Editing Semi-grand ensembles
Jump to navigation
Jump to search
The edit can be undone. Please check the comparison below to verify that this is what you want to do, and then publish the changes below to finish undoing the edit.
Latest revision | Your text | ||
Line 1: | Line 1: | ||
== General features == | |||
Semi-grand ensembles are used in Monte Carlo simulation of mixtures. In these ensembles the total number of molecules is fixed, but the composition can change. | |||
== Canonical ensemble: fixed volume, temperature and number(s) of molecules == | == Canonical ensemble: fixed volume, temperature and number(s) of molecules == | ||
We shall consider a system consisting of ''c'' components;. | We shall consider a system consisting of ''c'' components;. | ||
In the [[Canonical ensemble|canonical ensemble]], the differential | In the [[Canonical ensemble|canonical ensemble]], the differential | ||
Line 9: | Line 12: | ||
*<math> A </math> is the [[Helmholtz energy function]] | *<math> A </math> is the [[Helmholtz energy function]] | ||
*<math> \beta | *<math> \beta \equiv 1/k_B T </math> | ||
*<math> k_B</math> is the [[Boltzmann constant]] | *<math> k_B</math> is the [[Boltzmann constant]] | ||
*<math> T </math> is the absolute [[temperature]] | *<math> T </math> is the absolute [[temperature]] | ||
Line 39: | Line 42: | ||
where <math> \left. \mu_{i1} \equiv \mu_i - \mu_1 \right. </math>. | where <math> \left. \mu_{i1} \equiv \mu_i - \mu_1 \right. </math>. | ||
* Now considering the | * Now considering the thermodynamical potential: <math> \beta A - \sum_{i=2}^c \left( N_i \beta \mu_{i1} \right) </math> | ||
:<math> d \left[ \beta A - \sum_{i=2}^c ( \beta \mu_{i1} N_i ) \right] = E d \beta - \left( \beta p \right) d V + \beta \mu_{1} d N - | :<math> d \left[ \beta A - \sum_{i=2}^c ( \beta \mu_{i1} N_i ) \right] = E d \beta - \left( \beta p \right) d V + \beta \mu_{1} d N - | ||
Line 61: | Line 64: | ||
:<math> \beta G (\beta,\beta p, N_1, N_2, \cdots N_c ) \rightarrow \beta \Phi (\beta, \beta p, N, \beta \mu_{21}, \cdots, \beta \mu_{c1} ) </math>, | :<math> \beta G (\beta,\beta p, N_1, N_2, \cdots N_c ) \rightarrow \beta \Phi (\beta, \beta p, N, \beta \mu_{21}, \cdots, \beta \mu_{c1} ) </math>, | ||
where the ''new'' | where the ''new'' thermodynamical Potential <math> \beta \Phi </math> is given by: | ||
:<math> d (\beta \Phi) = d \left[ \beta G - \sum_{i=2}^c (\beta \mu_{i1} N_i ) \right] = E d \beta + V d (\beta p) + \beta \mu_1 d N | :<math> d (\beta \Phi) = d \left[ \beta G - \sum_{i=2}^c (\beta \mu_{i1} N_i ) \right] = E d \beta + V d (\beta p) + \beta \mu_1 d N |