Specific heat capacity

Specific heat capacity
Specific heat capacity
Other names	Specific heat
Common symbols	c
SI unit	J⋅kg−1⋅K−1
InSI base units	m2⋅K−1⋅s−2
Intensive?	Yes
Dimension	L2⋅T−2⋅K−1

Inthermodynamics,thespecific heat capacity(symbol $c$ ) of a substance is the amount ofheatthat must be added to one unit of mass of the substance in order to cause an increase of one unit intemperature.It is also referred to asmassic heat capacityor as thespecific heat.More formally it is theheat capacityof a sample of the substance divided by themassof the sample.^[1]TheSIunit of specific heat capacity isjouleperkelvinperkilogram,J⋅kg⁻¹⋅K⁻¹.^[2]For example, the heat required to raise the temperature of1 kgofwaterby1 Kis4184 joules,so the specific heat capacity of water is4184 J⋅kg⁻¹⋅K⁻¹.^[3]

Specific heat capacity often varies with temperature, and is different for eachstate of matter.Liquid water has one of the highest specific heat capacities among common substances, about4184 J⋅kg⁻¹⋅K⁻¹at 20 °C; but that of ice, just below 0 °C, is only2093 J⋅kg⁻¹⋅K⁻¹.The specific heat capacities ofiron,granite,andhydrogengas are about 449 J⋅kg⁻¹⋅K⁻¹,790 J⋅kg⁻¹⋅K⁻¹,and 14300 J⋅kg⁻¹⋅K⁻¹,respectively.^[4]While the substance is undergoing aphase transition,such as melting or boiling, its specific heat capacity is technically undefined, 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 capacityat constant pressure) than when it is heated in a closed vessel that prevents expansion (specific heat capacityat constant volume). These two values are usually denoted by $c_{p}$ and $c_{V}$ ,respectively; their quotient $\gamma =c_{p}/c_{V}$ is theheat capacity ratio.

The termspecific heatmay 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;^[5]much in the fashion ofspecific gravity.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 ofmoles,one gets themolar heat capacityinstead, whose SI unit is joule per kelvin per mole, J⋅mol⁻¹⋅K⁻¹.If the amount is taken to be thevolumeof the sample (as is sometimes done in engineering), one gets thevolumetric heat capacity,whose SI unit is joule per kelvin percubic meter,J⋅m⁻³⋅K⁻¹.

One of the first scientists to use the concept wasJoseph Black,an 18th-century medical doctor and professor of medicine atGlasgow University.He measured the specific heat capacities of many substances, using the termcapacity for heat.^[6]

Definition

The specific heat capacity of a substance, usually denoted by $c$ or $s$ ,is the heat capacity $C$ of a sample of the substance, divided by the mass $M$ of the sample:^[7] $c={\frac {C}{M}}={\frac {1}{M}}\cdot {\frac {\mathrm {d} Q}{\mathrm {d} T}},$ where $\mathrm {d} Q$ representsthe amount of heat needed to uniformly raise the temperature of the sample by a small increment $\mathrm {d} T$ .

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 $p$ 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⋅kg⁻¹⋅K⁻¹(15 °C). "^[8]When not specified, published values of the specific heat capacity $c$ generally are valid for somestandard 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 anintensive propertyof 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.^[9])

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 generateswork,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 measuredat constant pressure(orisobaric) and is often denoted $c_{p}$ .
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 measuredat constant volume(orisochoric) and denoted $c_{V}$ .

The value of $c_{V}$ 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 theheat capacity ratioof gases is typically between 1.3 and 1.67.^[10]

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 aphase change,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,and dividing by the sample's mass. Several techniques can be applied for estimating the heat capacity of a substance, such asfast differential scanning calorimetry.^[11]^[12]

Graph of temperature of phases of water heated from−100 °Cto200 °C– the dashed line example shows that melting and heating1 kgof ice at−50 °Cto water at40 °Cneeds600 kJ

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 thecoefficient of thermal expansionand thecompressibilityof the material, and compute the specific heat capacity at constant volume from these data according to the laws of thermodynamics.^{[citation needed]}

Units

International system

The SI unit for specific heat capacity is joule per kelvin per kilogram⁠J/kg⋅K⁠,J⋅K⁻¹⋅kg⁻¹.Since an increment of temperature of onedegree Celsiusis the same as an increment of one kelvin, that is the same as joule per degree Celsius per kilogram: J/(kg⋅°C). Sometimes thegramis used instead of kilogram for the unit of mass: 1 J⋅g⁻¹⋅K⁻¹= 1000 J⋅kg⁻¹⋅K⁻¹.

The specific heat capacity of a substance (per unit of mass) hasdimensionL²⋅Θ⁻¹⋅T⁻²,or (L/T)²/Θ. Therefore, the SI unit J⋅kg⁻¹⋅K⁻¹is equivalent tometresquared persecondsquared perkelvin(m²⋅K⁻¹⋅s⁻²).

