In ^{−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 ^{−1}⋅K^{−1}, 790 J⋅kg^{−1}⋅K^{−1}, and 14300 J⋅kg^{−1}⋅K^{−1}, respectively. While the substance is undergoing a

Encyclopedia.com

Columbia University Press. Accessed on 2019-04-11. much in the fashion of^{−1}⋅K^{−1}. If the amount is taken to be the ^{−3}⋅K^{−1}.
One of the first scientists to use the concept was Joseph Black, an 18th-century medical doctor and professor of medicine at

^{−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.
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 $(p,T)$, and approximates the specific heat capacity by a constant $c$ suitable for those ranges.
Specific heat capacity is an

^{−1}⋅kg^{−1}. Since an increment of temperature of one degree Celsius is the same as an increment of one kelvin, that is the same as joule per degree Celsius per kilogram: J/(kg⋅°C). Sometimes the ^{−1}⋅K^{−1} = 0.001 J⋅kg^{−1}⋅K^{−1}.
The specific heat capacity of a substance (per unit of mass) has ^{2}⋅Θ^{−1}⋅T^{−2}, or (L/T)^{2}/Θ. Therefore, the SI unit J⋅kg^{−1}⋅K^{−1} is equivalent to ^{2}⋅K^{−1}⋅s^{−2}).

^{−1}⋅kg^{−1}) is greater than that of an hypothetical monatomic 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.

^{−1}⋅mol^{−1})
*''N'' is the number of molecules in the body. (dimensionless)
*''k''_{B} is the Boltzmann constant (J⋅K^{−1})
Again, SI units shown for example.
Read more about the quantities of dimension one at BIPM
In the

_{f}''
:$S(T\_f)=\backslash int\_^\; \backslash frac\; =\backslash int\_0^\; \backslash frac\backslash frac\; =\backslash int\_0^\; C(T)\backslash ,\backslash frac.$
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. One of the strengths of the

^{−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)..

intensive property
Physical properties of materials and systems can often be categorized as being either intensive or extensive, according to how the property changes when the size (or extent) of the system changes. According to IUPAC, an intensive quantity is one ...

can be expressed on a per- mole basis instead of a per-mass basis by the following equations analogous to the per mass equations:
:$C\_\; =\; \backslash left(\backslash frac\backslash right)\_P\; =\; \backslash text$
:$C\_\; =\; \backslash left(\backslash frac\backslash right)\_V\; =\; \backslash text$
where ''n'' is the number of moles in the body or thermodynamic system. 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.

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 ...

(J/(K⋅mol)),
*''N'' is the number of molecules in the body (dimensionless),
*''k''_{B} is the Boltzmann constant (J/(K⋅molecule)).
In the

_{f}:
:$S(T\_\backslash text)\; =\; \backslash int\_^\; \backslash frac\; =\; \backslash int\_0^\; \backslash frac\backslash frac\; =\; \backslash int\_0^\; C(T)\backslash ,\backslash frac.$

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 am ...

, 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.

coefficient of thermal expansion
Thermal expansion is the tendency of matter to change its shape, area, volume, and density in response to a change in temperature, usually not including phase transitions.
Temperature is a monotonic function of the average molecular kine ...

,
*$\backslash beta\_T$ is the compressibility
In thermodynamics and fluid mechanics, the compressibility (also known as the coefficient of compressibility or, if the temperature is held constant, the isothermal compressibility) is a measure of the instantaneous relative volume change of a ...

,
both depending on the state $(T,\; P,\; \backslash nu)$.
The heat capacity ratio, 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.

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 ...

). 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. 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"

Link to Archiv e-print**Link to Hal e-print**

/ref>

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 am ...

, evaluating the partial derivatives above according to the equation of state
In physics, chemistry, and thermodynamics, an equation of state is a thermodynamic equation relating state variables, which describe the state of matter under a given set of physical conditions, such as pressure, volume, temperature, or inter ...

, 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 ...

