic equilibrium Thermodynamic equilibrium is an axiomatic concept of thermodynamics Thermodynamics is a branch of physics that deals wit ...
. One of the strengths of the
Debye model In thermodynamics Thermodynamics is a branch of physics that deals with heat, Work (thermodynamics), work, and temperature, and their relation to energy, radiation, and physical properties of matter. The behavior of these quantities is gover ...
is that (unlike the preceding Einstein model) it predicts the proper mathematical form of the approach of heat capacity toward zero, as absolute zero temperature is approached.

## Solid phase

The theoretical maximum heat capacity for larger and larger multi-atomic gases at higher temperatures, also approaches the Dulong–Petit limit of 3''R'', so long as this is calculated per mole of atoms, not molecules. The reason is that gases with very large molecules, in theory have almost the same high-temperature heat capacity as solids, lacking only the (small) heat capacity contribution that comes from potential energy that cannot be stored between separate molecules in a gas. The Dulong–Petit limit results from the
equipartition theorem In , the equipartition theorem relates the of a system to its average . The equipartition theorem is also known as the law of equipartition, equipartition of energy, or simply equipartition. The original idea of equipartition was that, in , e ...
, and as such is only valid in the classical limit of a microstate continuum, which is a high temperature limit. For light and non-metallic elements, as well as most of the common molecular solids based on carbon compounds at standard ambient temperature, quantum effects may also play an important role, as they do in multi-atomic gases. These effects usually combine to give heat capacities lower than 3''R'' per mole of ''atoms'' in the solid, although in molecular solids, heat capacities calculated ''per mole of molecules'' in molecular solids may be more than 3''R''. For example, the heat capacity of water ice at the melting point is about 4.6''R'' per mole of molecules, but only 1.5''R'' per mole of atoms. The lower than 3''R'' number "per atom" (as is the case with diamond and beryllium) results from the “freezing out” of possible vibration modes for light atoms at suitably low temperatures, just as in many low-mass-atom gases at room temperatures. Because of high crystal binding energies, these effects are seen in solids more often than liquids: for example the heat capacity of liquid water is twice that of ice at near the same temperature, and is again close to the 3''R'' per mole of atoms of the Dulong–Petit theoretical maximum. For a more modern and precise analysis of the heat capacities of solids, especially at low temperatures, it is useful to use the idea of
phonons In physics Physics is the natural science that studies matter, its Elementary particle, fundamental constituents, its Motion (physics), motion and behavior through Spacetime, space and time, and the related entities of energy and force. ...
. See
Debye model In thermodynamics Thermodynamics is a branch of physics that deals with heat, Work (thermodynamics), work, and temperature, and their relation to energy, radiation, and physical properties of matter. The behavior of these quantities is gover ...
.

## Theoretical estimation

The path integral Monte Carlo method is a numerical approach for determining the values of heat capacity, based on quantum dynamical principles. However, good approximations can be made for gases in many states using simpler methods outlined below. For many solids composed of relatively heavy atoms (atomic number > iron), at non-cryogenic temperatures, the heat capacity at room temperature approaches 3R = 24.94 joules per kelvin per mole of atoms (Dulong–Petit law, R is the gas constant). Low temperature approximations for both gases and solids at temperatures less than their characteristic Einstein temperatures or Debye temperatures can be made by the methods of Einstein and Debye discussed below. Water (liquid): CP = 4185.5 J⋅K−1⋅kg−1 (15 °C, 101.325 kPa) Water (liquid): CVH = 74.539 J⋅K−1⋅mol−1 (25 °C) For liquids and gases, it is important to know the pressure to which given heat capacity data refer. Most published data are given for standard pressure. However, different standard conditions for temperature and pressure have been defined by different organizations. The International Union of Pure and Applied Chemistry (IUPAC) changed its recommendation from one atmosphere to the round value 100 kPa (≈750.062 Torr)..

