Heat capacity: Difference between revisions

Revision as of 11:10, 21 June 2007

δ Q = d U + p d V

the heat capacity is given by

C = \frac{δ Q}{\partial T}

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

where U is the internal energy, T is the temperature, and V is the volume.

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

where p is the pressure.

We have

C_{p} - C_{V} = (p + {\frac{\partial U}{\partial V} |}_{T}) {\frac{\partial V}{\partial T} |}_{p}

@@ Line 7: / Line 7: @@
 :<math>C = \frac{\delta Q}{\partial T}</math>
 ==At constant volume==
-:<math>C_v = \left.\frac{\delta Q}{\partial T} \right\vert_V = \left. \frac{\partial U}{\partial T} \right\vert_V </math>
+:<math>C_V = \left.\frac{\delta Q}{\partial T} \right\vert_V = \left. \frac{\partial U}{\partial T} \right\vert_V </math>
 where ''U'' is the [[internal energy]], ''T'' is the temperature, and  ''V'' is the volume.
@@ Line 14: / Line 14: @@
 where ''p'' is the [[pressure]].
+We have
+:<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>
 [[category: classical thermodynamics]]