, for an ideal gasCengel, Yunus A. and Boles, Michael A. (2010) ''Thermodynamics: An Engineering Approach'', 7th Edition, McGraw-Hill .
:$P\; V\; =\; n\; R\; T,$
:$C\_P\; -\; C\_V\; =\; T\; \backslash left(\backslash frac\backslash right)\_\; \backslash left(\backslash frac\backslash right)\_,$
:$P\; =\; \backslash frac\; \backslash Rightarrow\; \backslash left(\backslash frac\backslash right)\_\; =\; \backslash frac,$
:$V\; =\; \backslash frac\; \backslash Rightarrow\; \backslash left(\backslash frac\backslash right)\_\; =\; \backslash frac.$
Substituting
:$T\; \backslash left(\backslash frac\backslash right)\_\; \backslash left(\backslash frac\backslash right)\_\; =\; T\; \backslash frac\; \backslash frac\; =\; \backslash frac\; \backslash frac\; =\; P\; \backslash frac\; =\; nR,$
this equation reduces simply to Mayer'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.

Phonon theory sheds light on liquid thermodynamics, heat capacity – Physics World**The phonon theory of liquid thermodynamics , Scientific Reports**

{{Authority control Physical quantities Thermodynamic properties

thermodynamics
Thermodynamics is a branch of physics that deals with heat, work, and temperature, and their relation to energy, entropy, and the physical properties of matter and radiation. The behavior of these quantities is governed by the four laws o ...

, the specific heat capacity (symbol ) of a substance is the heat capacity of a sample of the substance divided by the mass
Mass is an intrinsic property of a body. It was traditionally believed to be related to the quantity of matter in a physical body, until the discovery of the atom and particle physics. It was found that different atoms and different eleme ...

of the sample, also sometimes referred to as massic heat capacity. Informally, it is the amount 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 al ...

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 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 specific heat capacity is joule
The joule ( , ; symbol: J) is the unit of energy in the International System of Units (SI). It is equal to the amount of work done when a force of 1 newton displaces a mass through a distance of 1 metre in the direction of the force applied ...

per kelvin
The kelvin, symbol K, is the primary unit of temperature in the International System of Units (SI), used alongside its prefixed forms and the degree Celsius. It is named after the Belfast-born and University of Glasgow-based engineer and phys ...

per kilogram
The kilogram (also kilogramme) is the unit of mass in the International System of Units (SI), having the unit symbol kg. It is a widely used measure in science, engineering and commerce worldwide, and is often simply called a kilo colloquiall ...

, J⋅kgstate 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, ...

. 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 chemical element with symbol Fe (from la, ferrum) and atomic number 26. It is a metal that belongs to the first transition series and group 8 of the periodic table. It is, by mass, the most common element on Earth, right in fro ...

, granite
Granite () is a coarse-grained ( phaneritic) intrusive igneous rock composed mostly of quartz, alkali feldspar, and plagioclase. It forms from magma with a high content of silica and alkali metal oxides that slowly cools and solidifies unde ...

, and hydrogen
Hydrogen is the chemical element with the symbol H and atomic number 1. Hydrogen is the lightest element. At standard conditions hydrogen is a gas of diatomic molecules having the formula . It is colorless, odorless, tasteless, non-toxi ...

gas are about 449 J⋅kgphase 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 of ...

, 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 (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 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 $\backslash gamma\; =\; c\_p/c\_V$is the heat capacity ratio.
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 bEncyclopedia.com

Columbia University Press. Accessed on 2019-04-11. much in the fashion of

specific gravity
Relative density, or specific gravity, is the ratio of the density (mass of a unit volume) of a substance to the density of a given reference material. Specific gravity for liquids is nearly always measured with respect to water at its densest ...

. 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 chemical substance is the amount of energy that must be added, in the form of heat, to one mole of the substance in order to cause an increase of one unit in its temperature. Alternatively, it is the heat capacity of ...

instead, whose SI unit is joule per kelvin per mole, J⋅molvolume
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). The ...

of the sample (as is sometimes done in engineering), one gets the volumetric heat capacity, whose SI unit is joule per kelvin per cubic meter, J⋅mGlasgow University
, image = UofG Coat of Arms.png
, image_size = 150px
, caption = Coat of arms
Flag
, latin_name = Universitas Glasguensis
, motto = la, Via, Veritas, Vita
, ...

. 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: :$c\; =\; \backslash frac\; =\; \backslash frac\; \backslash cdot\; \backslash frac$ where $\backslash mathrm\; Q$ represents the amount of heat needed to uniformly raise the temperature of the sample by a small increment $\backslash mathrm\; T$. 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 thepressure
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 and ...

