Gibbs-Duhem integration: Difference between revisions
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* Consider that at given conditions of <math> T , p, \lambda </math> two phases of the systems are at equilibrium, this implies: | * Consider that at given conditions of <math> T , p, \lambda </math> two phases of the systems are at equilibrium, this implies: | ||
: <math> \mu_{ | : <math> \mu_{a} \left( T, p, \lambda \right) = \mu_{b} \left( T, p, \lambda \right) </math> | ||
Given the thermal equilibrium we can also write: | Given the thermal equilibrium we can also write: | ||
: <math> \beta \mu_{ | : <math> \beta \mu_{a} \left( \beta, \beta p, \lambda \right) = \beta \mu_{b} \left( \beta, \beta p, \lambda \right) </math> | ||
where | |||
* <math> \beta = 1/k_B T </math>, where <math> k_B </math> is the [[Boltzmann constant]] | |||
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: | ||
Revision as of 12:18, 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: Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a } and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle b } , at thermodynamic equilibrium at certain conditions. The thermodynamic equilibrium implies:
- Equal temperature in both phases: Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle T = T_{a} = T_{b} } , i.e. thermal equilbirum.
- Equal pressure in both phases Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle p = p_{a} = p_{b} } , i.e. mechanical equilbrium.
- Equal chemical potentials for the components Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mu_i = \mu_{ia} = \mu_{ib} } , i.e. material equilibrium.
In addition if we are dealing with a statistical mechanics model, with certain parameters that we can represent as Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \lambda } , 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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle T , p, \lambda } two phases of the systems are at equilibrium, this implies:
- Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mu_{a} \left( T, p, \lambda \right) = \mu_{b} \left( T, p, \lambda \right) }
Given the thermal equilibrium we can also write:
- Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \beta \mu_{a} \left( \beta, \beta p, \lambda \right) = \beta \mu_{b} \left( \beta, \beta p, \lambda \right) }
where
- Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \beta = 1/k_B T } , where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle k_B } is the Boltzmann constant
When a differential change of the conditions is performed we wil have for any phase:
- Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle d \mu = \left( \frac{ \partial \mu }{\partial T} \right)_{p,\lambda} d T + \left( \frac{ \partial \mu }{\partial p} \right)_{T,\lambda} d p + \left( \frac{ \partial \mu }{\partial \lambda} \right)_{T,p} d \lambda. }
Taking into account that Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mu } is the Gibbs free energy per particle:
TO BE CONTINUED .. soon
References
- 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)
- 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)