Heat capacity - Revision history

Carl McBride: /* References */ Added a recent publication

2017-07-28T10:31:56Z

References: Added a recent publication

← Older revision		Revision as of 12:31, 28 July 2017
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	==References==		==References==
	<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]]		[[category: classical thermodynamics]]

129.69.120.39: /* Excess heat capacity */ internal energy has other units than heat capacity. d_v=partial U/partialT

2015-11-25T12:00:11Z

Excess heat capacity: internal energy has other units than heat capacity. d_v=partial U/partialT

← Older revision		Revision as of 14:00, 25 November 2015
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	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)		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>		:<math>C_v^{ex} = C_v - \frac{3}{2}Nk_B</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>):		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>):

Carl McBride: /* Excess heat capacity */ Added the Rosenfeld-Tarazona expression

2013-11-11T11:43:48Z

Excess heat capacity: Added the Rosenfeld-Tarazona expression

← Older revision		Revision as of 13:43, 11 November 2013
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	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>.		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==		==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>.

Carl McBride: /* Liquids */ Added a little more

2012-06-19T10:41:45Z

Liquids: Added a little more

← Older revision		Revision as of 12:41, 19 June 2012
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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==

Carl McBride: /* Liquids */ tmp save

2012-06-15T14:54:48Z

Liquids: tmp save

← Older revision		Revision as of 16:54, 15 June 2012
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	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>.		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==
	<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>):


			:<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

	==Solids==		==Solids==
	====Petit and Dulong====		====Petit and Dulong====

Carl McBride: Added mention of the Adiabatic index

2012-05-23T13:49:52Z

Added mention of the Adiabatic index

← Older revision		Revision as of 15:49, 23 May 2012
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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>
			==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==		==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)		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)

Carl McBride: Added section on the excess heat capacity

2012-05-11T14:28:12Z

Added section on the excess heat capacity

@@ Line 16: / Line 16: @@
 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>
 ==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==
 ====Petit and Dulong====

Carl McBride: /* See also */ Added an internal link

2010-04-19T14:23:52Z

Carl McBride: Added a reference.

2009-11-05T13:25:51Z

Added a reference.

← Older revision		Revision as of 15:25, 5 November 2009
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	*[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)]		*[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)]
	==Solids==		==Solids==
	====Dulong ~~and Petit~~====		====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====		====Einstein====
	====Debye====		====Debye====
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	*[[Ideal gas: Heat capacity \| Heat capacity of an ideal gas]]		*[[Ideal gas: Heat capacity \| Heat capacity of an ideal gas]]
	==References==		==References==
			<references/>
	[[category: classical thermodynamics]]		[[category: classical thermodynamics]]

Nice and Tidy: /* Debye */ Added a formula.

2009-01-21T14:31:02Z

Debye: Added a formula.

← Older revision		Revision as of 16:31, 21 January 2009
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	====Einstein====		====Einstein====
	====Debye====		====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==		==See also==
	*[[Ideal gas: Heat capacity \| Heat capacity of an ideal gas]]		*[[Ideal gas: Heat capacity \| Heat capacity of an ideal gas]]
	==References==		==References==
	[[category: classical thermodynamics]]		[[category: classical thermodynamics]]

← Older revision		Revision as of 16:23, 19 April 2010
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	==See also==		==See also==
	*[[Ideal gas: Heat capacity \| Heat capacity of an ideal gas]]		*[[Ideal gas: Heat capacity \| Heat capacity of an ideal gas]]
			*[[Yang-Yang anomaly]]

	==References==		==References==
	<references/>		<references/>
	[[category: classical thermodynamics]]		[[category: classical thermodynamics]]