formulas:maxwell_relations
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formulas:maxwell_relations [2018/04/25 15:42] jakobadmin |
formulas:maxwell_relations [2018/12/19 11:01] (current) jakobadmin ↷ Links adapted because of a move operation |
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| <tabbox Concrete> | <tabbox Concrete> | ||
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| + | **Derivation** | ||
| The Maxwell relations follow directly from the fact that [[https://en.wikipedia.org/wiki/Symmetry_of_second_derivatives|partial derivatives commute]]: $\partial_x \partial_y = \partial_y \partial_x$. | The Maxwell relations follow directly from the fact that [[https://en.wikipedia.org/wiki/Symmetry_of_second_derivatives|partial derivatives commute]]: $\partial_x \partial_y = \partial_y \partial_x$. | ||
| - | If we have some function $U$ that depends on the entropy $S$ and the volume $V$, the total change of it is given by | + | If we have some function $U(S,V)$ (called the internal energy) that depends on the entropy $S$ and the volume $V$, the total change of it is given by |
| $$ dU = \frac{\partial U}{\partial S} \big |_V dS + \frac{\partial U}{\partial V} \big |_S dV, $$ | $$ dU = \frac{\partial U}{\partial S} \big |_V dS + \frac{\partial U}{\partial V} \big |_S dV, $$ | ||
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| The other Maxwell relations follow completely analogous, but with different functions instead of the internal energy $U$. | The other Maxwell relations follow completely analogous, but with different functions instead of the internal energy $U$. | ||
| - | For example, if we start with the Helmholtz free energy: | + | For example, if we start with the Helmholtz free energy $A(T,V)$: |
| $$ A = U -TS $$ | $$ A = U -TS $$ | ||
| - | and follow exactly the same steps, we can derive | + | and follow exactly the same steps ($dA= dU-d(TS) = (Tds-PdV)-(SdT+TdS)=-SdT-PdV$), we can derive |
| $$ \frac{\partial S}{\partial V} \big |_T = \frac{\partial P}{\partial T} \big |_V $$ | $$ \frac{\partial S}{\partial V} \big |_T = \frac{\partial P}{\partial T} \big |_V $$ | ||
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| + | The other Maxwell relations follow by starting with the enthalpy $H(S,P)$ or the Gibbs free energy $G(T,P)$. | ||
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| + | (These notions appear, since: | ||
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| + | * a system with fixed entropy and volume will choose the state with minimum internal energy $U$, | ||
| + | * a system with fixed temperature and volume will choose the state with minimum enthalpy $H$, | ||
| + | * a system with fixed entropy and pressure will choose the state with minimum Helmholtz free energy $A$, | ||
| + | * a system with fixed temperature and pressure will choose the state with minimum Gibbs free energy $G$.) | ||
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| ---- | ---- | ||
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| <tabbox Why is it interesting?> | <tabbox Why is it interesting?> | ||
| - | The Maxwell relations encode useful relationships between notions of [[theories:thermodynamics|thermodynamics]]. | + | The Maxwell relations encode useful relationships between notions of [[models:thermodynamics|thermodynamics]]. |
| </tabbox> | </tabbox> | ||
formulas/maxwell_relations.1524663751.txt.gz · Last modified: 2018/04/25 13:42 (external edit)
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