## Calculation from first principles

The path integral Monte Carlo method is a numerical approach for determining the values of heat capacity, based on quantum dynamical principles. However, good approximations can be made for gases in many states using simpler methods outlined below. For many solids composed of relatively heavy atoms (atomic number > iron), at non-cryogenic temperatures, the heat capacity at room temperature approaches 3''R'' = 24.94 joules per kelvin per mole of atoms (
Dulong–Petit law The Dulong–Petit law, a thermodynamic law proposed in 1819 by French physicists Pierre Louis Dulong and Alexis Thérèse Petit, states the classical expression for the molar Heat capacity, specific heat capacity of certain chemical elements. Expe ...
, ''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 The Boltzmann constant ( or ) is the proportionality f ...
). Low temperature approximations for both gases and solids at temperatures less than their characteristic Einstein temperatures or
Debye temperature In thermodynamics Thermodynamics is a branch of physics that deals with heat, Work (thermodynamics), work, and temperature, and their relation to energy, radiation, and physical properties of matter. The behavior of these quantities is gover ...
s can be made by the methods of Einstein and Debye discussed below.

## Relation between heat capacities

Measuring the specific heat capacity at constant volume can be prohibitively difficult for liquids and solids. That is, small temperature changes typically require large pressures to maintain a liquid or solid at constant volume, implying that the containing vessel must be nearly rigid or at least very strong (see
coefficient of thermal expansion Thermal expansion is the tendency of matter to change its shape A shape or figure is the form of an object or its external boundary, outline, or external surface File:Water droplet lying on a damask.jpg, Water droplet lying on a damask ...
and
compressibility In thermodynamics Thermodynamics is a branch of physics that deals with heat, Work (thermodynamics), work, and temperature, and their relation to energy, entropy, and the physical properties of matter and radiation. The behavior of these quan ...
). Instead, it is easier to measure the heat capacity at constant pressure (allowing the material to expand or contract freely) and solve for the heat capacity at constant volume using mathematical relationships derived from the basic thermodynamic laws. The
heat capacity ratio In thermal physics Example of a thermal column between the ground and a cumulus A thermal column (or thermal) is a column of rising air in the lower altitudes of Earth's atmosphere File:Atmosphere gas proportions.svg, Composition of Ear ...
, or adiabatic index, is the ratio of the heat capacity at constant pressure to heat capacity at constant volume. It is sometimes also known as the isentropic expansion factor.

### Ideal gas

For an
ideal gas An ideal gas is a theoretical gas Gas is one of the four fundamental states of matter In physics Physics is the natural science that studies matter, its Elementary particle, fundamental constituents, its Motion (physics), motion ...
, evaluating the partial derivatives above according to the
equation of state In physics Physics (from grc, φυσική (ἐπιστήμη), physikḗ (epistḗmē), knowledge of nature, from ''phýsis'' 'nature'), , is the natural science that studies matter, its Motion (physics), motion and behavior through S ...
, 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 The Boltzmann constant ( or ) is the proportionality f ...
, for an ideal gas :$P V = n R T,$ :$C_P - C_V = T \left\left(\frac\right\right)_ \left\left(\frac\right\right)_,$ :$P = \frac \Rightarrow \left\left(\frac\right\right)_ = \frac,$ :$V = \frac \Rightarrow \left\left(\frac\right\right)_ = \frac.$ Substituting :$T \left\left(\frac\right\right)_ \left\left(\frac\right\right)_ = T \frac \frac = \frac \frac = P \frac = nR,$ this equation reduces simply to 's relation: :$C_ - C_ = R.$ The differences in heat capacities as defined by the above Mayer relation is only exact for an ideal gas and would be different for any real gas.