$p$ applied to it. Therefore, it should be considered a function $c(p,T)$ 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⋅kgintensive property
Physical properties of materials and systems can often be categorized as being either intensive or extensive, according to how the property changes when the size (or extent) of the system changes. According to IUPAC, an intensive quantity is one ...

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 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 of gases is typically between 1.3 and 1.67.Lange's Handbook of Chemistry, 10th ed. page 1524Applicability

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 aphase change
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 of ...

, 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 acalorimeter
A calorimeter is an object used for calorimetry, or the process of measuring the heat of chemical reactions or physical changes as well as heat capacity. Differential scanning calorimeters, isothermal micro calorimeters, titration calorimet ...

, 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, area, volume, and density in response to a change in temperature, usually not including phase transitions.
Temperature is a monotonic function of the average molecular kine ...

and the compressibility
In thermodynamics and fluid mechanics, the compressibility (also known as the coefficient of compressibility or, if the temperature is held constant, the isothermal compressibility) is a measure of the instantaneous relative volume change of a ...

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⋅Kgram
The gram (originally gramme; SI unit symbol g) is a unit of mass in the International System of Units (SI) equal to one one thousandth of a kilogram.
Originally defined as of 1795 as "the absolute weight of a volume of pure water equal to th ...

is used instead of kilogram for the unit of mass: 1 J⋅gdimension
In physics and mathematics, the dimension of a mathematical space (or object) is informally defined as the minimum number of coordinates needed to specify any point within it. Thus, a line has a dimension of one (1D) because only one coordin ...

Lmetre
The metre ( British spelling) or meter ( American spelling; see spelling differences) (from the French unit , from the Greek noun , "measure"), symbol m, is the primary unit of length in the International System of Units (SI), though its pr ...

squared per second
The second (symbol: s) is the unit of time in the International System of Units (SI), historically defined as of a day – this factor derived from the division of the day first into 24 hours, then to 60 minutes and finally to 60 seconds e ...

squared per kelvin
The kelvin, symbol K, is the primary unit of temperature in the International System of Units (SI), used alongside its prefixed forms and the degree Celsius. It is named after the Belfast-born and University of Glasgow-based engineer and phys ...

(mImperial engineering units

Professionals inconstruction
Construction is a general term meaning the art and science to form objects, systems, or organizations,"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 co ...

, civil engineering
Civil engineering is a professional engineering discipline that deals with the design, construction, and maintenance of the physical and naturally built environment, including public works such as roads, bridges, canals, dams, airports, sewa ...

, chemical engineering
Chemical engineering is an engineering field 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 materials int ...

, 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 primarily located in North America. It consists of 50 states, a federal district, five major unincorporated territo ...

, 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 pound (lb = 0.45359237 kg) as the unit of mass, the degree Fahrenheit or Rankine (°R = K, about 0.555556 K) as the unit of temperature increment, and the British thermal unit (BTU ≈ 1055.056 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/lb⋅°R, or 1 = 4186.68. The BTU was originally defined so that the average specific heat capacity of water would be 1 BTU/lb⋅°F. Note the value's similarity to that of the calorie - 4187 J/kg⋅°C ≈ 4184 J/kg⋅°C (~.07%) - as they are essentially measuring the same energy, using water as a basis reference, scaled to their systems' respective lbs and °F, or kg and °C.
Calories

In chemistry, heat amounts were often measured incalorie
The calorie is a unit of energy. For historical reasons, two main definitions of "calorie" are in wide use. The large calorie, food calorie, or kilogram calorie was originally defined as the amount of heat needed to raise the temperature of o ...

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⋅g.
*The "grand calorie" (also "kilocalorie", "kilogram-calorie", or "food calorie"; "kcal" or "Cal") is 1000 small calories, that is, 4184 J, exactly. It was defined so that the specific heat capacity of water would be 1 Cal/°C⋅kg.
While these units are still used in some contexts (such as kilogram calorie in nutrition
Nutrition is the biochemical and physiological process by which an organism uses food to support its life. It provides organisms with nutrients, which can be metabolized to create energy and chemical structures. Failure to obtain sufficient ...

