Heat capacity

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Heat capacity (denoted as C) is a measurable physical quantity that characterizes the ability of a body to store heat. It is defined as the amount of heat needed at the given conditions and state of the body (foremost its temperature) to raise its temperature by one degree. Therefore, heat capacity is measured in units of joules per kelvin and is an extensive quantity. Dividing heat capacity by the body's mass yields a specific heat, which is an intensive quantity.

1 Definition
2 Heat capacity at absolute zero
3 Heat capacity of compressible bodies
4 Specific heat capacity
5 Dimensionless heat capacity
6 Theoretical models
- 6.1 Gas phase
- 6.2 Solid phase
7 See also

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 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. For a one-dimensional system the only thermodynamic variable is temperature and the short path is implicitly defined, but for higher dimensional systems it must be explicitly defined, since the value of heat capacity depends on which one is chosen. It should be remarked that the above definition requires a constraint on the system when the measurement is made, but the heat capacity corresponding to that constraint is a state function and does not require any constraints on the system in order to be defined for that system.

Of particular relevance are the values of heat capacity for constant volume, C_V, and constant pressure, C_P:

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

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

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 particular type of heat capacity is assumed to be known from the context of the discussion.

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 T_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 violating the third law of thermodynamics.

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 $δ Q$ is the infinitesimal amount of heat added, and $d T$ 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 $H = U + p V$ . The increment of enthalpy is

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

So the heat capacity at constant pressure is

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

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

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

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

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

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

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

A related parameter 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

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

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 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 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 Boltzman'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	C_V,m (J K⁻¹ mol⁻¹)	C_V,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 n_a

$f=3n_a \,$

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 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	C_V,m (J K⁻¹ mol⁻¹)	C_V,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 $B r 2$ . 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 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

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/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

Contents

Definition

Heat capacity at absolute zero

Heat capacity of compressible bodies

Specific heat capacity

Dimensionless heat capacity

Theoretical models

Gas phase

Solid phase

See also