Editing Computation of phase equilibria
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conditions <math> \left. N,P,T \right. </math>. Let <math> p_{eq} </math> the pressure at which the phase transition occurs. In such a | conditions <math> \left. N,P,T \right. </math>. Let <math> p_{eq} </math> the pressure at which the phase transition occurs. In such a | ||
case the following scenario is expected for <math> \left. P(V|N,p,T) \right. </math>: | case the following scenario is expected for <math> \left. P(V|N,p,T) \right. </math>: | ||
*<math> \left. P(V|N,p_{eq},T) \right. </math> has two maxima, corresponding to the liquid and vapor pure phases, with <math> \left. P(V_v|N,p_{eq},T) = P(V_l|N,p_{eq},T) = P_{v/l} \right. </math> | *<math> \left. P(V|N,p_{eq},T) \right. </math> has two maxima, corresponding to the liquid and vapor pure phases, with | ||
: <math> \left. P(V_v|N,p_{eq},T) = P(V_l|N,p_{eq},T) = P_{v/l} \right. </math> | |||
*The probability of a given intermediate volume at <math> \left. p_{eq} \right. </math> can be estimated (from macroscopic arguments) as: | *The probability of a given intermediate volume at <math> \left. p_{eq} \right. </math> can be estimated (from macroscopic arguments) as: | ||
:<math> \left. P(V|N,p_{eq},T) \simeq P_{v/l} \times \exp \left[ - \frac{ \gamma(T) \mathcal A }{k_B T } \right] \right. </math>, | : <math> \left. P(V|N,p_{eq},T) \simeq P_{v/l} \times \exp \left[ - \frac{ \gamma(T) \mathcal A }{k_B T } \right] \right. </math>, | ||
where <math> \left. \gamma(T) \right. </math> is the [[surface tension]] of the vapor-liquid interface, | where <math> \left. \gamma(T) \right. </math> is the [[surface tension]] of the vapor-liquid interface, | ||
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A similar procedure can be built up to compute <math> \left. p(\rho) \right. </math> | A similar procedure can be built up to compute <math> \left. p(\rho) \right. </math> | ||
from <math> \left. \mu(\rho) \right. </math>. | from <math> \left. \mu(\rho) \right. </math>. | ||
Once <math> \left. p(\rho) \right. </math> and <math> \left. \mu(\rho) \right. </math> are known it is straightforward to compute the coexistence point. | Once <math> \left. p(\rho) \right. </math> and <math> \left. \mu(\rho) \right. </math> are known it is straightforward to compute the coexistence point. | ||
==== Practical details ==== | ==== Practical details ==== | ||
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== Mixtures == | == Mixtures == | ||
=== Symmetric mixtures === | === Symmetric mixtures === | ||
Examples of symmetric [[mixtures]] can be found both in lattice of continuous model. The [[Ising Models | Examples of symmetric [[mixtures]] can be found both in lattice of continuous model. The [[Ising Models]] can be | ||
viewed as mixture of two different chemical species which de-mix at low | viewed as mixture of two different chemical species which de-mix at low | ||
temperature. The symmetry in the interactions can be exploited to simplify the calculation of phase diagrams. | temperature. The symmetry in the interactions can be exploited to simplify the calculation of phase diagrams. |