), 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 ("small calorie") = 1 = 1 ("large calorie") = 4184 = 4.184 .
Note that while cal is of a Cal or kcal, it is also per ''gram'' instead of kilo''gram'': ergo, in either unit, the specific heat capacity of water is approximately 1.
Physical basis

The temperature of a sample of a substance reflects the averagekinetic energy
In physics, the kinetic energy of an object is the energy that it possesses due to its motion.
It is defined as the work needed to accelerate a body of a given mass from rest to its stated velocity. Having gained this energy during its accele ...

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 classical statistical mechanics, the equipartition theorem relates the temperature of a system to its average energies. The equipartition theorem is also known as the law of equipartition, equipartition of energy, or simply equipartition. ...

.
Monatomic gases

Quantum mechanics
Quantum mechanics is a fundamental theory in physics that provides a description of the physical properties of nature at the scale of atoms and subatomic particles. It is the foundation of all quantum physics including quantum chemistry, ...

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, $c\_\; =\; 3R/2\; \backslash approx\; \backslash mathrm$ and $c\_\; =\; 5R/2\; \backslash approx\; \backslash mathrm$, where $R\; \backslash approx\; \backslash mathrm$ is the ideal gas unit (which is the product of Boltzmann conversion constant from kelvin
The kelvin, symbol K, is the primary unit of temperature in the International System of Units (SI), used alongside its prefixed forms and the degree Celsius. It is named after the Belfast-born and University of Glasgow-based engineer and phys ...

microscopic energy unit to the macroscopic energy unit joule
The joule ( , ; symbol: J) is the unit of energy in the International System of Units (SI). It is equal to the amount of work done when a force of 1 newton displaces a mass through a distance of 1 metre in the direction of the force applied ...

, and the Avogadro number).
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''; 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 ...

$A$. That is, approximately,
:$c\_V\; \backslash approx\; \backslash mathrm/A\; \backslash quad\backslash quad\backslash quad\; c\_p\; \backslash approx\; \backslash mathrm/A$
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 extradegrees 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 ...

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 with the symbol N and atomic number 7. Nitrogen is a nonmetal and the lightest member of group 15 of the periodic table, often called the pnictogens. It is a common element in the universe, estimated at sevent ...

at constant volume is $c\_\; =\; \backslash mathrm$ (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⋅KNIST-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 one can show, :$c\_p\; -\; c\_v\; =\; \backslash frac$ where, *$\backslash alpha$ is thecoefficient of thermal expansion
Thermal expansion is the tendency of matter to change its shape, area, volume, and density in response to a change in temperature, usually not including phase transitions.
Temperature is a monotonic function of the average molecular kine ...

,
*$\backslash beta\_T$ is the isothermal
In thermodynamics, an isothermal process is a type of thermodynamic process in which the temperature ''T'' of a system remains constant: Δ''T'' = 0. This typically occurs when a system is in contact with an outside thermal reservoir, and ...

compressibility
In thermodynamics and fluid mechanics, the compressibility (also known as the coefficient of compressibility or, if the temperature is held constant, the isothermal compressibility) is a measure of the instantaneous relative volume change of a ...

, and
*$\backslash rho$ is 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. Mathematica ...

.
A derivation is discussed in the article Relations between specific heats.
For an 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 am ...

, if $\backslash rho$ is expressed as molar density in the above equation, this equation reduces simply to Mayer's relation,
:$C\_\; -\; C\_\; =\; R\; \backslash !$
where $C\_$ and $C\_$ are intensive property
Physical properties of materials and systems can often be categorized as being either intensive or extensive, according to how the property changes when the size (or extent) of the system changes. According to IUPAC, an intensive quantity is one ...

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 *$\backslash rho\; =\; \backslash frac$ 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 isobaric (constant pressure, $dp\; =\; 0$) orisochoric
Isochoric may refer to:
*cell-transitive, in geometry
*isochoric process
In thermodynamics, an isochoric process, also called a constant-volume process, an isovolumetric process, or an isometric process, is a thermodynamic process during which ...

