When do formulas for adiabatic processes apply?

[phantomvommand]

Homework Statement

Please see the attached photo

Relevant Equations

Change in internal energy = nCvT

In this problem, the method used to solve the question is to equate pdV with change in internal energy. This implies an adiabatic process as Q = 0? (not sure about this claim) However, why is it not correct to simply apply the PV^ϒ = constant formula?

Thank you.

 

[haruspex]

why is it not correct to simply apply the PV^ϒ = constant formula?

What would you apply that to? You cannot apply it to the system as a whole since there are two initial pressures. You cannot apply it to one partition since there is heat flow between them.