Do processes $P\propto\frac{1}{V^2}$, $P\propto\frac{1}{V^3}$, $P\propto\frac{1}{V^4}$, etc., exist in the real world? Is there any real process in which $PV^n=C$ where $P,V$ stands for pressure, volume respectively. $C$ is a constant and $n$ is a positive integer?
I am familiar with Boyle's law that states that $P\propto\frac{1}{V}$ when the temperature is constant. But according to the first equation, since $n$ is any positive integer, there are systems where $P\propto\frac{1}{V^2}$ , $P\propto\frac{1}{V^3}$ etc. 
Do such systems in which pressure is inversely proportional to square or cube of the volume really exist? Can anyone explain with example? Does this have any application in Engineering? (I found this in an engineering textbook.)
 A: In principle, yes, you can create any such process. After all, you can control the volume of a gas sample more or less arbitrarily by changing the size of its container, and then you can add or remove heat to change the temperature and thus set the pressure to whatever value you want. (Obviously you have to stay within the boundaries of the gaseous region on the substance's phase diagram.) So with a suitable apparatus, you can force a gas to change its state while keeping $PV^n$ constant for any $n$.
However, you might also wonder whether such a thing tends to happen naturally. Whatever definition of "naturally" you go by, the answer will probably be no. There are a few classes that describe most of the processes gases "naturally" go through:


*

*Isochoric (constant volume) processes can be considered the $n\to\infty$ limit of $PV^n = \text{const}$

*Isobaric (constant pressure) processes have $n = 0$ in $PV^n = \text{const}$

*Isothermal (constant temperature) processes have $n = 1$ in $PV^n = \text{const}$

*Adiabatic processes (no heat exchange) have $PV^\gamma = \text{const}$ where $\gamma = C_p/C_v$ is in general not an integer


Outside of these special cases, though, there's no particular reason for a gas to follow a $P-V$ curve with an integer value of $n$. (And in fact even these special cases are approximations to real processes...)
A: Your formula is used for an adiabatic transformation. Air is a very poor heat conductor, so if you consider a transformation which is fast compared to the heat combustion time scale, you can use this formula (combined with pV=n*R*T). An example of such a transformation is an explosion followed by a fireball. But the barometric formula shows that even the atmosphere can be considered as adiabatic, though the heating by the sun/ground is slower.
A: You can always have an arbitrary continuous path connecting a pair of points on the P-V indicator diagram. You may break up each infinitesimal section of such a path into an isothermal and an adiabatic component to visualize the physical existence of the path.
Reference: Zemansky and Dittman,Heat and Thermodynamics,McGraw-Hill,7th Ed, Chapter 8[Entropy]Section 8.1,Figures 8-1 and 8-2
The authors prove that $\oint\frac{dQ}{T}=0$ for a reversible closed path using the above concept.
NB: An arbitrary path on the P-V diagram will be of the form $P=f(V)$
