Main Points
- A homogeneous phase can be spontaneously decomposed into two immiscible phases.
- Concentration gradients can be spontaneously formed at the border between immiscible phases when the system reaches its equilibrium state.
- At constant temperature and pressure the entropy of system in question spontaneously decreases during the separation process.
- In the adiabatic system the increase of the entropy leads to the phase separation from a homogeneous solution.
Introduction
Recently I have written a small text Schrödinger’s Order, Disorder and Entropy and discussed it on the biosemiotic list. During discussion, there was a suggestion to consider a solution with a miscibility gap. In this case, a homogeneous mixture spontaneously decomposes to two different solutions with different concentrations of components.
In this text, I will take a simple regular solution as an example and show that in an adiabatically isolated system, the increase of the entropy corresponds to the formation of two phases from a homogeneous mixture. I believe that this is a good point to think it over in what case there is more order or disorder: in a homogeneous system or in two immiscible phases.
Paper
I have tried to keep the mathematics as simple as possible. Still there are many equations in the text and it is available as pdf at http://evgenii.rudnyi.ru/doc/teaching/miscibility/miscibility.pdf
Auxiliary Files
In the paper there are figures. Some are computed according to analytical formulas, for some it is necessary to solve a nonlinear equation numerically. Below there is a description of files (http://evgenii.rudnyi.ru/doc/teaching/miscibility/) that I have used to make computations and produces figures.
01_fig.plt contains the Gnuplot script that produces figures ($ gnuplot 01_fig.plt). The computations except for Fig 5 are done directly in Gnuplot. The data for Fig 5 (pd.L1L2) are produced in TDLIB.
Attention. TDLIB’s files are SGML-like and a browser may not display them correctly. To view them, download them and open in a text editor.
02_sys.mod is a description of the regular solution in TDLIB. It could be extended to model more complicated solution phases.
03_prop.out.mod is for plotting of the properties of the solution defined in TDLIB. With
$ access 02_sys 03_prop.out -o prop
one gets files that could be plotted in Gnuplot similar to pd.L1L2.
04_pd.out.mod for computation and plotting the miscibility phase diagram. The command
$ access 02_sys 04_pd.out -o pd
produces file pd.L1L2 that was plotted in Gnuplot.
05_solve.py is a script to solve numerically Eq 24 and 23. Its use
$ python 05_solve.py [Tini [Cp [beta]]]
Without parameters the script uses default values of Tini = 700, Cp = 20, beta = 2000. Cp and beta are defined as Cp/R and beta/R.