# Combined gas equation in adiabatic process

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)

• There's the combined gas law, maybe? – Kyle Kanos Oct 23 '18 at 13:51
• In adiabatic processes, the temperature must change. – Bill N Oct 23 '18 at 18:05

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.