What is polytropic index? 
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*What is polytropic index? 

*What is the connection between it and work of an adiabatic system?
I tried surfing but didn't able to find a proper answer for that.
 A: The polytropic index is that defined via a polytropic equation of state of the form $P \propto \rho^{1 +1/n}$, where $P$ is pressure, $\rho$ is density, and $n$ is the polytropic index.
There is a relationship between the polytropic index and the adiabatic index. The latter is defined through
$PV^{\gamma} = $constant. If we consider adiabatic changes then $\gamma$ is the ratio of specific heats.
You can see that in this case $\gamma = 1 + 1/n$, so that $n = 1/(\gamma -1)$.
For an adiabatic change in an ideal monatomic gas $\gamma = 5/3$ and the polytropic index is $n=3/2$.
Unfortunately, whilst the above is the accepted definition of polytropic index in astrophysics, it appears from a brief www trawl that many also appear to refer to $\gamma$ as the polytropic index.
A: For a polytropic process path, one in which $pV^n$=constant along the path, n is the polytropic index.  Such a process path is typically regarded as reversible, and the temperature, volume, and pressure can be varying along the path. Also, heat exchange between the system and surroundings can be occurring.  A subset of polytropic paths are adiabatic reversible paths in which no heat exchange is occurring between the system and surroundings.  In that case, k is equal to $\gamma$, the ratio of the heat capacity at constant pressure to the heat capacity at constant volume.