## Specific heat capacity

The specific heat capacity of a material on a per mass basis is :$c = \frac,$ which in the absence of phase transitions is equivalent to :$c = E_m = \frac = \frac,$ where *$C$ is the heat capacity of a body made of the material in question, *$m$ is the mass of the body, *$V$ is the volume of the body, *$\rho = \frac$ is the density of the material. For gases, and also for other materials under high pressures, there is need to distinguish between different boundary conditions for the processes under consideration (since values differ significantly between different conditions). Typical processes for which a heat capacity may be defined include (constant pressure, $\textP = 0$) or (constant volume, $\textV = 0$) processes. The corresponding specific heat capacities are expressed as :$c_P = \left\left(\frac\right\right)_P,$ :$c_V = \left\left(\frac\right\right)_V.$ From the results of the previous section, dividing through by the mass gives the relation :$c_P - c_V = \frac.$ A related parameter to $c$ is $C/V$, the
volumetric heat capacity The volumetric heat capacity of a material is the heat capacity Heat capacity or thermal capacity is a physical property A physical property is any property Property (''latin: Res Privata'') in the Abstract and concrete, abstract is what be ...
. In engineering practice, $c_V$ for solids or liquids often signifies a volumetric heat capacity, rather than a constant-volume one. In such cases, the specific heat capacity is often explicitly written with the subscript $m$, as $c_m$. Of course, from the above relationships, for solids one writes :$c_m = \frac = \frac.$ For pure
homogeneous Homogeneity and heterogeneity are concepts often used in the sciences Science () is a systematic enterprise that Scientific method, builds and organizes knowledge in the form of Testability, testable explanations and predictions about th ...
chemical compound A chemical compound is a chemical substance composed of many identical molecules (or molecular entity, molecular entities) composed of atoms from more than one chemical element, element held together by chemical bonds. A homonuclear molecule, m ...
s with established molecular or molar mass, or a molar quantity, heat capacity as an
intensive propertyIn grammar, an intensive word form is one which denotes stronger, more forceful, or more concentrated action relative to the root on which the intensive is built. Intensives are usually lexical formations, but there may be a regular process for formi ...
can be expressed on a per-
mole Mole (or Molé) may refer to: Animals * Mole (animal) or "true mole", mammals in the family Talpidae, found in Eurasia and North America * Golden moles, southern African mammals in the family Chrysochloridae, similar to but unrelated to Talpidae ...
basis instead of a per-mass basis by the following equations analogous to the per mass equations: :$C_ = \left\left(\frac\right\right)_P = \text$ :$C_ = \left\left(\frac\right\right)_V = \text$ where ''n'' is the number of moles in the body or
thermodynamic system A thermodynamic system is a body of matter In classical physics and general chemistry, matter is any substance that has mass and takes up space by having volume. All everyday objects that can be touched are ultimately composed of atoms, whic ...
. One may refer to such a per-mole quantity as molar heat capacity to distinguish it from specific heat capacity on a per-mass basis.

## Polytropic heat capacity

The heat capacity is calculated at processes if all the thermodynamic properties (pressure, volume, temperature) change: :$C_ = \left\left(\frac\right\right) = \text$ The most important polytropic processes run between the adiabatic and the isotherm functions, the polytropic index is between 1 and the adiabatic exponent (''γ'' or ''κ'').

## Dimensionless heat capacity

The
dimensionless In dimensional analysis In engineering and science, dimensional analysis is the analysis of the relationships between different physical quantities by identifying their base quantity, base quantities (such as length, mass, time, and electric curre ...
heat capacity of a material is :$C^* = \frac = \frac,$ where *$C$ is the heat capacity of a body made of the material in question (J/K), *''n'' is the
amount of substance In chemistry Chemistry is the study of the properties and behavior of . It is a that covers the that make up matter to the composed of s, s and s: their composition, structure, properties, behavior and the changes they undergo during a ...
in the body ( mol), *''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 The Boltzmann constant ( or ) is the proportionality f ...
(J/(K⋅mol)), *''N'' is the number of molecules in the body (dimensionless), *''k''B is
Boltzmann's constant The Boltzmann constant ( or ) is the proportionality factor that relates the average relative kinetic energy of particles in a ideal gas, gas with the thermodynamic temperature of the gas. It occurs in the definitions of the kelvin and the gas ...
(J/(K⋅molecule)). In the
ideal gas An ideal gas is a theoretical gas Gas is one of the four fundamental states of matter In physics Physics is the natural science that studies matter, its Elementary particle, fundamental constituents, its Motion (physics), motion ...
article, dimensionless heat capacity $C^*$ is expressed as $\hat c$ and is related there directly to half the number of degrees of freedom per particle. This holds true for quadratic degrees of freedom, a consequence of the
equipartition theorem In , the equipartition theorem relates the of a system to its average . The equipartition theorem is also known as the law of equipartition, equipartition of energy, or simply equipartition. The original idea of equipartition was that, in , e ...
. More generally, the dimensionless heat capacity relates the logarithmic increase in temperature to the increase in the dimensionless entropy per particle $S^* = S / N k_\text$, measured in nats. :$C^* = \frac.$ Alternatively, using base-2 logarithms, $C^*$ relates the base-2 logarithmic increase in temperature to the increase in the dimensionless entropy measured in
bit The bit is a basic unit of information in computing Computing is any goal-oriented activity requiring, benefiting from, or creating computing machinery. It includes the study and experimentation of algorithm of an algorithm (Euclid's algo ...
s.

