How is $PV^n$ an adiabatic process if specific heat ratio, i.e n itself decreases with increase in temperature? I got this doubt while studying about air compressors.
n decreases with temperature increase
In adiabatic process compression from h-s graph, I can see that actual enthalpy change is more than in the isentropic process, the reason being increase in temperature due to compression.
So 1.So in air compressors we know that compression work is minimum in isothermal and max. in adiabatic...thus as polytropic index decreases input work required is reduced..
Now, n decreases with increase in temperature...now considering the new decreased n as new polytropic index..sti the work is reduced...
Thus increasing or decreasing the temperature gives the same result i.e in a way decreasing the polytropic index, thus reducing work input...Which process should i prefer and why?
Am i conceptually incorrect?If yes,please guide me.
 A: 
I start with same pressure and volume..end with same final pressure and
  different volumes. The 2 processes i am mainly comparing are adiabatic and polytropic

The diagram below qualitatively shows different polytropic processes, including the adiabatic process and isothermal process. Each is shown starting at the same initial equilibrium P1 and V1 and terminating in the same final pressure P2 (except the isobaric process) and different volumes (the intersections of P=P2 with the different curves).
You can see from inspection, based on the areas under the curves from the initial state to the different final volumes at the same final pressure, that the work done in the adiabatic process is less than the isothermal process and would be for processes in between (not shown). 

Which process should i prefer and why? Am i conceptually incorrect?If
  yes,please guide me.

For air as an ideal gas and a reversible adiabatic process we have
$$n=\frac{c_p}{c_v}$$
For air, although this decreases with temperature, it's not by much for practical temperature ranges.
For air at 300 K (approx room temperature), $n$ = 1.4. 
At 1500K (a 1200 C temperature rise), $n$ = 1.3. The difference isn't that much and the work will still be less for the adiabatic process compared to the isothermal or other processes in between.
Hope this helps.

