Entropy and Miscibility Gap: Tutorial for Biologists

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.


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.


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.

Python with SciPy

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.