Stochastic ordering: Difference between revisions

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{{Expert-subject|Physics|date=February 2008}}
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The '''excess chemical potential''' is defined as the difference between the [[chemical potential]] of a given species and that of an [[ideal gas]] under the same conditions (in particular, at the same [[pressure]], [[temperature]], and [[wikt:composition|composition]]).<ref>{{cite book | last = Frenkel | first = Daan | authorlink = Daan Frenkel | coauthors = Smit, Berend | title = Understanding Molecular Simulation : from algorithms to applications | publisher = [[Academic Press]] | year = 2001 | location = [[San Diego, California]] | pages = | url = | doi = | id = | isbn = 0-12-267351-4 }}</ref>
 
Chemical potential of pure fluid can be estimated by the [[Widom insertion method]].
 
For a system of diameter <math>L</math> and volume <math>V</math>, at constant temperature :<math>T</math>, the classical [[partition function (statistical mechanics)|partition function]]  <math>Q(N,V,T)=\frac{V^{N}}{\Lambda^{dN}N!}\int_{0}^{1}\ldots\int_{0}^{1}ds^{N}\exp[-\beta U(s^{N};L)]</math>
s is a scaled coordinate.
 
:<math>F(N,V,T)= -k_{B}T\ln Q=-k_{B}T\ln\left(\frac{V^{N}}{\Lambda^{dN}N!}\right)-k_{B}T \ln{\int ds^{N}\exp[-\beta U(s^{N};L)]}=</math>
::<math>\;=F_{id}(N,V,T)+F_{ex}(N,V,T)</math>
 
combining the above equation with the definition of chemical potential:  <math>\mu_{a}= \left(\frac{\partial G}{\partial N_{a}}\right)_{PTN}</math>
 
we get the chemical potential of a sufficient large system is :
 
:<math>\mu= -k_{B}T\ln(Q_{N+1}/Q_{N})=-k_{B}T\ln\left(\frac{V/\Lambda^{d}}{N+1}\right) - k_{B}T \ln{\frac{\int ds^{N+1}\exp[-\beta U(s^{N+1})]}{\int ds^{N}\exp[-\beta U(s^{N})]}}=\mu_{id}(\rho) + \mu_{ex}</math>
 
wherein the chemical potential of ideal gas can be evaluated analytically.  
Now let's focus on <math>\mu_{ex}</math>, since the potential energy of an N+1 particle system can be separated into the potential energy of an N [[particle system]] and the potential of the excess particle interacts with the N particle system, there is
 
:<math>\Delta U\equiv U(s^{N+1}) - U(s^{N})</math>
 
and  
 
:<math>\mu_{ex}= -k_{B}T \ln \int ds_{N+1} \langle \exp(-\beta\Delta U)\rangle_{N}</math>
 
thus far we converted the excess chemical potential into an ensemble average, and the integral in the above equation can be sampled by brute force [[Monte Carlo method]].
 
The calculating of excess chemical potential is not only limited to homogeneous system, it has also been extended to inhomogeneous system by [[Widom insertion method]], or other [[Statistical ensemble (mathematical physics)|ensembles]] such as NPT, [[Microcanonical ensemble|NVE]].
 
==See also==
[[Apparent molar property]]
 
==References==
* Note: the equations and presentation in this article are drawn from [http://www.pages.drexel.edu/~cfa22/msim/node39.html Excess Chemical Potential via the Widom Method]<!-- not a copyright violation--see talk page and [[Wikipedia talk:Copyright problems#Copyright violations in Excess chemical potential]] -->
{{Reflist}}
 
{{DEFAULTSORT:Excess Chemical Potential}}
[[Category:Potentials]]
[[Category:Thermodynamics]]
[[Category:Chemical thermodynamics]]
 
[[ro:Mărimi molare de exces]]

Latest revision as of 18:03, 23 April 2014

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