User Tools

Site Tools


formulas:maxwell_relations

Differences

This shows you the differences between two versions of the page.

Link to this comparison view

Both sides previous revision Previous revision
Next revision
Previous revision
formulas:maxwell_relations [2017/11/05 16:09]
jakobadmin [Why is it interesting?]
formulas:maxwell_relations [2018/12/19 11:01] (current)
jakobadmin ↷ Links adapted because of a move operation
Line 1: Line 1:
 ====== Maxwell Relations ====== ====== Maxwell Relations ======
  
-<tabbox Why is it interesting?> ​ 
-The Maxwell relations encode useful relationships between notions of thermodynamics. ​ 
  
-<​tabbox ​Layman+ 
 +<​tabbox ​Intuitive
  
 <note tip> <note tip>
Line 10: Line 9:
 </​note>​ </​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 ​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 abstractionthe 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>​ </​tabbox>​
  
  
formulas/maxwell_relations.1509894563.txt.gz · Last modified: 2017/12/04 08:01 (external edit)