Imperial engineering units

Professionals inconstruction,civil engineering,chemical engineering,and other technical disciplines, especially in theUnited States,may useEnglish Engineering unitsincluding thepound(lb = 0.45359237 kg) as the unit of mass, thedegree FahrenheitorRankine(°R =⁠5/9⁠K, about 0.555556 K) as the unit of temperature increment, and theBritish thermal unit(BTU ≈ 1055.056 J),^[13]^[14]as the unit of heat.

In those contexts, the unit of specific heat capacity is BTU/lb⋅°R, or 1⁠BTU/lb⋅°R⁠= 4186.68⁠J/kg⋅K⁠.^[15]The BTU was originally defined so that the average specific heat capacity of water would be 1 BTU/lb⋅°F.^[16]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 incalories.Confusingly, there are two common units with that name, respectively denotedcalandCal:

thesmall calorie(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).
Thegrand calorie(kilocalorie, kilogram-calorie, food calorie, kcal, Cal) is 1000 small calories, 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 innutrition), 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:

1⁠cal°C⋅g⁠= 1⁠Cal°C⋅kg⁠= 1⁠kcal°C⋅kg⁠= 4184⁠Jkg⋅K⁠^[17]= 4.184⁠kJkg⋅K⁠.

Note that while cal is1⁄1000of a Cal or kcal, it is also pergraminstead ofkilogram: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 energyof 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 theequipartition theorem.

Monatomic gases

Quantum mechanicspredicts 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 moleis the same for all monatomic gases (such as the noble gases). More precisely, $c_{V,\mathrm {m} }=3R/2\approx \mathrm {12.5\,J\cdot K^{-1}\cdot mol^{-1}}$ and $c_{P,\mathrm {m} }=5R/2\approx \mathrm {21\,J\cdot K^{-1}\cdot mol^{-1}}$ ,where $R\approx \mathrm {8.31446\,J\cdot K^{-1}\cdot mol^{-1}}$ is theideal gas unit(which is the product ofBoltzmann conversion constantfromkelvinmicroscopic energy unit to the macroscopic energy unitjoule,and theAvogadro number).

Therefore, the specific heat capacity (per gram, not per mole) of a monatomic gas will be inversely proportional to its (adimensional)atomic weight $A$ .That is, approximately,

$c_{V}\approx \mathrm {12470\,J\cdot K^{-1}\cdot kg^{-1}} /A\quad \quad \quad c_{p}\approx \mathrm {20785\,J\cdot K^{-1}\cdot kg^{-1}} /A$

For the noble gases, from helium to xenon, these computed values are

Gas	He	Ne	Ar	Kr	Xe
$A$	4.00	20.17	39.95	83.80	131.29
$c_{V}$ (J⋅K⁻¹⋅kg⁻¹)	3118	618.3	312.2	148.8	94.99
$c_{p}$ (J⋅K⁻¹⋅kg⁻¹)	5197	1031	520.3	248.0	158.3

Polyatomic gases

On the other hand, a polyatomic gas molecule (consisting of two or more atoms bound together) can store heat energy in kinetic energy, but also inrotationof the molecule and vibration of the atoms relative to each other (including internalpotential energy).

These extradegrees of freedomor "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 the other degrees of freedom. To achieve the same increase in temperature, more heat energy is needed for a gram of that substance than for a gram of a monatomic gas. Thus, the specific heat capacity per mole of a polyatomic gas depends both on the molecular mass and the number degrees of freedom of the molecules.^[18]^[19]^[20]

Quantum mechanics further says that each rotational or vibrational mode can only take or lose energy in certain discrete amounts (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 increases with temperature, sometimes in a step-like fashion as mode becomes unfrozen and starts absorbing part of the input heat energy.

For example, the molar heat capacity ofnitrogenN
₂at constant volume is $c_{V,\mathrm {m} }=\mathrm {20.6\,J\cdot K^{-1}\cdot mol^{-1}}$ (at 15 °C, 1 atm), which is $2.49R$ .^[21]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 $c_{V}$ ofN
₂(736 J⋅K⁻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 of⁠5/3⁠.

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⁻¹at 1500 °C, 36.9 at 2500 °C, and 37.5 at 3500 °C.^[22]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 thefundamental thermodynamic relationone can show,

$c_{p}-c_{v}={\frac {\alpha ^{2}T}{\rho \beta _{T}}}$

where

$\alpha$ is thecoefficient of thermal expansion,
$\beta _{T}$ is theisothermal compressibility,and
$\rho$ isdensity.

A derivation is discussed in the articleRelations between specific heats.

