The contact angle at $25\backslash$deg C of an oil drop at a smooth planar solid surface in contact with air is

$30\backslash$deg $.$ At this temperature: The surface tension of the oil is $25\mathrm{.}0$ mJ$/$m$?.$ The surface energy of the sold material is $30\mathrm{.}0$ mJ$/$m$?.$ The oil$-$water interfacial energy is $50\mathrm{.}0$ mJ$/$m$\text{'}.$ The solid$-$water interfacial energy is $65$ mJ$/$m$\text{'}.$ All phases can be considered totally immiscible.

$($a$)$ Calculate the interfacial energy of the oil$-$solid interface at $25$ C$.$

$($b$)$ Calculate the spreading coefficient for the oil at the solid$-$water interface for the given system. Does oil spread at the solid$-$water interface?

$($c$)$ When a surface$-$active substance $($a detergent$)$ is dissolved in the water, it adsorbs at both the oil$-$water and solid$-$water intefaces. The solid$-$water interfacial energy becomes effectively zero. If the aim of adding detergent is to remove an oil film from the solid surface, what is the maximum oil$-$water interfacial energy that the detergent$$should$$achieve$?$