# Heat capacity

The **heat capacity** is defined as the differential of heat with respect to the temperature ,

where is heat and is the entropy.

## At constant volume[edit]

From the first law of thermodynamics one has

thus at constant volume, denoted by the subscript , then ,

## At constant pressure[edit]

At constant pressure (denoted by the subscript ),

where is the enthalpy. The difference between the heat capacity at constant pressure and the heat capacity at constant volume is given by

## Adiabatic index[edit]

Sometimes the ratio of heat capacities is known as the *adiabatic index*:

## Excess heat capacity[edit]

In a classical system the excess heat capacity for a monatomic fluid is given by subtracting the ideal internal energy (which is kinetic in nature)

in other words the excess heat capacity is associated with the component of the internal energy due to the intermolecular potential, and for that reason it is also known as the *configurational* heat capacity. Given that the excess internal energy for a pair potential is given by (Eq. 2.5.20 in ^{[1]}):

where is the intermolecular pair potential and is the radial distribution function, one has

For many-body distribution functions things become more complicated ^{[2]}.

### Rosenfeld-Tarazona expression[edit]

Rosenfeld and Tarazona
^{[3]}
^{[4]}
used fundamental-measure theory to obtain a *unified analytical description* of classical bulk solids and fluids, one result being:

## Liquids[edit]

The calculation of the heat capacity in liquids is more difficult than in gasses or solids ^{[5]}.
Recently an expression for the energy of a liquid has been developed (Eq. 5 of ^{[6]}):

where is the Frenkel frequency, is the Debye frequency, is the Debye function, and
is the thermal expansion coefficient. The differential of this energy with respect to temperature provides the heat capacity.

## Solids[edit]

#### Petit and Dulong[edit]

^{[7]}

#### Einstein[edit]

#### Debye[edit]

A low temperatures on has

where is the Boltzmann constant, is the temperature and is an empirical parameter known as the Debye temperature.

## See also[edit]

## References[edit]

- ↑ J-P. Hansen and I. R. McDonald "Theory of Simple Liquids", Academic Press (2006) (Third Edition) ISBN 0-12-370535-5
- ↑ Ben C. Freasier, Adam Czezowski, and Richard J. Bearman "Multibody distribution function contributions to the heat capacity for the truncated Lennard‐Jones fluid", Journal of Chemical Physics
**101**pp. 7934-7938 (1994) - ↑ Yaakov Rosenfeld and Pedro Tarazona "Density functional theory and the asymptotic high density expansion of the free energy of classical solids and fluids", Molecular Physics
**95**pp. 141-150 (1998) - ↑ Trond S. Ingebrigtsen , Arno A. Veldhorst , Thomas B. Schrøder and Jeppe C. Dyre "Communication: The Rosenfeld-Tarazona expression for liquids’ specific heat: A numerical investigation of eighteen systems", Journal of Chemical Physics
**139**171101 (2013) - ↑ Claudio A. Cerdeiriña, Diego González-Salgado, Luis Romani, María del Carmen Delgado, Luis A. Torres and Miguel Costas "Towards an understanding of the heat capacity of liquids. A simple two-state model for molecular association", Journal of Chemical Physics
**120**pp. 6648-6659 (2004) - ↑ D. Bolmatov, V. V. Brazhkin and K. Trachenko "The phonon theory of liquid thermodynamics", Scientific Reports
**2**Article number: 421 (2012) - ↑ Alexis-Thérèse Petit and Pierre-Louis Dulong "Recherches sur quelques points importants de la Théorie de la Chaleur", Annales de Chimie et de Physique
**10**pp. 395-413 (1819)

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