# Specific heat capacity: Wikis

Note: Many of our articles have direct quotes from sources you can cite, within the Wikipedia article! This article doesn't yet, but we're working on it! See more info or our list of citable articles.

# Encyclopedia

Specific heat capacity (often shortened to specific heat) is the measure of heat or thermal energy required to increase the temperature of a unit quantity of a substance by one unit. For example, at a temperature of 15 °C, the heat required to raise the temperature of 1 kg of water by K (equivalent to 1 °C) is 4186 joules, meaning that the specific heat of water is 4.186 kJ·kg−1·K−1.[1] This measure was originally determined by mechanical means.

More heat is required to increase the temperature of a substance with high specific heat capacity than one with low specific heat capacity. For instance, eight times the energy is required to increase the temperature of a magnesium ingot (1.030 kJ·kg−1·K−1 at 25 °C) as is required for a lead ingot (130 J·kg−1·K−1 at 25 °C) of the same mass.[2] The specific heat capacity of virtually any substance can be measured, including chemical elements, compounds, alloys, solutions, and composites. The term originated primarily through the work of 18th-century medical doctor and professor of Medicine at Glasgow University, Joseph Black, who conducted various measurements and used the phrase capacity for heat.[3]

Temperature is the result of the average total kinetic energy of particles in matter. Heat is transfer of thermal energy; it flows from regions of high temperature to regions of low temperature. Thermal energy is stored as kinetic energy and, in molecules and solids, also as potential energy in the modes of vibration.[4] These represent degrees of freedom of movement for atoms. These degrees of freedom, and sometimes others, contribute to the heat capacity of a thermodynamic system. As the temperature approaches absolute zero, the specific heat capacity of a system also approaches zero.

Quantum theory can be used to quantitatively predict specific heat capacities in simple systems.

In the measurement of physical properties of matter, the term specific denotes a bulk property−an intensive property which is independent of the quantity of substance. Properties that do depend on the quantity of matter are extensive properties, such as the total heat capacity of a sample; this is also called the thermal mass.

## Metrology

Specific heat capacity is an intensive property and as such it is quoted in reference to a measurement that represents quantity of matter, such as mass, or number of atoms or molecules. Sometimes the term unit quantity is used as a term in general discussions. When measuring specific heat capacity in most science fields, the unit quantity of a substance is defined invariably in terms of mass, most often the gram or kilogram, both being units in the International System of Units. Especially in chemistry, the unit quantity of specific heat capacity may also be the mole, which is a certain number of particles. When the unit quantity is the mole, the quantity is termed molar heat capacity or mole-specific heat capacity.

When mass is the unit quantity, the symbol for specific heat capacity is lowercase c.[citation needed] When the mole is the unit quantity, the symbol is often uppercase C. These are not universal conventions in all fields. If the divisor unit is not mentioned, it is assumed to be mass. However, volume-specific heat capacity is used in engineering (where the divisor unit is volume of material) and the other intensive quantity is sometimes referred to in that case as mass-specific heat capacity (as the longer and more formal form of specific heat capacity).

The reference scale for temperature in the sciences, is the Kelvin scale. In the United States other units of measure for specific heat capacity may be quoted in disciplines such as construction, civil engineering, and chemical engineering. A still common system is the English Engineering Units in which the mass reference is pound mass and the temperature is specified in degrees Fahrenheit or Rankine. In construction, the unit of heat is the British thermal unit.

If temperature is expressed in natural rather than historical terms, i.e. as a rate of energy increase per unit increase in state uncertainty, then heat capacity is the number of bits of mutual information between system and surroundings lost per two-fold increase in absolute temperature[5]. Thus for instance, with each two-fold increase in absolute temperature 3/2 bits of mutual information per atom in a monatomic ideal gas is lost.

The heat capacity of most systems is not a constant, but depends on the state variables of the thermodynamic system under study. In particular it is dependent on temperature itself, as well as the pressure and on the volume of the system. Different measurements of specific heat capacity may therefore be performed, most commonly at constant pressure and constant volume. The so-measured values are usually sup-scripted (by p and V, respectively) to indicate the definition. Gases and liquids are typically also measured at constant volume. Measurements under constant pressure produce larger values than those at constant volume because work must be performed in the former. This difference is particularly notable in gases where values under constant pressure are typically 30% to 66.7% greater than those at constant volume.[citation needed]

The heat flow, i.e. cooling or heating, denoted Q, required to change the temperature of a sample by a certain difference ΔT is given by

$Q = m c \Delta T\,$
where c is the specific heat capacity and m is the mass (or other unit of quantity used) of the substance.

The modern SI units for measuring specific heat capacity are either the joule per kilogram-kelvin (J·kg−1·K−1) or the joule per mole-kelvin (J·mol−1·K−1). Multiples, as usually, are indicated by the various SI prefixes.

 At constant pressure At constant volume Unit quantity = mass cp cv Unit quantity = mole Cp Cv

The heat capacity of a thermodynamic system is typically derived from its internal energy, U, less the work performed on the system. When this external work, i.e. the product of pressure and volume, is recognized, meaning the system's energy is measured in terms of its enthalpy,

$H = U + pV,\,$

the resulting measurement is the specific enthalpy, often denoted with an additional suffix (H) after the variable symbol, e.g., CvH.

