# Heat capacity

Heat capacity (usually denoted by a capital C, often with subscripts) is a measurable physical quantity that characterizes the ability of a body to store heat as it changes in temperature. It is defined as the rate of change of temperature as heat is added to a body at the given conditions and state of the body (foremost its temperature). In the International System of Units, heat capacity is expressed in units of joules per kelvin. It is termed an "extensive quantity" because it is sensitive to the size of the object (for example, a bathtub of water has a greater heat capacity than a cup of water). Dividing heat capacity by the body's mass yields a specific heat capacity (also called more properly "mass-specific heat capacity" or more loosely "specific heat"), which is an "intensive quantity," meaning it is no longer dependent on amount of material, and is now more dependent on the type of material, as well as the physical conditions of heating.

##  Definition

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

$C = \left( \frac{\delta Q}{dT} \right)_{cond.} = T \left( \frac{d S}{d T} \right)_{cond.}$

For thermodynamic systems with more than one physical dimension, 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 though 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.

##  Heat capacity of compressible bodies

The state of a simple compressible body with fixed mass is described by two thermodynamic parameters such as temperature T and pressure P. Therefore as mentioned above, one may distinguish between heat capacity at constant volume, $C_V$, and heat capacity at constant pressure, $C_P$:

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

where

$\delta Q$ is the infinitesimal amount of heat added,
$dT$ is the subsequent rise in temperature.

The increment of internal energy is the heat added and the work added:

$dU=T\,dS-P\,dV$

So the heat capacity at constant volume is

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

The enthalpy is defined by $H=U+PV$. The increment of enthalpy is

$dH = dU + (PdV+VdP) \!$

which, after replacing dU with the equation above and cancelling the PdV terms reduces to:

$dH=T\,dS+V\,dP.$

So the heat capacity at constant pressure is

$C_P=\left(\frac{\partial H}{\partial T}\right)_P.$

Note that this last "definition" is a bit circular, since the concept of "enthalpy" itself was invented to be a measure of heat absorbed or produced at constant pressures (the conditions in which chemists usually work). As such, enthalpy merely accounts for the extra heat which is produced or absorbed by pressure-volume work at constant pressure. Thus, it is not surprising that constant-pressure heat capacities may be defined in terms of enthalpy, since "enthalpy" was defined in the first place to make this so.

##  Specific heat capacity

The specific heat capacity of a material is

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

which in the absence of phase transitions is equivalent to

$c=c_m={C \over m} = {C \over {\rho V}}$

where

C is the heat capacity of a body made of the material in question (J·K−1)
m is the mass of the body (kg)
V is the volume of the body (m3)
ρ = mV−1 is the density of the material (kg·m−3)

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, d$P = 0$) and isochoric (constant volume, d$V = 0$) processes, and one conventionally writes for gases:

$c_P=\left(\frac{\partial C}{\partial m}\right)_P$
$c_V=\left(\frac{\partial C}{\partial m}\right)_V$

Units shown are SI units but, of course, any consistent set of units may be used.

A related parameter to $c$ is CV−1, the volumetric heat capacity, (J·m-3·K-1 in SI units). 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:

$c_m={C \over m} = {c_V \over {\rho }}$

##  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−1)
n is the amount of matter in the body (mol)
R is the gas constant (J·K−1·mol−1)
nR=Nk is the amount of matter in the body (J·K−1)
N is the number of molecules in the body. (dimensionless)
k is Boltzmann's constant (J·K−1·molecule−1)

Again, SI units shown for example.

##  Theoretical models

###  Gas phase

According to the equipartition theorem from classical statistical mechanics, for a system made up of independent and quadratic degrees of freedom, any input of energy into a closed system composed of N molecules is evenly divided among the degrees of freedom available to each molecule. It can be shown that, in the classical limit of statistical mechanics, for each independent and quadratic degree of freedom, that

$E_i=\frac{k_B T}{2}$

where

$E_i$ is the mean energy (measured in joules) associated with degree of freedom i.
T is the temperature (measured in kelvins)

$k_B$ is Boltzmann's constant, (1.380 6505(24) × 10−23 J K−1)

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:

$C_V=\left(\frac{\partial U}{\partial T}\right)_V=\frac{3}{2}N\,k_B =\frac{3}{2}n\,R$
$C_{V,m}=\frac{C_V}{n}=\frac{3}{2}R = 1.5 R$

where

$C_V$ is the heat capacity at constant volume of the gas
$C_{V,m}$ 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 K−1mol−1). R is equal to the product of Boltzmann's constant $k_B$ 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 K−1 mol−1) CV, m/R
He12.51.50
Ne12.51.50
Ar12.51.50
Kr12.51.50
Xe12.51.50

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. 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 we shall only consider the existence of two degrees of rotational freedom for linear molecules. 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. We can easily 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, we expect that a diatomic molecule would 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

Diatomic gas CV, m (J K−1 mol−1) CV, m / R
H220.182.427
CO20.22.43
N219.92.39
Cl224.12.90
Br232.03.84

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 are large, the predicted molar constant volume heat capacity for a diatomic molecule becomes

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

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, the quantum vibrational energy level spacings become finer, which allows more excitations into higher vibrational levels at a fixed temperature.

###  Solid phase

Image:DebyeVSEinstein.jpg
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 dimensionless specific heat capacity assumes the value 3. Indeed, for solid metallic chemical elements at room temperature, heat capacities range from about 2.8 to 3.4 (beryllium being a notable exception at 2.0).

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 contibution 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 ambiant 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 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 number results from the "freezing out" of possible vibration modes for light atoms at suitably low temperatures, just as in many gases. These effects are seen in solids more often than liquids: for example the heat capacity of liquid water is again close to the theoretical 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.

##  Heat capacity at absolute zero

From the definition of entropy

$TdS=\delta Q\,$

we can calculate the absolute entropy by integrating from zero 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 an approach of heat capacity toward zero as zero temperature is approached, and also predicts the proper mathematical form of this approach.