Gibbs-Duhem integration: Difference between revisions

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When a differential change of the conditions is performed we wil have for any phase:
When a differential change of the conditions is performed we wil have for any phase:


: <math> d \mu = \left( \frac{ \partial \mu }{\partial T} \right)_{p,\lambda} d T +
: <math> d \left( \beta\mu \right) = \left[ \frac{ \partial (\beta \mu) }{\partial \beta} \right]_{\beta p,\lambda} d \beta +
\left( \frac{ \partial \mu }{\partial p} \right)_{T,\lambda} d p +  
\left[ \frac{ \partial (\beta \mu) }{\partial (\beta p)} \right]_{\beta,\lambda} d (\beta p) +  
\left( \frac{ \partial \mu }{\partial \lambda} \right)_{T,p} d \lambda.
\left[ \frac{ \partial (\beta \mu) }{\partial \lambda} \right]_{\beta,\beta p} d \lambda.
</math>  
</math>  


Taking into account that <math> \mu </math> is the [[Gibbs energy function|Gibbs free energy]] per particle:
Taking into account that <math> \mu </math> is the [[Gibbs energy function|Gibbs free energy]] per particle:
: <math> d \left( \beta\mu \right) =  \frac{E}{N}  d \beta +  \frac{ V }{N } d (\beta p)  +
\left[ \frac{ \partial (\beta \mu) }{\partial \lambda} \right]_{\beta,\beta p} d \lambda.
</math>


TO BE CONTINUED .. soon
TO BE CONTINUED .. soon

Revision as of 12:24, 2 March 2007

CURRENTLY THIS ARTICLE IS UNDER CONSTRUCTION

History

The so-called Gibbs-Duhem Integration referes to a number of methods that couple molecular simulation techniques with thermodynamic equations in order to draw phase coexistence lines.

The method was proposed by Kofke (Ref 1-2).

Basic Features

Consider two thermodynamic phases: and , at thermodynamic equilibrium at certain conditions. The thermodynamic equilibrium implies:

  • Equal temperature in both phases: , i.e. thermal equilbirum.
  • Equal pressure in both phases , i.e. mechanical equilbrium.
  • Equal chemical potentials for the components , i.e. material equilibrium.

In addition if we are dealing with a statistical mechanics model, with certain parameters that we can represent as , the model should be the same in both phases.

Example: phase equilibria of one-compoment system

Notice: The derivation that follows is just a particular route to perform the integration

  • Consider that at given conditions of two phases of the systems are at equilibrium, this implies:

Given the thermal equilibrium we can also write:

where

  • , where is the Boltzmann constant

When a differential change of the conditions is performed we wil have for any phase:

Taking into account that is the Gibbs free energy per particle:

TO BE CONTINUED .. soon

References

  1. David A. Kofke, Gibbs-Duhem integration: a new method for direct evaluation of phase coexistence by molecular simulation, Mol. Phys. 78 , pp 1331 - 1336 (1993)
  2. David A. Kofke, Direct evaluation of phase coexistence by molecular simulation via integration along the saturation line, J. Chem. Phys. 98 ,pp. 4149-4162 (1993)