The specific heat capacities of substances comprising molecules (distinct from the monatomic gases) are not fixed constants and vary somewhat depending on temperature. Accordingly, the temperature at which the measurement is made is usually also specified. Examples of two common ways to cite the specific heat of a substance are as follows:

Water (liquid): cp = 4.1855 J/(g·K) (25 °C)
Water (liquid): CvH = 74.539 J/(mol·K) (25 °C)

For liquids and gases, it is important to know the pressure for which given heat-capacity data refer. Most published data are given for standard pressure. However, quite 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]

The ratio of the specific heats (or Heat capacity ratio) is usually denoted by γ (gamma). It is often used in equations, such as for calculating speed of sound in an ideal gas.

## Derivations of heat capacity

### Definition of heat capacity

Heat capacity is defined as the ratio of a small amount of heat δQ added to the body, to the corresponding small increase in its temperature dT:

$C = \frac{\delta Q}{dT} = T \frac{d S}{d T}$

For thermodynamic systems with more than one state variable or physical property, the above definition does not give a single, unique quantity unless a particular infinitesimal path through the system’s phase space has been defined. This means that one needs to know at all times where all parts of the system are, how much mass they have, and how fast they are moving. This information is used to account for different ways that heat can be stored as kinetic energy (energy of motion) and potential energy (energy stored in force fields), as an object expands or contracts. For all real systems, the path through these changes must be explicitly defined, since the value of heat capacity depends on which path from one temperature to another, is chosen. Of particular usefulness in this context are the values of heat capacity for constant volume, CV, and constant pressure, CP. These will be defined below.

### Thermodynamic relation to internal energy

The state of a simple thermodynamic system with fixed mass may be described by three thermodynamic parameters, temperature T, pressure p, and volume V:

${\ f(T, p, V) = 0}.$

Therefore, the internal energy of the system is only a function of these state variables.

Most measurable thermodynamic properties are the second derivatives of the thermodynamic potentials. Heat capacity, however, may also be defined in terms of first derivatives of the internal energy, if the potential is expanded into terms of the state variables, T, p, and V.

When heat, dQ, is introduced into the system, the change of its internal energy, dU, is:[6]

${\ dU = dQ - p dV }.$

If the process is performed at constant volume, i.e. when the second term of this relation vanishes, one readily obtains

$\left(\frac{\delta Q}{\ dT}\right)_V = \left(\frac{\partial U}{\partial T}\right)_V = C_V$

This is defined as the heat capacity at constant volume.[6]

Most experimental conditions, especially for the condensed phases, involve working at constant pressure, rather than constant volume of the sample, and therefore the volume will change in general as heat is introduced into a system. Some of this heat is returned to the environment in performance of the work of expanding the sample and the change of the internal energy is reduced.

The corresponding heat capacity at constant pressure is

$C_p=\left(\frac{\delta Q}{\ dT}\right)_p = \left(\frac{\partial U + p \partial V}{\partial T}\right)_p$.

This introduces the enthalpy into the formalism, which is defined as H = U + pV. Enthalpy, like internal energy is another state function.

$C_p=\left(\frac{\partial H}{\partial T}\right)_p.$

The heat capacity at constant pressure is the change of the enthalpy of the sample with respect to temperature.

### Relation between specific heats

Measuring the 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 the containing vessel must be nearly rigid or at least very strong (see coefficient of thermal expansion and compressibility). Instead it is easier to measure the heat capacity at constant pressure (allowing the material to expand or contract as it wishes) and solve for the heat capacity at constant volume using mathematical relationships derived from the basic thermodynamic laws. Starting from the fundamental Thermodynamic Relation one can show,

$c_p - c_v = \frac{\alpha^2 T}{\rho \beta_T}$

where,

α is the coefficient of thermal expansion,
βT is the isothermal compressibility, and
ρ is density.

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

For an ideal gas, if ρ is expressed as molar density in the above equation, this equation reduces simply to Mayer's relation,

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

where Cp,m and Cv,m are intensive property 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

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

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 include isobaric (constant pressure, dp = 0) or isochoric (constant volume, dV = 0) processes. The corresponding specific heat capacities are expressed as

$c_p = \left(\frac{\partial C}{\partial m}\right)_p,$
$c_V = \left(\frac{\partial C}{\partial m}\right)_V.$

A related parameter to c is $CV^{-1}\,$, 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 (specific heat) 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 homogeneous chemical compounds with established molecular or molar mass or a molar quantity is established, heat capacity as an intensive property can be expressed on a per mole basis instead of a per mass basis by the following equations analogous to the per mass equations:

$C_{p,m} = \left(\frac{\partial C}{\partial n}\right)_p$ = molar heat capacity at constant pressure
$C_{V,m} = \left(\frac{\partial C}{\partial n}\right)_V$ = molar heat capacity at constant volume

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

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

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

The most important polytropic processes run between the adiabatic and the isotherm functions, the polytropic index is between 1 and the adiabatic exponent (γ or κ)

### Dimensionless heat capacity

The dimensionless heat capacity of a material is

$C^*={C \over nR} = {C \over {Nk}}$

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 (J/(K·mol)
N is the number of molecules in the body. (dimensionless)
k is Boltzmann’s constant (J/(K·molecule)

Again, SI units shown for example.

