Heat capacity

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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.

Contents

[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] Liquids

[edit] Solids

[edit] Petit and Dulong

[1]

[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 kB is the Boltzmann constant, T is the temperature and ΘD is an empirical parameter known as the Debye temperature.

[edit] See also

[edit] References

  1. ↑ 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)
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