Semi-grand ensembles: Difference between revisions
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(New page: == General Features == Semi-grand ensembles are used in Monte Carlo simulation of mixtures. In this ensembles the total number of molecules is fixed, but the composition can change. == F...) |
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In this ensembles the total number of molecules is fixed, but the composition can change. | In this ensembles the total number of molecules is fixed, but the composition can change. | ||
== | == Canonical Ensemble: fixed volume, temperature and number(s) of molecules == | ||
We will consider a binary system; | We will consider a binary system;. | ||
In the Canonical Ensemble, the differential | |||
equation energy for the [[Helmholtz energy function]] can be written as: | |||
: <math> d \left( \beta A \right) = E d \beta - (\beta p) d V + \sum_{i=1}^2 (\beta \mu_i) d N_i </math>, | |||
where: | |||
*<math> A </math> is the [[Helmholtz energy function]] | |||
*<math> \beta \equiv 1/k_B T </math> | |||
*<math> k_B</math> is the [[Boltzmann constant]] | |||
*<math> T </math> is the absolute temperature | |||
*<math> E </math> is the internal energy | |||
*<math> p </math> is the pressure | |||
*<math> \mu_i </math> is the chemical potential of the species "i" | |||
*<math> N_i </math> is the number of molecules of the species "i" | |||
== Semi-grand ensemble at fixed volume and temperature == | |||
Revision as of 13:00, 5 March 2007
General Features
Semi-grand ensembles are used in Monte Carlo simulation of mixtures.
In this ensembles the total number of molecules is fixed, but the composition can change.
Canonical Ensemble: fixed volume, temperature and number(s) of molecules
We will consider a binary system;. In the Canonical Ensemble, the differential equation energy for the Helmholtz energy function can be written as:
- Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle d\left(\beta A\right)=Ed\beta -(\beta p)dV+\sum _{i=1}^{2}(\beta \mu _{i})dN_{i}} ,
where:
- is the Helmholtz energy function
- 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 \equiv 1/k_B T }
- 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
- 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 } is the absolute temperature
- 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 E } is the internal energy
- 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 } is the pressure
- 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 } is the chemical potential of the species "i"
- 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 N_i } is the number of molecules of the species "i"
