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| :<math> \left. \partial Q \right\vert_p = - \left. \partial p \right\vert_Q = C_p </math> | | :<math> \left. \partial Q \right\vert_p = - \left. \partial p \right\vert_Q = C_p </math> |
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| :<math> \left. \partial W \right\vert_p = - \left. \partial p \right\vert_W = p\left. \frac{\partial V}{\partial T} \right\vert_p</math>
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| :<math> \left. \partial U \right\vert_p = - \left. \partial p \right\vert_U = C_p - p\left. \frac{\partial V}{\partial T} \right\vert_p</math>
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| :<math> \left. \partial H \right\vert_p = - \left. \partial p \right\vert_H = C_p </math>
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| :<math> \left. \partial G \right\vert_p = - \left. \partial p \right\vert_G = -S </math>
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| :<math> \left. \partial A \right\vert_p = - \left. \partial p \right\vert_A = -\left( S + p\left. \frac{\partial V}{\partial T} \right\vert_p \right)</math>
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| ====temperature====
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| :<math> \left. \partial V \right\vert_T = - \left. \partial T \right\vert_V = - \left. \frac{\partial V}{\partial p} \right\vert_T</math>
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| :<math> \left. \partial S \right\vert_T = - \left. \partial T \right\vert_S = \left. \frac{\partial V}{\partial T} \right\vert_p</math>
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| :<math> \left. \partial Q \right\vert_T = - \left. \partial T \right\vert_Q = T\left. \frac{\partial V}{\partial T} \right\vert_p</math>
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| :<math> \left. \partial W \right\vert_T = - \left. \partial T \right\vert_W = - p\left. \frac{\partial V}{\partial p} \right\vert_T</math>
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| :<math> \left. \partial U \right\vert_T = - \left. \partial T \right\vert_U = T\left. \frac{\partial V}{\partial T} \right\vert_p + p\left. \frac{\partial V}{\partial p} \right\vert_T</math>
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| :<math> \left. \partial H \right\vert_T = - \left. \partial T \right\vert_H = -V + T\left. \frac{\partial V}{\partial T} \right\vert_p </math>
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| :<math> \left. \partial G \right\vert_T = - \left. \partial T \right\vert_G = -V </math>
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| :<math> \left. \partial A \right\vert_T = - \left. \partial T \right\vert_A = p\left. \frac{\partial V}{\partial p} \right\vert_T</math>
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| ====volume====
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| :<math> \left. \partial S \right\vert_V = - \left. \partial V \right\vert_S = 1/T \left( C_p\left. \frac{\partial V}{\partial p} \right\vert_T + T\left. \left( \frac{\partial V}{\partial T} \right)^2 \right\vert_p \right)</math>
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| :<math> \left. \partial Q \right\vert_V = - \left. \partial V \right\vert_Q = C_p\left. \frac{\partial V}{\partial p} \right\vert_T + T\left. \left( \frac{\partial V}{\partial T} \right)^2 \right\vert_p </math>
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| :<math> \left. \partial W \right\vert_V = - \left. \partial V \right\vert_W = 0 </math>
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| :<math> \left. \partial U \right\vert_V = - \left. \partial V \right\vert_U = C_p\left. \frac{\partial V}{\partial p} \right\vert_T + T\left. \left( \frac{\partial V}{\partial T} \right)^2 \right\vert_p </math>
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| :<math> \left. \partial H \right\vert_V = - \left. \partial V \right\vert_H = C_p\left. \frac{\partial V}{\partial p} \right\vert_T + T\left. \left( \frac{\partial V}{\partial T} \right)^2 \right\vert_p - V\left. \frac{\partial V}{\partial T} \right\vert_p </math>
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| :<math> \left. \partial G \right\vert_V = - \left. \partial V \right\vert_G = - \left( V \left. \frac{\partial V}{\partial T} \right\vert_p + S\left. \frac{\partial V}{\partial p} \right\vert_T \right) </math>
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| :<math> \left. \partial A \right\vert_V = - \left. \partial V \right\vert_A = -S\left. \frac{\partial V}{\partial p} \right\vert_T </math>
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| ====entropy====
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| :<math> \left. \partial Q \right\vert_S = - \left. \partial S \right\vert_Q = 0 </math>
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| :<math> \left. \partial W \right\vert_S = - \left. \partial S \right\vert_W = -(p/T) \left( C_p\left. \frac{\partial V}{\partial p} \right\vert_T + T\left. \left( \frac{\partial V}{\partial T} \right)^2 \right\vert_p \right) </math>
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| :<math> \left. \partial U \right\vert_S = - \left. \partial S \right\vert_U = (p/T) \left( C_p\left. \frac{\partial V}{\partial p} \right\vert_T + T\left. \left( \frac{\partial V}{\partial T} \right)^2 \right\vert_p \right) </math>
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| :<math> \left. \partial H \right\vert_S = - \left. \partial S \right\vert_H = -VC_p/T </math>
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| :<math> \left. \partial G \right\vert_S = - \left. \partial S \right\vert_G = -(1/T) \left( VC_p -ST\left. \frac{\partial V}{\partial T} \right\vert_p \right) </math>