Read more about the quantities of dimension one[7] at BIPM

In the Ideal gas article, dimensionless heat capacity $C^* \,$ is expressed as $\hat c$ .

### Heat capacity at absolute zero

From the definition of entropy

$TdS=\delta Q\,$

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

$S(T_f)=\int_{T=0}^{T_f} \frac{\delta Q}{T} =\int_0^{T_f} \frac{\delta Q}{dT}\frac{dT}{T} =\int_0^{T_f} C(T)\,\frac{dT}{T}.$

The heat capacity must be zero at zero temperature in order for the above integral not to yield an infinite absolute entropy, thus violating the third law of thermodynamics. One of the strengths of the Debye model 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.

## Theory of heat capacity

### Factors that affect specific heat capacity

Molecules undergo many characteristic internal vibrations. Potential energy stored in these internal degrees of freedom contributes to a sample’s heat content, but not to its temperature. More internal degrees of freedom increase a substance's specific heat capacity.

#### Degrees of freedom

|Degrees of freedom (physics and chemistry)}} Molecules are quite different from the monatomic gases like helium and argon. With monatomic gases, thermal energy comprises only translational motions. Translational motions are ordinary, whole-body movements in 3D space whereby particles move about and exchange energy in collisions—like rubber balls in a vigorously shaken container (see animation here). These simple movements in the three X, Y, and Z–axis dimensions of space means individual atoms have three translational degrees of freedom. A degree of freedom is any form of energy in which heat transferred into an object can be stored. This can be in translational kinetic energy, rotational kinetic energy, or other forms such as potential energy in vibrational modes. Only three translational degrees of freedom are essential for individual atoms as vibrations don't apply to them. As to rotation about atom's axis, its energy is proportional to the square of atomic radius. Because the latter is relatively small, this contribution can be neglected in monatomic gases.

In molecules, however, rotational modes may become active due to higher moments of inertia about certain axes. Further internal vibrational degrees of freedom also may become active; molecules are complex objects their population of atoms that can move about within the molecule in different ways (see animation at right). Thermal energy is stored in these internal motions, but on a per-atom basis, the heat capacity of molecules does not exceed the heat capacity of monatomic gases, unless vibrational modes are brought into play.

For instance, nitrogen, which is a diatomic molecule, has five active degrees of freedom at room temperature: the three comprising translational motion plus two rotational degrees of freedom internally. Although the constant-volume molar heat capacity of nitrogen at this temperature is five-thirds that of monatomic gases, on a per-mole of atoms basis, it is only five-sixths that of a monatomic gas. Two separate nitrogen atoms would have a total of six degrees of freedom—the three translational degrees of freedom of each atom. When the atoms are bonded the molecule will still only have three translational degrees of freedom, as the two atoms in the molecule move as one. However, the molecule cannot be treated as a point object, and the moment of inertia has increased sufficiently about two axes to allow two rotational degrees of freedom to be active at room temperature to give five degrees of freedom. The moment of inertia about the third axis remains small, as this is the axis passing through the centres of the two atoms, and so is similar to that of a monatomic gas.[8]

At higher temperatures, however, nitrogen gas gains two more degrees of internal freedom, as the molecule is excited into higher vibrational modes which store thermal energy, and then the heat capacity per volume or mole of molecules approaches seven-thirds that of monatomic gases, or seven-sixths of monatomic, on a mole-of-atoms basis. This is now a higher heat capacity per atom than the monatomic figure, because the vibrational mode enables an extra degree of potential energy freedom per pair of atoms, which monatomic gases cannot possess.[9] See thermodynamic temperature for more information on translational motions, kinetic (heat) energy, and their relationship to temperature.

#### Per mole of...

##### Per mole of molecules

When the specific heat capacity, c, of a material is measured (lowercase c means the unit quantity is in terms of mass), different values arise because different substances have different molar masses (essentially, the weight of the individual atoms or molecules). Thermal energy arises, in part, due to the number of atoms or molecules that are vibrating.

##### Per mole of atoms

Conversely, for molecular-based substances (which also absorb heat into their internal degrees of freedom), massive, complex molecules with high atomic count—like octane—can store a great deal of energy per mole and yet are quite unremarkable on a mass basis, or on a per-atom basis. This is because, in fully excited systems, heat is stored independently by each atom in a substance, not primarily by the bulk motion of molecules.

Thus, it is the heat capacity per-mole-of-atoms, not per-mole-of-molecules, which comes closest to being a constant for all substances at high temperatures. This relationship was noticed empirically in 1819, and is called the Dulong-Petit law, after its two discoverers. [10] Historically, the fact that specific heat capacities are approximately equal when corrected by the presumed weight of the atoms of solids, was an important piece of data in favor of the atomic theory of matter.

Because of the connection of heat capacity to the number of atoms, some care should be taken to specify a mole-of-molecules basis vs. a mole-of-atoms basis, when comparing specific heat capacities of molecular solids and gases. Ideal gases have the same numbers of molecules per volume, so increasing molecular complexity adds heat capacity on a per-volume and per-mole-of-molecules basis, but may lower or raise heat capacity on a per-atom basis, depending on whether the temperature is sufficient to store energy as atomic vibration.

