We know that $PV^{\gamma}=constant $. There are also 2 more equations which relates $P$ & $T$ and $V$ & $R$. Is there any equation which relates all these 3 things like ideal gas equation?
Also does the adiabatic process obey the ideal gas equation? How can this be? How can $PV^{\gamma}=constant$ and $PV=constant$ be true at the same time? (When $T$ is cconstant)
The ideal gas law is not PV=const. The ideal gas law is PV=nRT. If T is varying through a process, PV is not constant.
In an adiabatic reversible process, the temperature is varying in tandem with the pressure and volume changes. So $PV^{\gamma}=const.$ and PV=nRT are both satisfied simultaneously at all points along the process path. Another way of expressing this is: $$PV^{\gamma}=P_iV_i^{\gamma}$$and $$\frac{PV}{T}=\frac{P_iV_i}{T_i}$$where the subscript i refers to the initial state of the system. So you have 2 equations and three parameters. Once you specify one of these parameters at a point along the adiabatic reversible process path, the other two parameters are uniquely determined.
Adiabatic process is a process where no heat exchange occurs between the system and the surroundings. An equation of state is defined only for a state, not an entire process. In an adiabatic process, the product $PV^\gamma$ will remain constant.
$PV$ is constant for an isothermal process not an adiabatic process. So both $PV$ and $PV^\gamma$ wont be constant for the same process.
