Q.If the piston of a container is pushed fast inward, will the ideal gas equation be valid in the intermediate stage? If not, why?

A quasi-static process is an infinitely slow process in which a thermodynamic system's variables (pressure, volume, temperature) change so gradually that the system remains in thermal, mechanical and chemical equilibrium with its surroundings at every single instant throughout the process. Because the system is essentially always arbitrarily close to an equilibrium state, its pressure, volume, temperature and internal energy all have well-defined, unique values at every stage, and the equation of state ($PV=NkT$ or $PV=\mu RT$) can be applied throughout. By contrast, if a process happens suddenly -- for instance, if a heavy mass is dropped abruptly onto a piston -- the gas passes through non-equilibrium intermediate states where pressure is not uniform throughout the gas, and the ideal gas equation cannot be used to describe those intermediate stages, only the initial and final equilibri …