# Heat capacity

Heat capacity is a measure of the ability of a body to store heat.

The heat capacity of a body at a certain temperature is the ratio of a small amount of heat energy added to the body to the corresponding small increase in temperature of the body.

Dimension: energy/temperature.

SI unit: J·K−1 (joule per kelvin).

Abbreviation: C.

Formula:

${\displaystyle C={\frac {\delta Q}{dT}}=T{\frac {dS}{dT}}.}$

For a system with more than one dimension, the above definition is not complete until a particular infinitesimal path through the space has been defined. By the above formula, C is equal to the temperature times the change in entropy over some very short path divided by the change in temperature over that same path. For a one-dimensional system, the only thermodynamic variable will be the temperature, and that short path is implicitly defined, but for a multiply dimensioned system, it must be explicitly defined because the value of C depends on which short path you choose. For example, in a two dimensional system, the path might be defined by holding the volume constant, in which case we get the specific heat at constant volume:

${\displaystyle C_{v}=T\left({\frac {\partial S}{\partial T}}\right)_{V}}$

or we may define the path by holding the pressure constant, yielding the specific heat at constant pressure:

${\displaystyle C_{P}=T\left({\frac {\partial S}{\partial T}}\right)_{P}}$

In the following discussion, C(T) will be used to specify the specific heat of a one-dimensional system, or of a multiple dimensional system in which the path is assumed to be known from the context of the discussion.

## Heat capacity at absolute zero

From the definition of entropy

${\displaystyle TdS=\delta Q\,}$

we can calculate the absolute entropy by integrating from zero temperature to the final temperature Tf

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

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

## Heat capacity of compressible bodies

The state of a compressible body with fixed mass is described two thermodynamic parameters such as temperature T and pressure P. Therefore as mentioned above, one may distinguish between heat capacity at constant volume, ${\displaystyle C_{V}}$, and heat capacity at constant pressure, ${\displaystyle C_{p}}$:

${\displaystyle C_{V}=\left({\frac {\delta Q}{dT}}\right)_{V}=T\left({\frac {\partial S}{\partial T}}\right)_{V}}$
${\displaystyle C_{p}=\left({\frac {\delta Q}{dT}}\right)_{p}=T\left({\frac {\partial S}{\partial T}}\right)_{p}}$

where ${\displaystyle \delta Q}$ is the infinitesimal amount of heat added, and ${\displaystyle dT}$ is the subsequent rise in temperature.

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

${\displaystyle dU=T\,dS-p\,dV}$

So the heat capacity at constant volume is

${\displaystyle C_{V}=\left({\frac {\partial U}{\partial T}}\right)_{V}}$

The enthalpy is defined by ${\displaystyle H=U+pV}$. The increment of enthalpy is

${\displaystyle dH=T\,dS+V\,dp.}$

So the heat capacity at constant pressure is

${\displaystyle C_{p}=\left({\frac {\partial H}{\partial T}}\right)_{p}.}$

## Specific heat capacity

The specific heat capacity of a material is

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

which in the absence of phase transitions is equivalent to

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

• 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)

One has to distinguish between different boundary conditions for the processes under consideration. Typical processes for which a heat capacity may be defined include isobaric (constant pressure, d${\displaystyle p=0}$) and isochoric (constant volume, d${\displaystyle V=0}$) processes, and one conventionally writes

${\displaystyle c_{p}=\left({\frac {\partial C}{\partial m}}\right)_{p}}$
${\displaystyle c_{V}=\left({\frac {\partial C}{\partial m}}\right)_{V}}$

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

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

## Dimensionless heat capacity

The dimensionless heat capacity of a material is

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

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.

## Gas phase heat capacities

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

${\displaystyle E_{i}={\frac {k_{B}T}{2}}}$

where

${\displaystyle E_{i}}$ is the mean energy (measured in joules) associated with degree of freedom i.

T is the temperature (measured in kelvins)

${\displaystyle k_{B}}$ is Boltzman'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 6 degrees of freedom, all of a translational type. No energy is attached to the degrees of freedom attached to the position of atoms, while degrees of freedom corresponding to the momentums of atoms are quadratic degrees of freedom. N atoms thus correspond to 3N degrees of freedom, leading to the equation

${\displaystyle C_{v}={\frac {\partial E}{\partial T}}={\frac {3}{2}}N\,k_{B}={\frac {3}{2}}n\,R}$
${\displaystyle c_{v,m}={\frac {C_{v}}{n}}={\frac {3}{2}}R}$

where

${\displaystyle C_{v}}$ is the heat capacity at constant volume of the gas

${\displaystyle c_{v,m}}$ 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 Boltzman's constant ${\displaystyle k_{B}}$ 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 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. 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, there are three degrees of freedom f per atom in the molecule na

${\displaystyle f=3n_{a}\,}$

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 essentially zero. 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

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

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 ${\displaystyle R}$ in 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 constant volume heat capacity of

${\displaystyle C_{v}={\frac {3R}{2}}+R+R={\frac {7R}{2}}}$

where the terms originate from the translational, rotational, and vibrational degrees of freedom, respectively. The following is a table of some constant volume heat capacities of various diatomics

 Diatomic gas Cv,m (J K−1 mol−1), Cv,m/R H2 20.18 2.427 CO 20.2 2.43 N2 19.9 2.39 Cl2 24.1 2.90 Br2 32.0 3.84

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 ${\displaystyle Br_{2}}$. 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 interlevel energy spacings are large, the predicted constant volume heat capacity for a diatomic molecule becomes

${\displaystyle C_{v}={\frac {3R}{2}}+R={\frac {5R}{2}}}$

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 heat capacities

File: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 states that the dimensionless specific heat capacity assumes the value 3.

The Dulong-Petit law is however based on the equipartition theorem, and as such is only valid in the classical limit of a microstate continuum, which is a high temperature limit. For most of the common solids however, and at standard ambiant temperature, and contrarly to gas, quantum effects play an important role. For a more modern and precise analysis it is useful to use the idea of phonons. See Debye model.