Bridgman thermodynamic formulas: Difference between revisions

Revision as of 12:12, 6 October 2011

Notation used (from Table I):

$p$ is the pressure.
$T$ is the temperature (in Kelvin).
$V$ is the volume.
$S$ is the entropy.
$Q$ is the heat.
$W$ is the work.
$U$ is the internal energy.
$H$ is the enthalpy
$G$ is the Gibbs energy function
$A$ is the Helmholtz energy function.

Bridgman thermodynamic formulas ^[1]

Table II

pressure

{\partial T |}_{p} = - {\partial p |}_{T} = 1

{\partial V |}_{p} = - {\partial p |}_{V} = {\frac{\partial V}{\partial T} |}_{p}

{\partial S |}_{p} = - {\partial p |}_{S} = C_{p} / T

{\partial Q |}_{p} = - {\partial p |}_{Q} = C_{p}

{\partial W |}_{p} = - {\partial p |}_{W} = p {\frac{\partial V}{\partial T} |}_{p}

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

{\partial H |}_{p} = - {\partial p |}_{H} = C_{p}

{\partial G |}_{p} = - {\partial p |}_{G} = - S

{\partial A |}_{p} = - {\partial p |}_{A} = - (S + p {\frac{\partial V}{\partial T} |}_{p})

temperature

{\partial V |}_{T} = - {\partial T |}_{V} = - {\frac{\partial V}{\partial p} |}_{T}

{\partial S |}_{T} = - {\partial T |}_{S} = {\frac{\partial V}{\partial T} |}_{p}

{\partial Q |}_{T} = - {\partial T |}_{Q} = T {\frac{\partial V}{\partial T} |}_{p}

{\partial W |}_{T} = - {\partial T |}_{W} = - p {\frac{\partial V}{\partial p} |}_{T}

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

{\partial H |}_{T} = - {\partial T |}_{H} = - V + T {\frac{\partial V}{\partial T} |}_{p}

{\partial G |}_{T} = - {\partial T |}_{G} = - V

{\partial A |}_{T} = - {\partial T |}_{A} = p {\frac{\partial V}{\partial p} |}_{T}

volume

{\partial S |}_{V} = - {\partial V |}_{S} = 1 / T (C_{p} {\frac{\partial V}{\partial p} |}_{T} + T {{(\frac{\partial V}{\partial T})}^{2} |}_{p})

{\partial Q |}_{V} = - {\partial V |}_{Q} = C_{p} {\frac{\partial V}{\partial p} |}_{T} + T {{(\frac{\partial V}{\partial T})}^{2} |}_{p}

{\partial W |}_{V} = - {\partial V |}_{W} = 0

{\partial U |}_{V} = - {\partial V |}_{U} = C_{p} {\frac{\partial V}{\partial p} |}_{T} + T {{(\frac{\partial V}{\partial T})}^{2} |}_{p}

{\partial H |}_{V} = - {\partial V |}_{H} = C_{p} {\frac{\partial V}{\partial p} |}_{T} + T {{(\frac{\partial V}{\partial T})}^{2} |}_{p} - V {\frac{\partial V}{\partial T} |}_{p}

{\partial G |}_{V} = - {\partial V |}_{G} = - (V {\frac{\partial V}{\partial T} |}_{p} + S {\frac{\partial V}{\partial p} |}_{T})

{\partial A |}_{V} = - {\partial V |}_{A} = - S {\frac{\partial V}{\partial p} |}_{T}

entropy

{\partial Q |}_{S} = - {\partial S |}_{Q} = 0

{\partial W |}_{S} = - {\partial S |}_{W} = - (p / T) (C_{p} {\frac{\partial V}{\partial p} |}_{T} + T {{(\frac{\partial V}{\partial T})}^{2} |}_{p})

{\partial U |}_{S} = - {\partial S |}_{U} = (p / T) (C_{p} {\frac{\partial V}{\partial p} |}_{T} + T {{(\frac{\partial V}{\partial T})}^{2} |}_{p})

{\partial H |}_{S} = - {\partial S |}_{H} = - V C_{p} / T

{\partial G |}_{S} = - {\partial S |}_{G} = - (1 / T) (V C_{p} - S T {\frac{\partial V}{\partial T} |}_{p})

