Heat capacity: Difference between revisions
Carl McBride (talk | contribs) (Added section on the excess heat capacity) |
Carl McBride (talk | contribs) m (Added mention of the Adiabatic index) |
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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) | ||
Revision as of 14:49, 23 May 2012
The heat capacity is defined as the differential of heat with respect to the temperature Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle T} ,
- Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle C := \frac{\delta Q}{\partial T} = T \frac{\partial S}{\partial T}}
where is heat and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle S} is the entropy.
At constant volume
From the first law of thermodynamics one has
- Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \left.\delta Q\right. = dU + pdV}
thus at constant volume, denoted by the subscript Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle V} , then Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle dV=0} ,
- Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle C_V := \left.\frac{\delta Q}{\partial T} \right\vert_V = \left. \frac{\partial U}{\partial T} \right\vert_V }
At constant pressure
At constant pressure (denoted by the subscript Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle p} ),
- Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 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}
where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle H} is the enthalpy. The difference between the heat capacity at constant pressure and the heat capacity at constant volume is given by
- Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 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}
Adiabatic index
Sometimes the ratio of heat capacities is known as the adiabatic index:
- Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \gamma = \frac{C_p}{C_V}}
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)
- Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle C_v^{ex} = C_v - \frac{3}{2}Nk_BT}
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]):
- Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle U^{ex} = 2\pi N \rho \int_0^{\infty} \Phi_{12}(r) g(r) r^2 ~{\rm d}{\mathbf r}}
where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Phi_{12}(r)} is the intermolecular pair potential and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle g(r)} is the radial distribution function, one has
- Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 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} }
For many-body distribution functions things become more complicated [2].
Liquids
Solids
Petit and Dulong
Einstein
Debye
A low temperatures on has
- Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle c_v = \frac{12 \pi^4}{5} n k_B \left( \frac{T}{\Theta_D} \right)^3}
where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle k_B} is the Boltzmann constant, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle T} is the temperature and is an empirical parameter known as the Debye temperature.
See also
References
- ↑ 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)
- ↑ 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)
- ↑ 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)