(constant volume, $dV\; =\; 0$) processes. The corresponding specific heat capacities are expressed as
:$c\_p\; =\; \backslash left(\backslash frac\backslash right)\_p,$
:$c\_V\; =\; \backslash left(\backslash frac\backslash right)\_V.$
A related parameter to $c$ is $CV^$, the volumetric heat capacity. 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
:$c\_m\; =\; \backslash frac\; =\; \backslash frac.$
For pure homogeneous chemical compound
A chemical compound is a chemical substance composed of many identical molecules (or molecular entities) containing atoms from more than one chemical element held together by chemical bonds. A molecule consisting of atoms of only one element ...

s with established molecular or molar mass or a molar quantity is established, heat capacity as an intensive property
Physical properties of materials and systems can often be categorized as being either intensive or extensive, according to how the property changes when the size (or extent) of the system changes. According to IUPAC, an intensive quantity is one ...

can be expressed on a per mole basis instead of a per mass basis by the following equations analogous to the per mass equations:
:$C\_\; =\; \backslash left(\backslash frac\backslash right)\_p\; =\; \backslash text$
:$C\_\; =\; \backslash left(\backslash frac\backslash right)\_V\; =\; \backslash text$
where ''n'' = number of moles in the body or thermodynamic system. 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

Thepolytropic
A polytropic process is a thermodynamic process that obeys the relation:
p V^ = C
where ''p'' is the pressure, ''V'' is volume, ''n'' is the polytropic index, and ''C'' is a constant. The polytropic process equation describes expansion and com ...

heat capacity is calculated at processes if all the thermodynamic properties (pressure, volume, temperature) change
:$C\_\; =\; \backslash left(\backslash frac\backslash right)\; =\; \backslash text$
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

Thedimensionless
A dimensionless quantity (also known as a bare quantity, pure quantity, or scalar quantity as well as quantity of dimension one) is a quantity to which no physical dimension is assigned, with a corresponding SI unit of measurement of one (or 1) ...

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 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, expressed in units of energy per temperature increment per ...

(J⋅KIdeal 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 am ...

article, dimensionless heat capacity $C^*$ is expressed as $\backslash hat\; c$ .
Heat capacity at absolute zero

From the definition ofentropy
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 thermodyna ...

:$TdS=\backslash delta\; Q$
the absolute entropy can be calculated by integrating from zero kelvins temperature to the final temperature ''TDebye model
In thermodynamics and solid-state physics, the Debye model is a method developed by Peter Debye in 1912 for estimating the phonon contribution to the specific heat (Heat capacity) in a solid. It treats the vibrations of the atomic lattice (hea ...

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 theequipartition theorem
In classical statistical mechanics, the equipartition theorem relates the temperature of a system to its average energies. The equipartition theorem is also known as the law of equipartition, equipartition of energy, or simply equipartition. ...

, 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, a phonon is a collective excitation in a periodic, Elasticity (physics), elastic arrangement of atoms or molecules in condensed matter physics, condensed matter, specifically in solids and some liquids. A type of quasiparticle, a phon ...

. See Debye model
In thermodynamics and solid-state physics, the Debye model is a method developed by Peter Debye in 1912 for estimating the phonon contribution to the specific heat (Heat capacity) in a solid. It treats the vibrations of the atomic lattice (hea ...

.
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⋅KCalculation 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 by French physicists Pierre Louis Dulong and Alexis Thérèse Petit, states that the classical expression for the molar specific heat capacity of certain chemical elements is constant for tem ...

, ''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 ...

). 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.
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 (seecoefficient of thermal expansion
Thermal expansion is the tendency of matter to change its shape, area, volume, and density in response to a change in temperature, usually not including phase transitions.
Temperature is a monotonic function of the average molecular kine ...

and compressibility
In thermodynamics and fluid mechanics, the compressibility (also known as the coefficient of compressibility or, if the temperature is held constant, the isothermal compressibility) is a measure of the instantaneous relative volume change of a ...

). 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, 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 anideal 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 am ...

, evaluating the partial derivatives above according to the equation of state
In physics, chemistry, and thermodynamics, an equation of state is a thermodynamic equation relating state variables, which describe the state of matter under a given set of physical conditions, such as pressure, volume, temperature, or inter ...

, 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 ...