In solids, the quantitative limit of heat capacity in general is about 3 R per mole of atoms, where R is the ideal gas constant. This 3 R value is about 24.9 J/mole.K. Six degrees of freedom (three kinetic and three potential) are available to each atom. Each of these six contributes 12R specific heat capacity per mole of atoms. [11] This limit of 3 R per mole specific heat capacity is approached at room temperature for most solids, with significant departures at this temperature only for solids composed of the lightest atoms which are bound very strongly, such as beryllium (where the value is only of 66% of 3 R), or diamond (where it is only 24% of 3 R). These large departures are due to quantum effects which prevent full distribution of heat into all vibrational modes, when the energy difference between vibrational quantum states is very large compared to the average energy available to each atom from the ambient temperature.

For monatomic gases, the specific heat is only half of 3 R per mole, i.e. (32R per mole) due to loss of all potential energy degrees of freedom in these gases. For polyatomic gases, the heat capacity will be intermediate between these values on a per-mole-of-atoms basis, and (for heat-stable molecules) would approach the limit of 3 R per mole of atoms, for gases composed of complex molecules, and at higher temperatures at which all vibrational modes accept excitational energy. This is because large, complex gas molecules may be thought of as large blocks of solid matter which have lost only a small fraction of degrees of freedom, as compared to a fully-integrated solid.

#### Corollaries of these considerations for solids (volume-specific heat capacity)

Since the bulk density of a solid chemical element is strongly related to its molar mass (usually about 3 R per mole, as noted above), there exists noticeable inverse correlation between a solid’s density and its specific heat capacity on a per-mass basis. This is due to a very approximate tendency of atoms of most elements to be about the same size, despite much wider variations in density and atomic weight. These two factors (constancy of atomic volume and constancy of mole-specific heat capacity) result in a good correlation between the volume of any given solid chemical element and its total heat capacity. Another way of stating this, is that the volume-specific heat capacity (volumetric heat capacity) of solid elements is roughly a constant. The molar volume of solid elements is very roughly constant, and (even more reliably) so also is the molar heat capacity for most solid substances. These two factors determine the volumetric heat capacity, which as a bulk property may be striking in consistancy. For example, the element uranium is a metal which has a density almost 36 times that of the metal lithium, but uranium's specific heat capacity on a volumetric basis (i.e. per given volume of metal) is only 18% larger than lithium's.

Since the volume-specific corollary of the Dulong-Petit specific heat capacity relationship requires that atoms of all elements take up (on average) the same volume in solids, there are many departures from it, with most of these due to variations in atomic size. For instance, arsenic, which is only 14.5% less dense than antimony, has nearly 59% more specific heat capacity on a mass basis. In other words; even though an ingot of arsenic is only about 17% larger than an antimony one of the same mass, it absorbs about 59% more heat for a given temperature rise. The heat capacity ratios of the two substances closely follows the ratios of their molar volumes (the ratios of numbers of atoms in the same volume of each substance); the departure from the correlation to simple volumes in this case is due to lighter arsenic atoms being significantly more closely-packed than antimony atoms, instead of similar size. In other words, similar-sized atoms would cause a mole of arsenic to be 63% larger than a mole of antimony, with a correspondingly lower density, allowing its volume to more closely mirror its heat capacity behavior.

#### Other factors

##### Hydrogen bonds

Hydrogen-containing polar molecules like ethanol, ammonia, and water have powerful, intermolecular hydrogen bonds when in their liquid phase. These bonds provide another place where heat may be stored as potential energy of vibration, even at comparatively low temperatures. Hydrogen bonds account for the fact that liquid water stores nearly the theoretical limit of 3 R per mole of atoms, even at relatively low temperatures (i.e. near the freezing point of water).

##### Impurities

In the case of alloys, there are several conditions in which small impurity concentrations can greatly affect the specific heat. Alloys may exhibit marked difference in behaviour even in the case of small amounts of impurities being one element of the alloy; for example impurities in semiconducting ferromagnetic alloys may lead to quite different specific heat properties.[12]

### The simple case of the monatomic gas

In the case of a monatomic gas such as helium under constant volume, if it assumed that no electronic or nuclear quantum excitations occur, each atom in the gas has only 3 degrees of freedom, all of a translational type. No energy dependence is associated with the degrees of freedom which define the position of the atoms. While, in fact, the degrees of freedom corresponding to the momenta of the atoms are quadratic, and thus contribute to the heat capacity. There are N atoms, each of which has 3 components of momentum, which leads to 3N total degrees of freedom. This gives:

$C_V=\left(\frac{\partial U}{\partial T}\right)_V=\frac{3}{2}N\,k_B =\frac{3}{2}n\,R$
$C_{V,m}=\frac{C_V}{n}=\frac{3}{2}R$

where

CV is the heat capacity at constant volume of the gas
CV,m is the molar heat capacity at constant volume of the gas
N is the total number of atoms present in the container
n is the number of moles of atoms present in the container (n is the ratio of N and Avogadro’s number)
R is the ideal gas constant, (8.314570[70] J/(mol·K). R is equal to the product of Boltzmann’s constant kB and Avogadro’s number

The following table shows experimental molar constant volume heat capacity measurements taken for each noble monatomic gas (at 1 atm and 25 °C):

Monatomic gas CV, m (J/(mol·K)) CV, m/R
He 12.5 1.50
Ne 12.5 1.50
Ar 12.5 1.50
Kr 12.5 1.50
Xe 12.5 1.50

It is apparent from the table that the experimental heat capacities of the monatomic noble gases agrees with this simple application of statistical mechanics to a very high degree.

