Long $-$ long ago, in a galaxy far $-$ far away, a young Chemical Engineering graduate student

was investigating the $$extraction$$ of petroleum oils from oilfield produced water using supercritical CO$2$

$.$ The first step in this operation was the resolution of a very stable oil$-$in$-$water

emulsion. The graduate student observed that high$-$pressure CO$2$ was quite effective at

breaking this emulsion and believed that the process was controlled by the thermodynamics

$($chemical reaction equilibria$)$ and the chemical reaction kinetics of CO$2$ dissolution in the

produced water. Furthermore the graduate student proposed that the emulsion$-$breaking

success was a result of the acid$-$nature of CO$2$

$.$ Please help this struggling graduate

student by estimating the pH of water in contact with pure super$-$critical CO$2$ at $32$ and

$100$ bar.