, for an ideal gas
:$P\; V\; =\; n\; R\; T,$
:$C\_P\; -\; C\_V\; =\; T\; \backslash left(\backslash frac\backslash right)\_\; \backslash left(\backslash frac\backslash right)\_,$
:$P\; =\; \backslash frac\; \backslash Rightarrow\; \backslash left(\backslash frac\backslash right)\_\; =\; \backslash frac,$
:$V\; =\; \backslash frac\; \backslash Rightarrow\; \backslash left(\backslash frac\backslash right)\_\; =\; \backslash frac.$
Substituting
:$T\; \backslash left(\backslash frac\backslash right)\_\; \backslash left(\backslash frac\backslash right)\_\; =\; T\; \backslash frac\; \backslash frac\; =\; \backslash frac\; \backslash frac\; =\; P\; \backslash frac\; =\; nR,$
this equation reduces simply to Mayer'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 :$c\; =\; \backslash frac,$ which in the absence of phase transitions is equivalent to :$c\; =\; E\_m\; =\; \backslash frac\; =\; \backslash frac,$ 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, *$\backslash rho\; =\; \backslash frac$ 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 isobaric (constant pressure, $\backslash textP\; =\; 0$) orisochoric
Isochoric may refer to:
*cell-transitive, in geometry
*isochoric process
In thermodynamics, an isochoric process, also called a constant-volume process, an isovolumetric process, or an isometric process, is a thermodynamic process during which ...

(constant volume, $\backslash textV\; =\; 0$) processes. The corresponding specific heat capacities are expressed as
:$c\_P\; =\; \backslash left(\backslash frac\backslash right)\_P,$
:$c\_V\; =\; \backslash left(\backslash frac\backslash right)\_V.$
From the results of the previous section, dividing through by the mass gives the relation
:$c\_P\; -\; c\_V\; =\; \backslash frac.$
A related parameter to $c$ is $C/V$, the volumetric heat capacity. 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
:$c\_m\; =\; \backslash frac\; =\; \backslash frac.$
For pure homogeneous
Homogeneity and heterogeneity are concepts often used in the sciences and statistics relating to the uniformity of a substance or organism. A material or image that is homogeneous is uniform in composition or character (i.e. color, shape, size ...

chemical compound
A chemical compound is a chemical substance composed of many identical molecules (or molecular entities) containing atoms from more than one chemical element held together by chemical bonds. A molecule consisting of atoms of only one element ...

s with established molecular or molar mass, or a molar quantity, heat capacity as an Polytropic heat capacity

Thepolytropic
A polytropic process is a thermodynamic process that obeys the relation:
p V^ = C
where ''p'' is the pressure, ''V'' is volume, ''n'' is the polytropic index, and ''C'' is a constant. The polytropic process equation describes expansion and com ...

heat capacity is calculated at processes if all the thermodynamic properties (pressure, volume, temperature) change:
:$C\_\; =\; \backslash left(\backslash frac\backslash right)\; =\; \backslash text$
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

Thedimensionless
A dimensionless quantity (also known as a bare quantity, pure quantity, or scalar quantity as well as quantity of dimension one) is a quantity to which no physical dimension is assigned, with a corresponding SI unit of measurement of one (or 1) ...

heat capacity of a material is
:$C^*\; =\; \backslash frac\; =\; \backslash frac,$
where
*$C$ is the heat capacity of a body made of the material in question (J/K),
*''n'' is the amount of substance in the body ( mol),
*''R'' is the 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 am ...

article, dimensionless heat capacity $C^*$ is expressed as $\backslash hat\; c$ 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 classical statistical mechanics, the equipartition theorem relates the temperature of a system to its average energies. The equipartition theorem is also known as the law of equipartition, equipartition of energy, or simply equipartition. ...

.
More generally, the dimensionless heat capacity relates the logarithmic increase in temperature to the increase in the dimensionless entropy per particle $S^*\; =\; S\; /\; N\; k\_\backslash text$, measured in nats.
:$C^*\; =\; \backslash frac.$
Alternatively, using base-2 logarithms, $C^*$ relates the base-2 logarithmic increase in temperature to the increase in the dimensionless entropy measured in bits.
Heat capacity at absolute zero

From the definition ofentropy
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 thermodyna ...

:$T\; \backslash ,\; \backslash textS\; =\; \backslash delta\; Q,$
the absolute entropy can be calculated by integrating from zero to the final temperature ''T''Thermodynamic derivation

In theory, the specific heat capacity of a substance can also be derived from its abstract thermodynamic modeling by anequation of state
In physics, chemistry, and thermodynamics, an equation of state is a thermodynamic equation relating state variables, which describe the state of matter under a given set of physical conditions, such as pressure, volume, temperature, or inter ...