### Diatomic gas

In the somewhat more complex case of an ideal gas of diatomic molecules, the presence of internal degrees of freedom are apparent. In addition to the three translational degrees of freedom, there are rotational and vibrational degrees of freedom. In general, the number of degrees of freedom, f, in a molecule with na atoms is 3na:

$f=3n_a \,$

Mathematically, there are a total of three rotational degrees of freedom, one corresponding to rotation about each of the axes of three dimensional space. However, in practice only the existence of two degrees of rotational freedom for linear molecules will be considered. This approximation is valid because the moment of inertia about the internuclear axis is vanishingly small with respect other moments of inertia in the molecule (this is due to the extremely small radii of the atomic nuclei, compared to the distance between them in a molecule). Quantum mechanically, it can be shown that the interval between successive rotational energy eigenstates is inversely proportional to the moment of inertia about that axis. Because the moment of inertia about the internuclear axis is vanishingly small relative to the other two rotational axes, the energy spacing can be considered so high that no excitations of the rotational state can possibly occur unless the temperature is extremely high. It is easy to calculate the expected number of vibrational degrees of freedom (or vibrational modes). There are three degrees of translational freedom, and two degrees of rotational freedom, therefore

$f_\mathrm{vib}=f-f_\mathrm{trans}-f_\mathrm{rot}=6-3-2=1 \,$

Each rotational and translational degree of freedom will contribute R/2 in the total molar heat capacity of the gas. Each vibrational mode will contribute R to the total molar heat capacity, however. This is because for each vibrational mode, there is a potential and kinetic energy component. Both the potential and kinetic components will contribute R/2 to the total molar heat capacity of the gas. Therefore, a diatomic molecule would be expected to have a molar constant-volume heat capacity of

$C_{V,m}=\frac{3R}{2}+R+R=\frac{7R}{2}=3.5 R$

where the terms originate from the translational, rotational, and vibrational degrees of freedom, respectively.

The following is a table of some molar constant-volume heat capacities of various diatomic gasses at standard temperature (25 oC = 298 K)

Diatomic gas CV, m (J/(mol·K)) CV, m / R
H2 20.18 2.427
CO 20.2 2.43
N2 19.9 2.39
Cl2 24.1 3.06
Br2 (vapour) 28.2 3.39

From the above table, clearly there is a problem with the above theory. All of the diatomics examined have heat capacities that are lower than those predicted by the equipartition theorem, except Br2. However, as the atoms composing the molecules become heavier, the heat capacities move closer to their expected values. One of the reasons for this phenomenon is the quantization of vibrational, and to a lesser extent, rotational states. In fact, if it is assumed that the molecules remain in their lowest energy vibrational state because the inter-level energy spacings for vibration-energies are large, the predicted molar constant volume heat capacity for a diatomic molecule becomes just that from the contributions of translation and rotation:

$C_{V,m}=\frac{3R}{2}+R=\frac{5R}{2}=2.5R$

which is a fairly close approximation of the heat capacities of the lighter molecules in the above table. If the quantum harmonic oscillator approximation is made, it turns out that the quantum vibrational energy level spacings are actually inversely proportional to the square root of the reduced mass of the atoms composing the diatomic molecule. Therefore, in the case of the heavier diatomic molecules such as chlorine or bromine, the quantum vibrational energy level spacings become finer, which allows more excitations into higher vibrational levels at lower temperatures. This limit for storing heat capacity in vibrational modes, as discussed above, becomes 7"R" /2 = 3.5 R per mole, which is fairly consistent with the measured value for Br2 at room temperature. As temperatures rise, all diatomic gases approach this value.

### General gas phase

The specific heat of the gas is best conceptualized in terms of the degrees of freedom of an individual molecule. The different degrees of freedom correspond to the different ways in which the molecule may store energy. The molecule may store energy in its translational motion according to the formula:

$E=\frac{1}{2}\,m\left(v_x^2+v_y^2+v_z^2\right)$

where m  is the mass of the molecule and [vx,vy,vz] is velocity of the center of mass of the molecule. Each direction of motion constitutes a degree of freedom, so that there are three translational degrees of freedom.

In addition, a molecule may have rotational motion. The kinetic energy of rotational motion is generally expressed as

$E=\frac{1}{2}\,\left(I_1\omega_1^2+I_2\omega_2^2+I_3\omega_3^2\right)$

where I  is the moment of inertia tensor of the molecule, and 123] is the angular velocity pseudo-vector (in a coordinate system aligned with the principle axes of the molecule). In general, then, there will be three additional degrees of freedom corresponding to the rotational motion of the molecule, (For linear molecules one of the inertia tensor terms vanishes and there are only two rotational degrees of freedom). The degrees of freedom corresponding to translations and rotations are called the rigid degrees of freedom, since they do not involve any deformation of the molecule.

