Heat capacity: Difference between revisions

Revision as of 11:56, 2 December 2008

The heat capacity is defined as the differential of heat with respect to the temperature $T$ ,

C : = \frac{δ Q}{\partial T} = T \frac{\partial S}{\partial T}

where $Q$ is heat and $S$ is the entropy.

δ Q = d U + p d V

thus at constant volume, denoted by the subscript $V$ , then $d V = 0$ ,

C_{V} : = {\frac{δ Q}{\partial T} |}_{V} = {\frac{\partial U}{\partial T} |}_{V}

At constant pressure (denoted by the subscript $p$ ),

C_{p} : = {\frac{δ Q}{\partial T} |}_{p} = {\frac{\partial H}{\partial T} |}_{p} = {\frac{\partial U}{\partial T} |}_{p} + p {\frac{\partial V}{\partial T} |}_{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} = (p + {\frac{\partial U}{\partial V} |}_{T}) {\frac{\partial V}{\partial T} |}_{p}

@@ Line 1: / Line 1: @@
-From the [[first law of thermodynamics]] one has
+The '''heat capacity''' is defined as the differential of [[heat]] with respect to the [[temperature]] <math>T</math>,
-:<math>\left.\delta Q\right. = dU + pdV</math>
-where <math>Q</math> is the [[heat]], <math>U</math> is the [[internal energy]], <math>p</math> is the [[pressure]] and <math>V</math> is the volume.
-The '''heat capacity''' is given by the differential of the heat with respect to the [[temperature]],
 :<math>C := \frac{\delta Q}{\partial T} = T \frac{\partial S}{\partial T}</math>
-where <math>S</math> is the [[entropy]].
+where <math>Q</math> is [[heat]] and  <math>S</math> is the [[entropy]].
 ==At constant volume==
-At constant volume (denoted by the subscript <math>V</math>),
+From the [[first law of thermodynamics]] one has
+:<math>\left.\delta Q\right. = dU + pdV</math>
+thus at constant volume, denoted by the subscript <math>V</math>, then <math>dV=0</math>,
 :<math>C_V := \left.\frac{\delta Q}{\partial T} \right\vert_V = \left. \frac{\partial U}{\partial T} \right\vert_V </math>
 ==At constant pressure==