Heat capacity: Difference between revisions

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The difference between the heat capacity at constant pressure and the heat capacity at constant volume is given by
The difference between the heat capacity at constant pressure and the heat capacity at constant volume is given by
:<math>C_p -C_V = \left( p + \left. \frac{\partial U}{\partial V} \right\vert_T \right) \left. \frac{\partial V}{\partial T} \right\vert_p</math>
:<math>C_p -C_V = \left( p + \left. \frac{\partial U}{\partial V} \right\vert_T \right) \left. \frac{\partial V}{\partial T} \right\vert_p</math>
==Excess heat capacity==
In a classical system the excess heat capacity for a monatomic fluid is given by subtracting the  [[Ideal gas: Energy |ideal internal energy]]  (which is kinetic in nature)
:<math>C_v^{ex} = C_v - \frac{3}{2}Nk_BT</math>
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 <ref>J-P. Hansen and I. R. McDonald "Theory of Simple Liquids", Academic Press (2006) (Third Edition) ISBN 0-12-370535-5 </ref>):
:<math>U^{ex} = 2\pi N \rho \int_0^{\infty} \Phi_{12}(r) g(r) r^2  ~{\rm d}{\mathbf r}</math>
where <math>\Phi_{12}(r)</math> is the [[intermolecular pair potential]] and <math>g(r)</math> is the [[radial distribution function]],
one has
:<math>C_v^{ex} = 2\pi N \rho \int_0^{\infty} \Phi_{12}(r)  \left. \frac{\partial g(r)}{\partial T} \right\vert_V  r^2  ~{\rm d}{\mathbf r} </math>
For many-body distribution functions things become more complicated <ref>[http://dx.doi.org/10.1063/1.468220  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)]</ref>.
==Liquids==
==Liquids==
*[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>[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>
==Solids==
==Solids==
====Petit and Dulong====
====Petit and Dulong====

Revision as of 15:28, 11 May 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

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

[3]

Solids

Petit and Dulong

[4]

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