The motions of the atoms in a molecule which are not part of its gross translational motion or rotation may be classified as vibrational motions. It can be shown that if there are n atoms in the molecule, there will be as many as v = 3n − 3 − nr  vibrational degrees of freedom, where nr is the number of rotational degrees of freedom. A vibrational degree of freedom corresponds to a specific way in which all the atoms of a molecule can vibrate. The actual number of possible vibrations may be less than this maximal one, due to various symmetries.

For example, triatomic nitrous oxide N20 will have only 2 degrees of rotational freedom (since it is a linear molecule) and contains n=3 atoms: thus the number of possible vibrational degrees of freedom will be v = (3*3)-3-2 = 4. There are four ways or "modes" in which the three atoms can vibrate, corresponding to 1) A mode in which an atom at each end of the molecule moves away from, or towards, the center atom at the same time, 2) a mode in which either end atom moves asynchronously with regard to the other two, and 3) and 4) two modes in which the molecule bends out of line, from the center, in the two possible planar directions that are orthogonal to its axis. Each vibrational degree of freedom confers TWO total degrees of freedom, since vibrational energy mode partitions into 1 kinetic and 1 potential mode. This would give nitrous oxide 3 translational, 2 rotational, and 4 vibrational modes (but these last giving 8 vibrational degrees of freedom), for storing energy. This is a total of f = 3+2+8 = 13 total energy-storing degrees of freedom, for N20.

For a bent molecule like water H22O, a similar calculation gives 9-3-3 = 3 modes of vibration, and 3 (translational) + 3 (rotational) + 6(vibratonal) = 12 degrees of freedom.

### The storage of energy into degrees of freedom

If the molecule could be entirely described using classical mechanics, then the theorem of equipartition of energy could be used to predict that each degree of freedom would have an average energy in the amount of (1/2)kT  where k  is Boltzmann’s constant and T  is the temperature. Our calculation of the constant-volume heat content would be straightforward. Each molecule would be holding, on average, an energy of (f/2)kT  where f  is the total number of degrees of freedom in the molecule. Note that Nk = R if N is Avogadro's number, which is the case in considering the heat capacity of a mole of molecules. Thus, the total internal energy of the gas would be (f/2)NkT  where N  is the total number of molecules. The heat capacity (at constant volume) would then be a constant (f/2)Nk  the mole-specific heat capacity would be (f/2)R  the molecule-specific heat capacity would be (f/2)k  and the dimensionless heat capacity would be just f/2. Here again, each vibrational degree of freedom contributes 2f. Thus, a mole of nitrous oxide would have a total constant-volume heat capacity (including vibration) of (13/2)R by this calculation.

In summary, the molar heat capacity (mole-specific heat capacity) of an ideal gas with f degrees of freedom is given by

$C_{V,m}=\frac{f}{2} R$

This equation applies to all polyatomic gases, if the degrees of freedom are known.[13]

The constant-pressure heat capacity for any gas would exceed this by an extra factor of R (see Mayer's relation, above). As example Cp would be a total of (15/2)R/mole for nitrous oxide.

### The effect of quantum energy levels in storing energy in degrees of freedom

The various degrees of freedom cannot generally be considered to obey classical mechanics, however. Classically, the energy residing in each degree of freedom is assumed to be continuous—it can take on any positive value, depending on the temperature. In reality, the amount of energy that may reside in a particular degree of freedom is quantized: It may only be increased and decreased in finite amounts. A good estimate of the size of this minimum amount is the energy of the first excited state of that degree of freedom above its ground state. For example, the first vibrational state of the hydrogen chloride (HCl) molecule has an energy of about 5.74 × 10−20 joule. If this amount of energy were deposited in a classical degree of freedom, it would correspond to a temperature of about 4156 K.

If the temperature of the substance is so low that the equipartition energy of (1/2)kT  is much smaller than this excitation energy, then there will be little or no energy in this degree of freedom. This degree of freedom is then said to be “frozen out". As mentioned above, the temperature corresponding to the first excited vibrational state of HCl is about 4156 K. For temperatures well below this value, the vibrational degrees of freedom of the HCl molecule will be frozen out. They will contain little energy and will not contribute to the heat content or heat capacity of HCl gas.

### Energy storage mode "freeze-out" temperatures

It can be seen that for each degree of freedom there is a critical temperature at which the degree of freedom “unfreezes” and begins to accept energy in a classical way. In the case of translational degrees of freedom, this temperature is that temperature at which the thermal wavelength of the molecules is roughly equal to the size of the container. For a container of macroscopic size (e.g. 10 cm) this temperature is extremely small and has no significance, since the gas will certainly liquify or freeze before this low temperature is reached. For any real gas translational degrees of freedom may be considered to always be classical and contain an average energy of (3/2)kT  per molecule.

