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.

At constant volume[edit]

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

At constant pressure[edit]

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

Adiabatic index[edit]

Sometimes the ratio of heat capacities is known as the adiabatic index:

\gamma = \frac{C_p}{C_V}

Excess heat capacity[edit]

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_B

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]):

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 [2].

Rosenfeld-Tarazona expression[edit]

Rosenfeld and Tarazona [3] [4] used fundamental-measure theory to obtain a unified analytical description of classical bulk solids and fluids, one result being:

C_v^{ex} \propto T^{-2/5}

Liquids[edit]

The calculation of the heat capacity in liquids is more difficult than in gasses or solids [5]. Recently an expression for the energy of a liquid has been developed (Eq. 5 of [6]):


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)


where \omega_F is the Frenkel frequency, \omega_D is the Debye frequency, D is the Debye function, and \alpha is the thermal expansion coefficient. The differential of this energy with respect to temperature provides the heat capacity.

Solids[edit]

Petit and Dulong[edit]

[7]

Einstein[edit]

Debye[edit]

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.

See also[edit]

References[edit]

Related reading