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 volume $\backslash nu\; =\; V/M$, where $V$ is the volume of the sample. (This quantity is the reciprocal $1/\backslash rho$ of the material'sdensity
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. Mathematica ...

$\backslash rho\; =\; M/V$.) Like $T$ and $P$, the specific volume $\backslash nu$ 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, chemistry, and thermodynamics, an equation of state is a thermodynamic equation relating state variables, which describe the state of matter under a given set of physical conditions, such as pressure, volume, temperature, or inter ...

relating those three variables: $F(T,\; P,\; \backslash nu)\; =\; 0.$ 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 $U(T,\; P,\; \backslash nu)$ of these state variables, that is also specific of the material. The total internal energy in the sample then will be $M\; \backslash ,\; U(T,P,\backslash nu)$.
For some simple materials, like an 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 thelaw of conservation of energy
In physics and chemistry, the law of conservation of energy states that the total energy of an isolated system remains constant; it is said to be ''conserved'' over time. This law, first proposed and tested by Émilie du Châtelet, means that ...

, any infinitesimal increase $M\; \backslash ,\; \backslash mathrmU$ in the total internal energy $M\; U$ must be matched by the net flow of heat energy $\backslash mathrmQ$ into the sample, plus any net mechanical energy provided to it by enclosure or surrounding medium on it. The latter is $-P\; \backslash ,\; \backslash mathrmV$, where $\backslash mathrm\; V$ 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 textbook based on some lectures by Richard Feynman, a Nobel laureate who has sometimes been called "The Great Explainer". The lectures were presented before undergraduate students at the Cali ...

'', Vol. 1, Ch. 45 Therefore
:$\backslash mathrmQ\; -\; P\; \backslash ,\; \backslash mathrm\; V\; =\; M\; \backslash ,\; \backslash mathrmU$
hence
:$\backslash frac\; -\; P\; \backslash ,\; \backslash mathrm\backslash nu\; =\; \backslash mathrmU$
If the volume of the sample (hence the specific volume of the material) is kept constant during the injection of the heat amount $\backslash mathrmQ$, then the term $P\; \backslash ,\; \backslash mathrm\backslash nu$ is zero (no mechanical work is done). Then, dividing by $\backslash mathrm\; T$,
:$\backslash frac\; =\; \backslash frac$
where $\backslash mathrmT$ 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 thermodynamic system, is the sum of the system's internal energy and the product of its pressure and volume. It is a state function used in many measurements in chemical, biological, and physical systems at a constant p ...

of the system as the sum $h(T,\; P,\; \backslash nu)\; =\; U(T,\; P,\; \backslash nu)\; +\; P\; \backslash nu$. An infinitesimal change in the specific enthalpy will then be
:$\backslash mathrmh\; =\; \backslash mathrmU\; +\; V\; \backslash ,\; \backslash mathrmP\; +\; P\; \backslash ,\; \backslash mathrmV$
therefore
:$\backslash frac\; +\; V\; \backslash ,\; \backslash mathrmP\; =\; \backslash mathrmh$
If the pressure is kept constant, the second term on the left-hand side is zero, and
:$\backslash frac\; =\; \backslash frac$
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 $\backslash mathrmT,\; \backslash mathrmP,\; \backslash mathrmV,\; \backslash mathrmU$ are constrained by the equation of state and the specific internal energy function. Namely, :$\backslash begin\; \backslash displaystyle\; \backslash mathrmT\; \backslash frac(T,P,V)\; +\; \backslash mathrmP\; \backslash frac(T,P,V)\; +\; \backslash mathrmV\; \backslash frac(T,P,V)\; \&=\&\; 0\backslash \backslash ;\; href="/html/ALL/l/ex.html"\; ;"title="ex">ex$ Here $(\backslash partial\; F/\backslash partial\; T)(T,P,V)$ 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 $(T,P,V)$ 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 $\backslash mathrmQ$ is injected into the sample, namely byheat conduction
Conduction is the process by which heat is transferred from the hotter end to the colder end of an object. The ability of the object to conduct heat is known as its ''thermal conductivity'', and is denoted .
Heat spontaneously flows along a tem ...

, irradiation, electromagnetic induction, 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 consid ...

