Heat capacity: Difference between revisions

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==Liquids==
==Liquids==
The calculation of the heat capacity in liquids is more difficult than in gasses or solids <ref>[http://dx.doi.org/10.1063/1.1667469 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)]</ref>.
The calculation of the heat capacity in liquids is more difficult than in gasses or solids <ref>[http://dx.doi.org/10.1063/1.1667469 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)]</ref>.
Recently an expression for the energy of a liquid has been developed, taking into account... (Eq. 5 of <ref>[http://dx.doi.org/10.1038/srep00421 D. Bolmatov, V. V. Brazhkin and K. Trachenko "The phonon theory of liquid thermodynamics", Scientific Reports '''2''' Article number: 421 (2012)]</ref>):
Recently an expression for the energy of a liquid has been developed (Eq. 5 of <ref>[http://dx.doi.org/10.1038/srep00421 D. Bolmatov, V. V. Brazhkin and K. Trachenko "The phonon theory of liquid thermodynamics", Scientific Reports '''2''' Article number: 421 (2012)]</ref>):




:<math>E = NT \left(  1 + \frac{\alpha T}{2}\right)  \left(  3D \left( \frac{\hbar \omega_D}{T} \right) -\left( \frac{\omega_F}{\omega_D} \right)^3 D\left(  \frac{\hbar \omega_F}{T}\right)  \right)</math>
:<math>E = NT \left(  1 + \frac{\alpha T}{2}\right)  \left(  3D \left( \frac{\hbar \omega_D}{T} \right) -\left( \frac{\omega_F}{\omega_D} \right)^3 D\left(  \frac{\hbar \omega_F}{T}\right)  \right)</math>


from which
 
where <math>\omega_F</math> is the [[Frenkel frequency]], <math>\omega_D</math> is the [[Debye frequency]], <math>D</math> is the [[Debye function]], and <math>\alpha</math>
is the [[thermal expansion coefficient]]. The differential of this energy with respect to temperature provides the heat capacity.


==Solids==
==Solids==

Revision as of 11:41, 19 June 2012

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

From the first law of thermodynamics one has

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

At constant pressure

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

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

Excess heat capacity

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].

Liquids

The calculation of the heat capacity in liquids is more difficult than in gasses or solids [3]. Recently an expression for the energy of a liquid has been developed (Eq. 5 of [4]):



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

Petit and Dulong

[5]

Einstein

Debye

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

References