A foam film was generated by drainage from a supporting frame. It consist of a layer of solution sandwiched between two layers of amphiphilic molecules $("$soap$").$ In the vertical geometry shown below they are initially very thick but the solution drains rapidly and the film thins with time and eventually becomes too thin to reflect any significant amount of light and appears colourless. This is called a "black" film. Because of a balance between the Van der Waals attractive force and the double layer repulsive forces from the electrolyte solution in the film a metastable black film can often be generated. Such a metastable film of thickness $10$ nm could be obtained when the electrolyte concentration was $0\mathrm{.}1$ mol$/$lt $*$ a$)$Calculate the van der Waals interaction energy per unit area if the Hamaker constant for the interaction of two slabs of air across a thin film of water $($A$131)$ is taken to be approximately $6$ x $10-20$ J$.*$ b$)$What is the Debye length across the film? $*$ c$)$Calculate the repulsive energy per unit area within foam film for values of surface potential of $20$mV taking T$=300$K$.($Hint:Use DLVO theory$)$