, etc.
Relation between heat capacities

For any specific volume $\backslash nu$, denote $p\_\backslash nu(T)$ 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 $\backslash nu$. Analogously, for any pressure $P$, let $\backslash nu\_P(T)$ 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 $$F(T,\; p\_\backslash nu(T),\; \backslash nu)\; =\; 0$$and$$F(T,\; P,\; \backslash nu\_P(T))=\; 0$$ for any values of $T,P,\backslash nu$. In other words, the graphs of $p\_\backslash nu(T)$ and $\backslash nu\_P(T)$ are slices of the surface defined by the state equation, cut by planes of constant $\backslash nu$ and constant $P$, respectively. Then, from the fundamental thermodynamic relation it follows that :$c\_P(T,P,\backslash nu)\; -\; c\_V(T,P,\backslash nu)\; =\; T\; \backslash left;\; href="/html/ALL/l/frac(T)\backslash right.html"\; ;"title="frac(T)\backslash right">frac(T)\backslash right$isothermal
In thermodynamics, an isothermal process is a type of thermodynamic process in which the temperature ''T'' of a system remains constant: Δ''T'' = 0. This typically occurs when a system is in contact with an outside thermal reservoir, and ...

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 by French physicists Pierre Louis Dulong and Alexis Thérèse Petit, states that the classical expression for the molar specific heat capacity of certain chemical elements is constant for tem ...

, ''R'' is the Link to Archiv e-print

/ref>

Ideal gas

For anSee also

* Specific heat of melting (Enthalpy of fusion) * Specific heat of vaporization (Enthalpy of vaporization) * Frenkel line * Heat capacity ratio *Heat equation
In mathematics and physics, the heat equation is a certain partial differential equation. Solutions of the heat equation are sometimes known as caloric functions. The theory of the heat equation was first developed by Joseph Fourier in 1822 for t ...

* Heat transfer coefficient
In thermodynamics, the heat transfer coefficient or film coefficient, or film effectiveness, is the proportionality constant between the heat flux and the thermodynamic driving force for the flow of heat (i.e., the temperature difference, ). I ...

* 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 in much of science and technology, its history is finely wov ...

* 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- ...

(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, during a constant-temperature process — usually a first-order phase transition.
Latent heat can be underst ...

* 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. Examples for a simple 1-component system ...

* Quantum statistical mechanics
Quantum statistical mechanics is statistical mechanics applied to quantum mechanical systems. In quantum mechanics a statistical ensemble (probability distribution over possible quantum states) is described by a density operator ''S'', which is a ...

* R-value (insulation)
* Specific heat of vaporization
* Specific melting heat
* Statistical mechanics
* Table of specific heat capacities
The table of specific heat capacities gives the volumetric heat capacity as well as the specific heat capacity of some substances and engineering materials, and (when applicable) the molar heat capacity.
Generally, the most notable constant param ...

* Thermal mass
In building design, thermal mass is a property of the mass of a building that enables it to store heat and provide inertia against temperature fluctuations. It is sometimes known as the thermal flywheel effect. The thermal mass of heavy structura ...

* Thermodynamic databases for pure substances
Thermodynamic databases contain information about thermodynamic properties for substances, the most important being enthalpy, entropy, and Gibbs free energy. Numerical values of these thermodynamic properties are collected as tables or are calcu ...

* Thermodynamic equations
* Volumetric heat capacity
Notes

References

Further reading

* Emmerich Wilhelm & Trevor M. Letcher, Eds., 2010, ''Heat Capacities: Liquids, Solutions and Vapours'', Cambridge, U.K.:Royal Society of Chemistry, . A very recent outline of selected traditional aspects of the title subject, including a recent specialist introduction to its theory, Emmerich Wilhelm, "Heat Capacities: Introduction, Concepts, and Selected Applications" (Chapter 1, pp. 1–27), chapters on traditional and more contemporary experimental methods such as photoacoustic methods, e.g., Jan Thoen & Christ Glorieux, "Photothermal Techniques for Heat Capacities," and chapters on newer research interests, including on the heat capacities of proteins and other polymeric systems (Chs. 16, 15), of liquid crystals (Ch. 17), etc.External links

*(2012-05may-24Phonon theory sheds light on liquid thermodynamics, heat capacity – Physics World

{{Authority control Physical quantities Thermodynamic properties