The thermodynamic law, PV^Y = constant, is not applicable in case of

Since $$P{V^Y} = $$   constant is valid only for reversible process but since as free expansion is irreversible because it is working on a cycle by taking heat from a single reservoir and producing net expansion work when considered reversible and thus violating Kelvin-planck statement when considered reversible hence we can conclude free expansion is an irreversible process. And hence $$P{V^Y} = $$   constant can’t be valid.