# Heat capacity

Heat capacity (abbreviated Cth or just C, also called thermal capacity) is the ability of matter to store heat. The heat capacity of a certain amount of matter is the quantity of heat (measured in joules) required to raise its temperature by one kelvin. The SI unit for heat capacity is J/K (joule per kelvin).

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## Specific heat capacity

Main article: Specific heat capacity.

The heat capacity is the specific heat capacity multiplied by the mass.

C = m · c

where

C is the heat capacity (in SI units, measured in J/K)

m is mass (measured in kilograms)

c is specific heat capacity (measured in J/(kg·K))

## Thermal capacitance

Heat capacity is related to thermal capacitance by the formula

Cth = V · ρ · cp

or, more simply,

Cth = m · cp

where

Cth is the thermal capacitance or heat capacity

V is the volume (measured in m3)

ρ is the density (measured in kg/m3)

cp is the specific heat capacity (measured in J/(kg·K) at constant pressure)

m is the mass (measured in kg)

The product ρ · cp is known as volumetric heat capacity, thermal capacitance or (confusingly) thermal capacity, and has units of J/(m3·K). Dulong and Petit predicted in 1818 that ρcp would be constant for all solids (the Dulong-Petit law). In fact, the quantity varies from about 1.2 to 4.5 J/(m3K). For fluids it is in the range 1.3 to 1.9, and for ideal monatomic gases it is a constant 0.001 J/(m3K).

## Gas phase heat capacities

According to the Equipartition Theorem from classical statistical mechanics, 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, for each degree of freedom, that

[itex]\partial E=\frac{R}{2}\partial T[itex]

where

E is the input energy (measured in joules)

T is the temperature (measured in kelvins)

R is the ideal gas constant, (8.314570[70] J K-1mol-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 molecule has only 3 degrees of translational freedom. This is because there is one for each of the vector components of momentum in the x, y, and z directions. This leads to the equation

[itex]C_v=\frac{\partial E}{\partial T}=\frac{3}{2}nR[itex]

[itex]C_{v,m}=\frac{C_v}{n}=\frac{3}{2}R[itex]

where

n is the number of moles of molecules present in the container

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

[itex]f=3n_a \,[itex]

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

[itex]f_{vib}=f-f_{trans}-f_{rot}=6-3-2=1 \,[itex]

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 [itex]R[itex] 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

[itex]C_v=\frac{3R}{2}+R+R=\frac{7R}{2}[itex]

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 [itex]Br_2[itex]. 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

[itex]C_v=\frac{3R}{2}+R=\frac{5R}{2}[itex]

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

For matter in a crystalline solid phase, it is useful to use the idea of phonons. See Debye Approximation.

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