## Heat capacity at absolute zero

From the definition of
entropy Entropy is a scientific concept as well as a measurable physical property that is most commonly associated with a state of disorder, randomness, or uncertainty. The term and the concept are used in diverse fields, from classical thermodynamic ...
:$T \, \textS = \delta Q,$ the absolute entropy can be calculated by integrating from zero to the final temperature ''T''f: :$S\left(T_\text\right) = \int_^ \frac = \int_0^ \frac\frac = \int_0^ C\left(T\right)\,\frac.$

# Thermodynamic derivation

In theory, the specific heat capacity of a substance can also be derived from its abstract thermodynamic modeling by an
equation of state In physics Physics (from grc, φυσική (ἐπιστήμη), physikḗ (epistḗmē), knowledge of nature, from ''phýsis'' 'nature'), , is the natural science that studies matter, its Motion (physics), motion and behavior through S ...
and an internal energy function.

## State of matter in a homogeneous sample

To apply the theory, one considers the sample of the substance (solid, liquid, or gas) for which the specific heat capacity can be defined; in particular, that it has homogeneous composition and fixed mass $M$. Assume that the evolution of the system is always slow enough for the internal pressure $P$ and temperature $T$ be considered uniform throughout. The pressure $P$ would be equal to the pressure applied to it by the enclosure or some surrounding fluid, such as air. The state of the material can then be specified by three parameters: its temperature $T$, the pressure $P$, and its
specific volumeIn thermodynamics Thermodynamics is a branch of physics that deals with heat, Work (thermodynamics), work, and temperature, and their relation to energy, entropy, and the physical properties of matter and radiation. The behavior of these quantiti ...
$\nu = V/M$, where $V$ is the volume of the sample. (This quantity is the reciprocal $1/\rho$ of the material's
density The density (more precisely, the volumetric mass density; also known as specific mass), of a substance is its per unit . The symbol most often used for density is ''ρ'' (the lower case Greek letter ), although the Latin letter ''D'' can also ...

$\rho = M/V$.) Like $T$ and $P$, the specific volume $\nu$ is an intensive property of the material and its state, that does not depend on the amount of substance in the sample. Those variables are not independent. The allowed states are defined by an
equation of state In physics Physics (from grc, φυσική (ἐπιστήμη), physikḗ (epistḗmē), knowledge of nature, from ''phýsis'' 'nature'), , is the natural science that studies matter, its Motion (physics), motion and behavior through S ...
relating those three variables: $F\left(T, P, \nu\right) = 0.$ The function $F$ depends on the material under consideration. The specific internal energy stored internally in the sample, per unit of mass, will then be another function $U\left(T, P, \nu\right)$ of these state variables, that is also specific of the material. The total internal energy in the sample then will be $M \, U\left(T,P,\nu\right)$. For some simple materials, like an
ideal gas An ideal gas is a theoretical gas Gas is one of the four fundamental states of matter In physics Physics is the natural science that studies matter, its Elementary particle, fundamental constituents, its Motion (physics), motion ...
, one can derive from basic theory the equation of state $F = 0$ and even the specific internal energy $U$ In general, these functions must be determined experimentally for each substance.