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| :<math> \left. \partial A \right\vert_S = - \left. \partial S \right\vert_A = (1/T) \left( p\left( C_p \left. \frac{\partial V}{\partial p} \right\vert_T + T\left. \left( \frac{\partial V}{\partial T} \right)^2 \right\vert_p \right) + ST\left. \frac{\partial V}{\partial T} \right\vert_p \right) </math>
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| ====heat====
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| :<math> \left. \partial W \right\vert_Q = - \left. \partial Q \right\vert_W = -p \left( C_p\left. \frac{\partial V}{\partial p} \right\vert_T + T\left. \left( \frac{\partial V}{\partial T} \right)^2 \right\vert_p \right) </math>
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| :<math> \left. \partial U \right\vert_Q = - \left. \partial Q \right\vert_U = p \left( C_p\left. \frac{\partial V}{\partial p} \right\vert_T + T\left. \left( \frac{\partial V}{\partial T} \right)^2 \right\vert_p \right) </math>
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| :<math> \left. \partial H \right\vert_Q = - \left. \partial Q \right\vert_H = -VC_p </math>
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| :<math> \left. \partial G \right\vert_Q = - \left. \partial Q \right\vert_G = - \left( ST \left. \frac{\partial V}{\partial T} \right\vert_p -VC_p \right) </math>
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| :<math> \left. \partial A \right\vert_Q = - \left. \partial Q \right\vert_A = p \left( C_p\left. \frac{\partial V}{\partial p} \right\vert_T + T\left. \left( \frac{\partial V}{\partial T} \right)^2 \right\vert_p \right) + ST \left. \frac{\partial V}{\partial T} \right\vert_p</math>
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| ====work====
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| :<math> \left. \partial U \right\vert_W = - \left. \partial W \right\vert_U = p \left( C_p\left. \frac{\partial V}{\partial p} \right\vert_T + T\left. \left( \frac{\partial V}{\partial T} \right)^2 \right\vert_p \right) </math>
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| :<math> \left. \partial H \right\vert_W = - \left. \partial W \right\vert_H = p \left( C_p\left. \frac{\partial V}{\partial p} \right\vert_T + T\left. \left( \frac{\partial V}{\partial T} \right)^2 \right\vert_p - V \left. \frac{\partial V}{\partial T} \right\vert_p \right) </math>
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| :<math> \left. \partial G \right\vert_W = - \left. \partial W \right\vert_G = -p \left( V\left. \frac{\partial V}{\partial p} \right\vert_T + S \left. \frac{\partial V}{\partial p} \right\vert_T \right) </math>
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| :<math> \left. \partial A \right\vert_W = - \left. \partial W \right\vert_A = -pS \left. \frac{\partial V}{\partial p} \right\vert_T </math>
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| ====internal energy====
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| :<math> \left. \partial H \right\vert_U = - \left. \partial U \right\vert_H = -V \left( C_p - p\left. \frac{\partial V}{\partial T} \right\vert_p \right) - p \left( C_p\left. \frac{\partial V}{\partial p} \right\vert_T + T\left. \left( \frac{\partial V}{\partial T} \right)^2 \right\vert_p \right) </math>
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| :<math> \left. \partial G \right\vert_U = - \left. \partial U \right\vert_G = -V \left( C_p - p\left. \frac{\partial V}{\partial T} \right\vert_p \right) +S \left( T\left. \frac{\partial V}{\partial T} \right\vert_p + p\left. \frac{\partial V}{\partial p} \right\vert_T \right) </math>
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| :<math> \left. \partial A \right\vert_U = - \left. \partial U \right\vert_A = p \left( C_p\left. \frac{\partial V}{\partial p} \right\vert_T + T\left. \left( \frac{\partial V}{\partial T} \right)^2 \right\vert_p \right) </math>
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| ====enthalpy====
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| :<math> \left. \partial G \right\vert_H = - \left. \partial H \right\vert_G = -V(C_p+S) + TS \left. \frac{\partial V}{\partial T} \right\vert_p </math>
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| :<math> \left. \partial A \right\vert_H = - \left. \partial H \right\vert_A = -\left(S+p \left. \frac{\partial V}{\partial T} \right\vert_p \right) \left(V-T \left. \frac{\partial V}{\partial T} \right\vert_p \right) + p \left. \frac{\partial V}{\partial p} \right\vert_T </math>
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| ====Gibbs energy function====
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| :<math> \left. \partial A \right\vert_G = - \left. \partial G \right\vert_A = -S\left(V+p \left. \frac{\partial V}{\partial p} \right\vert_T \right) - pV \left. \frac{\partial V}{\partial T} \right\vert_p </math>
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| ==See also== | | ==See also== |
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| ==References== | | ==References== |
| <references/> | | <references/> |
| ;Related reading
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| *[http://arxiv.org/abs/1102.1540 James B. Cooper and T. Russell "On the Mathematics of Thermodynamics", arXiv:1102.1540v1 Tue, 8 Feb (2011)]
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| *[http://arxiv.org/abs/1108.4760 James B. Cooper "Thermodynamical identities - a systematic approach", arXiv:1108.4760v1 Wed, 24 Aug (2011)]
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| [[Category: Classical thermodynamics]] | | [[Category: Classical thermodynamics]] |