{\partial A |}_{S} = - {\partial S |}_{A} = (1 / T) (p (C_{p} {\frac{\partial V}{\partial p} |}_{T} + T {{(\frac{\partial V}{\partial T})}^{2} |}_{p}) + S T {\frac{\partial V}{\partial T} |}_{p})

heat

{\partial W |}_{Q} = - {\partial Q |}_{W} = - p (C_{p} {\frac{\partial V}{\partial p} |}_{T} + T {{(\frac{\partial V}{\partial T})}^{2} |}_{p})

{\partial U |}_{Q} = - {\partial Q |}_{U} = p (C_{p} {\frac{\partial V}{\partial p} |}_{T} + T {{(\frac{\partial V}{\partial T})}^{2} |}_{p})

{\partial H |}_{Q} = - {\partial Q |}_{H} = - V C_{p}

{\partial G |}_{Q} = - {\partial Q |}_{G} = - (S T {\frac{\partial V}{\partial T} |}_{p} - V C_{p})

{\partial A |}_{Q} = - {\partial Q |}_{A} = p (C_{p} {\frac{\partial V}{\partial p} |}_{T} + T {{(\frac{\partial V}{\partial T})}^{2} |}_{p}) + S T {\frac{\partial V}{\partial T} |}_{p}

work

{\partial U |}_{W} = - {\partial W |}_{U} = p (C_{p} {\frac{\partial V}{\partial p} |}_{T} + T {{(\frac{\partial V}{\partial T})}^{2} |}_{p})

{\partial H |}_{W} = - {\partial W |}_{H} = p (C_{p} {\frac{\partial V}{\partial p} |}_{T} + T {{(\frac{\partial V}{\partial T})}^{2} |}_{p} - V {\frac{\partial V}{\partial T} |}_{p})

{\partial G |}_{W} = - {\partial W |}_{G} = - p (V {\frac{\partial V}{\partial p} |}_{T} + S {\frac{\partial V}{\partial p} |}_{T})

{\partial A |}_{W} = - {\partial W |}_{A} = - p S {\frac{\partial V}{\partial p} |}_{T}

internal energy

{\partial H |}_{U} = - {\partial U |}_{H} = - V (C_{p} - p {\frac{\partial V}{\partial T} |}_{p}) - p (C_{p} {\frac{\partial V}{\partial p} |}_{T} + T {{(\frac{\partial V}{\partial T})}^{2} |}_{p})

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

{\partial A |}_{U} = - {\partial U |}_{A} = p (C_{p} {\frac{\partial V}{\partial p} |}_{T} + T {{(\frac{\partial V}{\partial T})}^{2} |}_{p})

enthalpy

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

{\partial A |}_{H} = - {\partial H |}_{A} = - (S + p {\frac{\partial V}{\partial T} |}_{p}) (V - T {\frac{\partial V}{\partial T} |}_{p}) + p {\frac{\partial V}{\partial p} |}_{T}

Gibbs energy function

{\partial A |}_{G} = - {\partial G |}_{A} = - S (V + p {\frac{\partial V}{\partial p} |}_{T}) - p V {\frac{\partial V}{\partial T} |}_{p}

References

↑ P. W. Bridgman "A Complete Collection of Thermodynamic Formulas", Physical Review 3 pp. 273-281 (1914)

[1] P. W. Bridgman "A Complete Collection of Thermodynamic Formulas", Physical Review 3 pp. 273-281 (1914)

[1]

@@ Line 105: / Line 105: @@
 ====internal energy====
-:<math> \left. \partial H \right\vert_Q  =  - \left. \partial Q \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>
+:<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>
-:<math> \left. \partial G \right\vert_Q  =  - \left. \partial Q \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>
+:<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>
-:<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) </math>
+:<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>
+====enthalpy====
+:<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>
+:<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>
 ====Gibbs energy function====
+:<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>
 ==See also==

Bridgman thermodynamic formulas: Difference between revisions

Revision as of 12:12, 6 October 2011

Contents

Table II

pressure

temperature

volume

entropy

heat

work

internal energy

enthalpy

Gibbs energy function

See also

References

Navigation menu

Bridgman thermodynamic formulas: Difference between revisions

Revision as of 12:12, 6 October 2011

Table II

pressure

temperature

volume

entropy

heat

work

internal energy

enthalpy

Gibbs energy function

See also

References

Navigation menu

Search