## Conservation of energy

The absolute value of this quantity is undefined, and (for the purposes of thermodynamics) the state of "zero internal energy" can be chosen arbitrarily. However, by the
law of conservation of energy In physics Physics (from grc, φυσική (ἐπιστήμη), physikḗ (epistḗmē), knowledge of nature, from ''phýsis'' 'nature'), , is the natural science that studies matter, its Motion (physics), motion and behavior through ...
, any infinitesimal increase $M \, \mathrmU$ in the total internal energy $M U$ must be matched by the net flow of heat energy $\mathrmQ$ into the sample, plus any net mechanical energy provided to it by enclosure or surrounding medium on it. The latter is $-P \, \mathrmV$, where $\mathrm V$ is the change in the sample's volume in that infinitesimal step.Feynman, Richard ''
The Feynman Lectures on Physics ''The Feynman Lectures on Physics'' is a physics Physics is the natural science that studies matter, its Elementary particle, fundamental constituents, its Motion (physics), motion and behavior through Spacetime, space and time, and th ...

'', Vol. 1, Ch. 45
Therefore :$\mathrmQ - P \, \mathrm V = M \, \mathrmU$ hence :$\frac - P \, \mathrm\nu = \mathrmU$ If the volume of the sample (hence the specific volume of the material) is kept constant during the injection of the heat amount $\mathrmQ$, then the term $P \, \mathrm\nu$ is zero (no mechanical work is done). Then, dividing by $\mathrm T$, :$\frac = \frac$ where $\mathrmT$ is the change in temperature that resulted from the heat input. The left-hand side is the specific heat capacity at constant volume $c_V$ of the material. For the heat capacity at constant pressure, it is useful to define the
specific enthalpy Enthalpy , a property of a , is the sum of the system's and the product of its pressure and volume. It is a used in many measurements in chemical, biological, and physical systems at a constant pressure, which is conveniently provided by the la ...
of the system as the sum $h\left(T, P, \nu\right) = U\left(T, P, \nu\right) + P \nu$. An infinitesimal change in the specific enthalpy will then be :$\mathrmh = \mathrmU + V \, \mathrmP + P \, \mathrmV$ therefore :$\frac + V \, \mathrmP = \mathrmh$ If the pressure is kept constant, the second term on the left-hand side is zero, and :$\frac = \frac$ The left-hand side is the specific heat capacity at constant pressure $c_P$ of the material.

## Connection to equation of state

In general, the infinitesimal quantities $\mathrmT, \mathrmP, \mathrmV, \mathrmU$ are constrained by the equation of state and the specific internal energy function. Namely, :$\begin \displaystyle \mathrmT \frac\left(T,P,V\right) + \mathrmP \frac\left(T,P,V\right) + \mathrmV \frac\left(T,P,V\right) &=& 0\\$ \displaystyle \mathrmT \frac(T,P,V) + \mathrmP \frac(T,P,V) + \mathrmV \frac(T,P,V) &=& \mathrmU \end Here $\left(\partial F/\partial T\right)\left(T,P,V\right)$ denotes the (partial) derivative of the state equation $F$ with respect to its $T$ argument, keeping the other two arguments fixed, evaluated at the state $\left(T,P,V\right)$ in question. The other partial derivatives are defined in the same way. These two equations on the four infinitesimal increments normally constrain them to a two-dimensional linear subspace space of possible infinitesimal state changes, that depends on the material and on the state. The constant-volume and constant-pressure changes are only two particular directions in this space. This analysis also holds no matter how the energy increment $\mathrmQ$ is injected into the sample, namely by
heat conduction Thermal conduction is the transfer of internal energy The internal energy of a thermodynamic system A thermodynamic system is a body of matter In classical physics and general chemistry, matter is any substance that has mass and takes u ...

electromagnetic induction Electromagnetic or magnetic induction is the production of an electromotive force In electromagnetism Electromagnetism is a branch of physics involving the study of the electromagnetic force, a type of physical interaction that occurs b ...

,
radioactive decay Radioactive decay (also known as nuclear decay, radioactivity, radioactive disintegration or nuclear disintegration) is the process by which an unstable atomic nucleus loses energy by radiation. A material containing unstable nuclei is consi ...

