Maxwell Relations [The Physics Travel Guide]

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--- formulas:maxwell_relations [2017/10/27 11:00]
jakobadmin ↷ Page moved from statistical_mechanics:maxwell_relations to theories:statistical_mechanics:maxwell_relations
+++ formulas:maxwell_relations [2018/12/19 11:01] (current)
jakobadmin ↷ Links adapted because of a move operation
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 ====== Maxwell Relations ======
-<tabbox Why is it interesting?>
-<tabbox Layman>
+<tabbox Intuitive>
 <note tip>
@@ Line 9: / Line 9: @@
 </note>
-<tabbox Student>
+<tabbox Concrete>
-For a great explanation, why the Maxwell relations are "just a sneaky way of saying that the mixed partial derivatives of the function U commute". see https://johncarlosbaez.wordpress.com/2012/01/19/classical-mechanics-versus-thermodynamics-part-1/
+**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 fact that partial derivatives commute is known as Schwarz' theorem (see  https://en.wikipedia.org/wiki/Maxwell_relations . Schwarz' theorem is simply a way of stating "the fact that a function S doesn’t change when we go around a parallelogram" https://johncarlosbaez.wordpress.com/2012/01/23/classical-mechanics-versus-thermodynamics-part-2/)
+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
-<tabbox Researcher>
-<note tip>
+$$ dU = \frac{\partial U}{\partial S} \big |_V dS + \frac{\partial U}{\partial V} \big |_S dV,  $$
-The motto in this section is: //the higher the level of abstraction, the better//.
-</note>
---> Common Question 1#
+where $\big |_V$ means that we keep $V$ fixed.
+Then, we introduce two definitions:
-<--
---> Common Question 2#
+$$ \frac{\partial U}{\partial S} \big |_V \equiv  T , \quad -\frac{\partial U}{\partial V} \big |_S \equiv  P.$$
+The minus sign here is just a convention and can be understood as follows: The internal energy usually gets smaller when we increase the volume. Thus, if we want to work with positive pressure $P$ in most situations, we need to include the minus sign here.
-<--
+Using these definitions equation reads
-<tabbox Examples>
---> Example1#
+$$ dU = T dS - P dV,  $$
+which is the [[https://en.wikipedia.org/wiki/Fundamental_thermodynamic_relation|fundamental thermodynamic relation]].
-<--
---> Example2:#
+Next, we use that partial derivative commute:
+$$ \frac{\partial^2 U}{\partial V\partial S} = \frac{\partial^2 U}{\partial S\partial V} $$
-<--
+and put in our two definitions from above:
-<tabbox History>
+$$ \frac{\partial T}{\partial S} \big |_V = -\frac{\partial P}{\partial V}\big |_S $$
+This is one of the Maxwell relations.
+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 $A(T,V)$:
+$$ A = U -TS $$
+and follow exactly the same steps ($dA= dU-d(TS) = (Tds-PdV)-(SdT+TdS)=-SdT-PdV$), we can derive
+$$ \frac{\partial S}{\partial V} \big |_T = \frac{\partial P}{\partial T} \big |_V $$
+The other Maxwell relations follow by starting with the enthalpy $H(S,P)$ or the Gibbs free energy $G(T,P)$.
+(These notions appear, since:
+  * 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 $G$.)
+----
+For a great explanation, why the Maxwell relations are "just a sneaky way of saying that the mixed partial derivatives of the function U commute". see https://johncarlosbaez.wordpress.com/2012/01/19/classical-mechanics-versus-thermodynamics-part-1/
+(The fact that partial derivatives commute is known as Schwarz' theorem (see  https://en.wikipedia.org/wiki/Maxwell_relations . Schwarz' theorem is simply a way of stating "the fact that a function S doesn’t change when we go around a parallelogram" https://johncarlosbaez.wordpress.com/2012/01/23/classical-mechanics-versus-thermodynamics-part-2/)
+<tabbox Abstract>
+<note tip>
+The motto in this section is: //the higher the level of abstraction, the better//.
+</note>
+<tabbox Why is it interesting?>
+The Maxwell relations encode useful relationships between notions of [[models:thermodynamics|thermodynamics]].
 </tabbox>

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