For anideal gas,if $\rho$ is expressed asmolardensity in the above equation, this equation reduces simply toMayer's relation,

$C_{p,m}-C_{v,m}=R\!$

where $C_{p,m}$ and $C_{v,m}$ areintensive propertyheat 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

$c={\partial C \over \partial m},$

which in the absence of phase transitions is equivalent to

$c=E_{m}={C \over m}={C \over {\rho V}},$

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
$\rho ={\frac {m}{V}}$ 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 includeisobaric(constant pressure, $dp=0$ ) orisochoric(constant volume, $dV=0$ ) processes. The corresponding specific heat capacities are expressed as

${\begin{aligned}c_{p}&=\left({\frac {\partial C}{\partial m}}\right)_{p},\\c_{V}&=\left({\frac {\partial C}{\partial m}}\right)_{V}.\end{aligned}}$

A related parameter to $c$ is $CV^{-1}$ ,thevolumetric 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

$c_{m}={\frac {C}{m}}={\frac {c_{V}}{\rho }}.$

For pure homogeneouschemical compoundswith establishedmolecular or molar massor amolar quantityis established, heat capacity as anintensive propertycan be expressed on a permolebasis instead of a per mass basis by the following equations analogous to the per mass equations:

${\begin{alignedat}{3}C_{p,m}=\left({\frac {\partial C}{\partial n}}\right)_{p}&={\text{molar heat capacity at constant pressure}}\\C_{V,m}=\left({\frac {\partial C}{\partial n}}\right)_{V}&={\text{molar heat capacity at constant volume}}\end{alignedat}}$

wheren= number of moles in the body orthermodynamic system.One may refer to such aper molequantity as molar heat capacity to distinguish it from specific heat capacity on a per-mass basis.

Polytropic heat capacity

Thepolytropicheat capacity is calculated at processes if all the thermodynamic properties (pressure, volume, temperature) change

$C_{i,m}=\left({\frac {\partial C}{\partial n}}\right)={\text{molar heat capacity at polytropic process}}$

The most important polytropic processes run between the adiabatic and the isotherm functions, the polytropic index is between1and the adiabatic exponent (γorκ)

Dimensionless heat capacity

Thedimensionlessheat capacity of a material is

$C^{*}={\frac {C}{nR}}={\frac {C}{Nk_{\text{B}}}}$

where

Cis the heat capacity of a body made of the material in question (J/K)
nis theamount of substancein the body (mol)
Ris thegas constant(J⋅K⁻¹⋅mol⁻¹)
Nis the number of molecules in the body. (dimensionless)
k_Bis theBoltzmann constant(J⋅K⁻¹)

Again,SIunits shown for example.

Read more about the quantities of dimension one^[23]at BIPM

In theIdeal gasarticle, dimensionless heat capacity $C^{*}$ is expressed as ${\hat {c}}$ .

Heat capacity at absolute zero

From the definition ofentropy

$TdS=\delta Q$

the absolute entropy can be calculated by integrating from zero kelvins temperature to the final temperatureT_f

$S(T_{f})=\int _{T=0}^{T_{f}}{\frac {\delta Q}{T}}=\int _{0}^{T_{f}}{\frac {\delta Q}{dT}}{\frac {dT}{T}}=\int _{0}^{T_{f}}C(T)\,{\frac {dT}{T}}.$

The heat capacity must be zero at zero temperature in order for the above integral not to yield an infinite absolute entropy, thus violating thethird law of thermodynamics.One of the strengths of theDebye modelis 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 3R,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,and as such is only valid in the classical limit of amicrostate 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 atstandard 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 3Rper mole ofatomsin the solid, although in molecular solids, heat capacities calculatedper mole of moleculesin molecular solids may be more than 3R.For example, the heat capacity of water ice at the melting point is about 4.6Rper mole of molecules, but only 1.5Rper mole of atoms. The lower than 3Rnumber "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 3Rper 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 ofphonons.SeeDebye model.

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).^{[notes 1]}

Calculation from first principles

Thepath integral Monte Carlomethod 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,Ris thegas constant). Low temperature approximations for both gases and solids at temperatures less than their characteristicEinstein temperaturesorDebye temperaturescan 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 expansionandcompressibility). 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.

Theheat 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,evaluating the partial derivatives above according to theequation of state,whereRis thegas constant,for an ideal gas^[24]

${\begin{alignedat}{3}PV&=nRT,&\\C_{P}-C_{V}&=T\left({\frac {\partial P}{\partial T}}\right)_{V,n}\left({\frac {\partial V}{\partial T}}\right)_{P,n},&\\P&={\frac {nRT}{V}}\Rightarrow \left({\frac {\partial P}{\partial T}}\right)_{V,n}&={\frac {nR}{V}},\\V&={\frac {nRT}{P}}\Rightarrow \left({\frac {\partial V}{\partial T}}\right)_{P,n}&={\frac {nR}{P}}.\end{alignedat}}$

Substituting

$T\left({\frac {\partial P}{\partial T}}\right)_{V,n}\left({\frac {\partial V}{\partial T}}\right)_{P,n}=T{\frac {nR}{V}}{\frac {nR}{P}}={\frac {nRT}{V}}{\frac {nR}{P}}=P{\frac {nR}{P}}=nR,$

this equation reduces simply toMayer's relation:

$C_{P,m}-C_{V,m}=R.$

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.