The rotational degrees of freedom are the next to “unfreeze". In a diatomic gas, for example, the critical temperature for this transition is usually a few tens of kelvins, although with a very light molecule such as hydrogen the rotational energy levels will be spaced so widely that rotational heat capacity may not completely "unfreeze" until considerably higher temperatures are reached. Finally, the vibrational degrees of freedom are generally the last to unfreeze. As an example, for diatomic gases, the critical temperature for the vibrational motion is usually a few thousands of kelvins, and thus for the nitrogen in our example at room temperature, no vibration modes would be exited, and the constant-volume heat capacity at room temperature is (5/2)R/mole, not (7/2)R/mole. As seen above, with some unusually heavy gases such as iodine gas Cl2, or bromine gas Br2, some vibrational heat capacity may be observed even at room temperatures.

It should be noted that it has been assumed that atoms have no rotational or internal degrees of freedom. This is in fact untrue. For example, atomic electrons can exist in excited states and even the atomic nucleus can have excited states as well. Each of these internal degrees of freedom are assumed to be frozen out due to their relatively high excitation energy. Nevertheless, for sufficiently high temperatures, these degrees of freedom cannot be ignored. In a few exceptional cases, such molecular electronic transitions are of sufficiently low energy that they contribute to heat capacity at room temperature, or even at cryogenic temperatures. One example of an electronic transition degree of freedom which contributes heat capacity at standard temperature is that of nitric oxide (NO), in which the single electron in an anti-bonding molecular orbital has energy transitions which contribute to the heat capacity of the gas even at room temperature.

An example of a nuclear magnetic transition degree of freedom which is of importance to heat capacity, is the transition which converts the spin isomers of hydrogen gas to each other. At room temperature, the proton spins of hydrogen gas are aligned 75% of the time, resulting in orthohydrogen. However, at liquid hydrogen temperatures, the parahydrogen form of H2 in which spins are anti-aligned predominates, and the heat capacity of the transition is sufficient to boil the hydrogen if this is heat is not removed with a catalyst, after the gas has been condensed. This example also illustrates the fact that some modes of storage of heat may not be in constant equilibrium with each other in substances, and heat absorbed or released from such phase changes may "catch up" with temperature changes of substances, only after a certain time.

### Solid phase

The dimensionless heat capacity divided by three, as a function of temperature as predicted by the Debye model and by Einstein’s earlier model. The horizontal axis is the temperature divided by the Debye temperature. Note that, as expected, the dimensionless heat capacity is zero at absolute zero, and rises to a value of three as the temperature becomes much larger than the Debye temperature. The red line corresponds to the classical limit of the Dulong-Petit law

For matter in a crystalline solid phase, the Dulong-Petit law, which was discovered empirically, states that the mole-specific heat capacity assumes the value 3 R. Indeed, for solid metallic chemical elements at room temperature, molar heat capacities range from about 2.8 R to 3.4 R. Large exceptions involve solids composed of light, tightly-bonded atoms such as beryllium at 2.0 R, and diamond at only 0.735 R. The latter conditions create large quantum vibrational energy spacing, so that many vibrational modes are not available (are frozen out) at room temperature.

The theoretical maximum heat capacity for larger and larger multi-atomic gases at higher temperatures, also approaches the Dulong-Petit limit of 3 R, so long as this is calculated per mole of atoms, not molecules. The reason is that gases with very large molecules, in theory have almost the same high-temperature heat capacity as solids, lacking only the (small) heat capacity contribution that comes from potential energy that cannot be stored between separate molecules in a gas.

The Dulong-Petit limit results from the equipartition theorem, 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. See Debye model.

## Table of specific heat capacities

Note that especially high values, as for paraffin, water and ammonia, result from calculating specific heats in terms of moles of molecules. If specific heat is expressed per mole of atoms for these substances, few constant-volume values exceed the theoretical Dulong-Petit limit of 25 J/(mol·K) = 3 R per mole of atoms.

Table of specific heat capacities at 25 °C unless otherwise noted Notable minima and maxima are shown in maroon
Substance Phase Cp
J/(g·K)
Cp,m
J/(mol·K)
Cv,m
J mol−1·K−1
Volumetric
heat capacity

