A piston is placed inside a cylindrical container, which has adiabatic walls. Initially, the piston divides the container into two equal parts, with volumes \(V\) each, both containing an ideal gas \(G\) at a pressure \(P\). Then the piston is moved very slowly by an external force \(F_{ext}\) till the volume on one side of the piston becomes thrice that on the other side.

Let the work done by \(F_{ext}\) in doing so be \(W\). \[W=kPV\]

If the heat capacity ratio of \(G\) is \(\dfrac{5}{3}\), then find \(k\).

**Note :** Take the piston to be insulating.

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