, etc.

## Relation between heat capacities

For any specific volume $\nu$, denote $p_\nu\left(T\right)$ the function that describes how the pressure varies with the temperature $T$, as allowed by the equation of state, when the specific volume of the material is forcefully kept constant at $\nu$. Analogously, for any pressure $P$, let $\nu_P\left(T\right)$ be the function that describes how the specific volume varies with the temperature, when the pressure is kept constant at $P$. Namely, those functions are such that $F(T, p_\nu(T), \nu) = 0$and$F(T, P, \nu_P(T))= 0$ for any values of $T,P,\nu$. In other words, the graphs of $p_\nu\left(T\right)$ and $\nu_P\left(T\right)$ are slices of the surface defined by the state equation, cut by planes of constant $\nu$ and constant $P$, respectively. Then, from the
fundamental thermodynamic relation In thermodynamics, the fundamental thermodynamic relation are four fundamental equations which demonstrate how four important thermodynamic quantities depend on variables that can be controlled and measured experimentally. Thus, they are essentiall ...
it follows that :
coefficient of thermal expansion Thermal expansion is the tendency of matter to change its shape A shape or figure is the form of an object or its external boundary, outline, or external surface File:Water droplet lying on a damask.jpg, Water droplet lying on a damask ...
, *$\beta_T$ is the
isothermal In thermodynamics, an isothermal process is a type of thermodynamic process in which the temperature ''T'' of a Thermodynamic system, system remains constant: Δ''T'' = 0. This typically occurs when a system is in contact with an outside the ...

compressibility In thermodynamics Thermodynamics is a branch of physics that deals with heat, Work (thermodynamics), work, and temperature, and their relation to energy, entropy, and the physical properties of matter and radiation. The behavior of these quan ...
, both depending on the state $\left(T, P, \nu\right)$. The
heat capacity ratio In thermal physics Example of a thermal column between the ground and a cumulus A thermal column (or thermal) is a column of rising air in the lower altitudes of Earth's atmosphere File:Atmosphere gas proportions.svg, Composition of Ear ...
, or adiabatic index, is the ratio $c_P/c_V$ of the heat capacity at constant pressure to heat capacity at constant volume. It is sometimes also known as the isentropic expansion factor.

## Calculation from first principles

The path integral Monte Carlo method is a numerical approach for determining the values of heat capacity, based on quantum dynamical principles. However, good approximations can be made for gases in many states using simpler methods outlined below. For many solids composed of relatively heavy atoms (atomic number > iron), at non-cryogenic temperatures, the heat capacity at room temperature approaches 3''R'' = 24.94 joules per kelvin per mole of atoms (
Dulong–Petit law The Dulong–Petit law, a thermodynamic law proposed in 1819 by French physicists Pierre Louis Dulong and Alexis Thérèse Petit, states the classical expression for the molar Heat capacity, specific heat capacity of certain chemical elements. Expe ...
, ''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 The Boltzmann constant ( or ) is the proportionality f ...
). Low temperature approximations for both gases and solids at temperatures less than their characteristic Einstein temperatures or
Debye temperature In thermodynamics Thermodynamics is a branch of physics that deals with heat, Work (thermodynamics), work, and temperature, and their relation to energy, radiation, and physical properties of matter. The behavior of these quantities is gover ...
s can be made by the methods of Einstein and Debye discussed below. However, attention should be made for the consistency of such ab-initio considerations when used along with an equation of state for the considered material.S. Benjelloun, "Thermodynamic identities and thermodynamic consistency of Equation of States"
/ref>