J/(cm3·K)-
Air (Sea level, dry, 0 °C) gas 1.0035 29.07 20.7643 0.001297
Air (typical room conditionsA) gas 1.012 29.19 20.85
Aluminium solid 0.897 24.2 2.422
Ammonia liquid 4.700 80.08 3.263
Animal (and human) tissue[14] mixed 3.5 - 3.7*
Antimony solid 0.207 25.2 1.386
Argon gas 0.5203 20.7862 12.4717
Arsenic solid 0.328 24.6 1.878
Beryllium solid 1.82 16.4 3.367
Bismuth[15] solid 0.123 25.7 1.20
Cadmium solid 0.231 --.-- ----
Carbon dioxide CO2[13] gas 0.839* 36.94 28.46
Chromium solid 0.449 --.-- ----
Copper solid 0.385 24.47 3.45
Diamond solid 0.5091 6.115 1.782
Ethanol liquid 2.44 112 1.925
Gasoline liquid 2.22 228 1.64
Glass[15] solid 0.84
Gold solid 0.129 25.42 2.492
Granite[15] solid 0.790 2.17
Graphite solid 0.710 8.53 1.534
Helium gas 5.1932 20.7862 12.4717
Hydrogen gas 14.30 28.82
Hydrogen sulfide H2S[13] gas 1.015* 34.60
Iron solid 0.450 25.1[citation needed] 3.537
Lead solid 0.127 26.4 1.44
Lithium solid 3.58 24.8 1.912
Magnesium solid 1.02 24.9 1.773
Mercury liquid 0.1395 27.98 1.888
Methane 275K gas 2.191
Nitrogen gas 1.040 29.12 20.8
Neon gas 1.0301 20.7862 12.4717
Oxygen gas 0.918 29.38
Paraffin wax solid 2.5 900 2.325
Polyethylene (rotomolding grade)[16] solid 2.3027
Polyethylene (rotomolding grade)[16] liquid 2.9308
Silica (fused) solid 0.703 42.2 1.547
Silver[15] solid 0.233 24.9 2.44
Tin solid 0.227 ----- ----
Tungsten[15] solid 0.134 24.8 2.58
Uranium solid 0.116 27.7 2.216
Water at 100 °C (steam) gas 2.080 37.47 28.03
Water at 25 °C liquid 4.1813 75.327 74.53 4.186
Water at -10 °C (ice)[15] solid 2.05 38.09 1.938
Zinc[15] solid 0.387 25.2 2.76
Substance Phase Cp
J/(g·K)
Cp,m
J/(mol·K)
Cv,m
J/(mol·K)
Volumetric
heat capacity

J/(cm3·K)

A Assuming an altitude of 194 metres above mean sea level (the world–wide median altitude of human habitation), an indoor temperature of 23 °C, a dewpoint of 9 °C (40.85% relative humidity), and 760 mm–Hg sea level–corrected barometric pressure (molar water vapor content = 1.16%).
*Derived data by calculation. This is for water-rich tissues such as brain. The whole-body average figure for mammals is approximately 2.9 J/(cm3·K) [17]

## Specific heat capacity of building materials

(Usually of interest to builders and solar designers)

Specific heat capacity of building materials
Substance Phase cp
J/(g·K)
Asphalt solid 0.920
Brick solid 0.840
Concrete solid 0.880
Glass, silica solid 0.840
Glass, crown solid 0.670
Glass, flint solid 0.503
Glass, pyrex solid 0.753
Granite solid 0.790
Gypsum solid 1.090
Marble, mica solid 0.880
Sand solid 0.835
Soil solid 0.800
Wood solid 1.7 (1.2 to 2.3)
Substance Phase cp
J/(g·K)

## Notes

1. ^ IUPAC.org, Gold Book, Standard Pressure. Besides being a round number, this had a very practical effect: relatively few people live and work at precisely sea level; 100 kPa equates to the mean pressure at an altitude of about 112 metres (which is closer to the 194–metre, world–wide median altitude of human habitation).

## References

1. ^ R. Nave. "Specific Heat". HyperPhysics. Georgia State University. Retrieved 2010-02-18.
2. ^ "Specific Heat Capacity". Science & Engineering Encyclopaedia. DirectDelta.co.uk. Retrieved 2010-02-18.
3. ^ Laider, Keith, J. (1993). The World of Physical Chemistry. Oxford University Press. ISBN 0-19-855919-4.
4. ^ Feynman, R., Lectures in Physics, vol. I, chapter 40, pp. 7-8
5. ^ P. Fraundorf (2003) Heat capacity in bits, American Journal of Physics 71:11, 1142-1151. (arXiv:cond-mat/9711074)
6. ^ a b Richard Feinman, Lectures on Physics, Vol.1 (45)
7. ^ About the unit one
8. ^ Quantum Physics and the Physics of large systems, Part 1A Physics, University of Cambridge, C.G. Smith, 2008.
9. ^ The comparison must be made under constant-volume conditions—CvH—so that no work is performed. Nitrogen’s CvH (100 kPa, 20 °C) = 20.8 J mol–1 K–1 vs. the monatomic gases which equal 12.4717 J mol–1 K–1. Citations: W.H. Freeman’s Physical Chemistry, Part 3: Change (422 kB PDF, here), Exercise 21.20b, Pg. 787. Also Georgia State University’s Molar Specific Heats of Gases.
10. ^ Petit A.-T., Dulong P.-L.: Recherches sur quelques points importants de la Théorie de la Chaleur. In: Annales de Chimie et de Physique 10, 395-413 (1819) (Translation)
11. ^ [1] Discussion of heat capacity of solids in terms of degrees of freedom.
12. ^ C. Michael Hogan, (1969) Density of States of an Insulating Ferromagnetic Alloy Phys. Rev. 188, 870–874, [Issue 2 – December 1969
13. ^ a b c Textbook: Young and Geller College Physics, 8e, Pearson Education, 2008
14. ^ Page 183 in: Medical biophysics. Flemming Cornelius. 6th Edition, 2008. (also giving a density of 1.06 kg/l)
15. ^ a b c d e f g Table of Specific Heats
16. ^ a b R.J. Crawford, Rotational molding of plastics
17. ^ Faber P., Garby L., Fat content affects heat capacity: a study in mice. Acta Physiol Scand 1995;153:187–5.