ANS:B - free expansion of an ideal gas.

The thermodynamic law 𝑃𝑉𝑦=constantPVy=constant represents a polytropic process, where 𝑃P is pressure, 𝑉V is volume, and 𝑦y is a constant exponent. Let's analyze each scenario:

1. Ideal compression of air: In this scenario, the compression process could be approximated by a polytropic process, so the law could be applicable.
2. Free expansion of an ideal gas: During free expansion, the gas undergoes an irreversible process with no work done. Since there's no defined pressure or volume relationship during free expansion, the polytropic law does not apply here.
3. Adiabatic expansion of steam in a turbine: Adiabatic expansion involves the expansion of a fluid without heat exchange with the surroundings. In the case of steam, it could be better described by the ideal gas law or the steam tables, but not necessarily by a polytropic process.
4. Adiabatic compression of a perfect gas: Adiabatic compression is a process where a gas is compressed without heat exchange with the surroundings. This process is typically described by the polytropic equation, making the law applicable.