Heat capacity
The heat capacity is defined as the differential of heat with respect to the temperature \(T\),
\[C := \frac{\delta Q}{\partial T} = T \frac{\partial S}{\partial T}\]
where \(Q\) is heat and \(S\) is the entropy.
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[edit] At constant volume
From the first law of thermodynamics one has \[\left.\delta Q\right. = dU + pdV\] thus at constant volume, denoted by the subscript \(V\), then \(dV=0\), \[C_V := \left.\frac{\delta Q}{\partial T} \right\vert_V = \left. \frac{\partial U}{\partial T} \right\vert_V \]
[edit] At constant pressure
At constant pressure (denoted by the subscript \(p\)), \[C_p := \left.\frac{\delta Q}{\partial T} \right\vert_p =\left.\frac{\partial H}{\partial T} \right\vert_p= \left. \frac{\partial U}{\partial T} \right\vert_p + p \left.\frac{\partial V}{\partial T} \right\vert_p\]
where \(H\) is the enthalpy. The difference between the heat capacity at constant pressure and the heat capacity at constant volume is given by \[C_p -C_V = \left( p + \left. \frac{\partial U}{\partial V} \right\vert_T \right) \left. \frac{\partial V}{\partial T} \right\vert_p\]
[edit] 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)
\[C_v^{ex} = C_v - \frac{3}{2}Nk_BT\]
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
\[U^{ex} = 2\pi N \rho \int_0^{\infty} \Phi_{12}(r) g(r) r^2 ~{\rm d}{\mathbf r}\]
where \(\Phi_{12}(r)\) is the intermolecular pair potential and \(g(r)\) is the radial distribution function, one has
\[C_v^{ex} = 2\pi N \rho \int_0^{\infty} \Phi_{12}(r) \left. \frac{\partial g(r)}{\partial T} \right\vert_V r^2 ~{\rm d}{\mathbf r} \]
For many-body distribution functions things become more complicated
[edit] Liquids
[edit] Solids
[edit] Petit and Dulong
[edit] Einstein
[edit] Debye
A low temperatures on has
\[c_v = \frac{12 \pi^4}{5} n k_B \left( \frac{T}{\Theta_D} \right)^3\]
where \(k_B\) is the Boltzmann constant, \(T\) is the temperature and \(\Theta_D\) is an empirical parameter known as the Debye temperature.
[edit] See also
[edit] 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)
