Editing Heat capacity

The '''heat capacity''' is defined as the differential of [[heat]] with respect to the [[temperature]] <math>T</math>,

:<math>C := \frac{\delta Q}{\partial T} = T \frac{\partial S}{\partial T}</math>

where <math>Q</math> is [[heat]] and  <math>S</math> is the [[entropy]].
==At constant volume==
From the [[first law of thermodynamics]] one has
:<math>\left.\delta Q\right. = dU + pdV</math>
thus at constant volume, denoted by the subscript <math>V</math>, then <math>dV=0</math>,
:<math>C_V := \left.\frac{\delta Q}{\partial T} \right\vert_V = \left. \frac{\partial U}{\partial T} \right\vert_V </math>
==At constant pressure==
At constant [[pressure]] (denoted by the subscript <math>p</math>),
:<math>C_p := \left.\frac{\delta Q}{\partial T} \right\vert_p =\left.\frac{\partial H}{\partial T} \right\vert_p= \left. \frac{\partial U}{\partial T} \right\vert_p + p \left.\frac{\partial V}{\partial T} \right\vert_p</math>

where <math>H</math> is the [[enthalpy]].
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>
==Adiabatic index==
Sometimes the ratio of heat capacities is known as the ''adiabatic index'':

:<math>\gamma = \frac{C_p}{C_V}</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>.
===Rosenfeld-Tarazona expression===
Rosenfeld and Tarazona 
<ref>[http://dx.doi.org/10.1080/00268979809483145 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)]</ref>
<ref>[http://dx.doi.org/10.1063/1.4827865  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)]</ref>
used [[fundamental-measure theory]] to obtain a ''unified analytical description'' of classical bulk solids and fluids, one result being:

:<math>C_v^{ex} \propto T^{-2/5}</math>

==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>.
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>


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==
====Petit and Dulong====
<ref>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)</ref>
====Einstein====
====Debye====
A low temperatures on has 

:<math>c_v = \frac{12 \pi^4}{5} n k_B \left( \frac{T}{\Theta_D} \right)^3</math>

where <math>k_B</math> is the [[Boltzmann constant]], <math>T</math> is the [[temperature]] and <math>\Theta_D</math> is an empirical parameter known as the Debye temperature.

==See also==
*[[Ideal gas: Heat capacity | Heat capacity of an ideal gas]]
*[[Yang-Yang anomaly]]

==References==
<references/>
;Related reading
*[http://dx.doi.org/10.1063/1.4993572 William R. Smith, Jan Jirsák, Ivo Nezbeda, and Weikai Qi "Molecular simulation of caloric properties of fluids modelled by force fields with intramolecular contributions: Application to heat capacities", Journal of Chemical Physics '''147''' 034508 (2017)]

[[category: classical thermodynamics]]