### Ideal gas

For an
ideal gas An ideal gas is a theoretical gas Gas is one of the four fundamental states of matter In physics Physics is the natural science that studies matter, its Elementary particle, fundamental constituents, its Motion (physics), motion ...
, evaluating the partial derivatives above according to the
equation of state In physics Physics (from grc, φυσική (ἐπιστήμη), physikḗ (epistḗmē), knowledge of nature, from ''phýsis'' 'nature'), , is the natural science that studies matter, its Motion (physics), motion and behavior through S ...
, 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 The Boltzmann constant ( or ) is the proportionality f ...
, for an ideal gasCengel, Yunus A. and Boles, Michael A. (2010) ''Thermodynamics: An Engineering Approach'', 7th Edition, McGraw-Hill . :$P V = n R T,$ :$C_P - C_V = T \left\left(\frac\right\right)_ \left\left(\frac\right\right)_,$ :$P = \frac \Rightarrow \left\left(\frac\right\right)_ = \frac,$ :$V = \frac \Rightarrow \left\left(\frac\right\right)_ = \frac.$ Substituting :$T \left\left(\frac\right\right)_ \left\left(\frac\right\right)_ = T \frac \frac = \frac \frac = P \frac = nR,$ this equation reduces simply to 's relation: :$C_ - C_ = R.$ The differences in heat capacities as defined by the above Mayer relation is only exact for an ideal gas and would be different for any real gas.

* Specific heat of melting (Enthalpy of fusion) * Specific heat of vaporization (Enthalpy of vaporization) * Frenkel line *
Heat capacity ratio In thermal physics Example of a thermal column between the ground and a cumulus A thermal column (or thermal) is a column of rising air in the lower altitudes of Earth's atmosphere File:Atmosphere gas proportions.svg, Composition of Ear ...
*
Heat equation In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...
*
Heat transfer coefficient The heat transfer coefficient or film coefficient, or film effectiveness, in thermodynamics Thermodynamics is a branch of physics that deals with heat, Work (thermodynamics), work, and temperature, and their relation to energy, entropy, and th ...
*
History of thermodynamics The history of thermodynamics is a fundamental strand in the history of physics, the history of chemistry, and the history of science in general. Owing to the relevance of thermodynamics Thermodynamics is a branch of physics that deals with ...
*
Joback method The Joback method (often named Joback/Reid method) predicts eleven important and commonly used pure component thermodynamic properties from molecular structure only. Basic principles Group-contribution method The Joback method is a group-c ...
(Estimation of heat capacities) *
Latent heat Latent heat (also known as latent energy or heat of transformation) is energy released or absorbed, by a body or a thermodynamic system A thermodynamic system is a body of matter In classical physics and general chemistry, matter is any su ...
*
Material properties (thermodynamics) The thermodynamic properties of materials are intensive thermodynamic parameters which are specific to a given material. Each is directly related to a second order differential of a thermodynamic potential A thermodynamic potential (or more accur ...
*
Quantum statistical mechanics Quantum statistical mechanics is statistical mechanics In physics, statistical mechanics is a mathematical framework that applies Statistics, statistical methods and probability theory to large assemblies of microscopic entities. It does not as ...
*
R-value (insulation) Aerogel is an extremely good thermal insulator, which at a pressure of one-tenth of an atmosphere has an R-value of R-20 per inch of thickness, compared to R-3.5/inch for a fiberglass blanket. In the context of Construction, building and construc ...
* Specific heat of vaporization * Specific melting heat *
Statistical mechanics In physics Physics is the that studies , its , its and behavior through , and the related entities of and . "Physical science is that department of knowledge which relates to the order of nature, or, in other words, to the regular ...
*
Thermal mass In building design, thermal mass is a property of the mass of a building which enables it to store heat, providing "inertia" against temperature fluctuations. It is sometimes known as the thermal flywheel effect. For example, when outside tempera ...
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Thermodynamic databases for pure substances Thermodynamic databases contain information about thermodynamic properties for substances, the most important being enthalpy, entropy Entropy is a scientific concept, as well as a measurable physical property that is most commonly associate ...
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Thermodynamic equations Thermodynamics Thermodynamics is a branch of physics Physics is the that studies , its , its and behavior through , and the related entities of and . "Physical science is that department of knowledge which relates to the order of ...
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Volumetric heat capacity The volumetric heat capacity of a material is the heat capacity Heat capacity or thermal capacity is a physical property A physical property is any property Property (''latin: Res Privata'') in the Abstract and concrete, abstract is what be ...