# 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.)

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You can always connect a pair of points$(P_1,V_1)$ and $(P_2,V_2)$ on the P-V indicator diagram by some arbitrary continuous path.Each infinitesimal element of this path may be decomposed into an isothermal and an adiabatic component to visualize the thing,a standard technique. –  Anamitra Palit Aug 18 '12 at 14:16

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...)

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Thanks for answering. Just one thing, you said that we can force a system to follow any relation we want by simultaneously controlling the container volume & the temperature. Does this imply that $PV^n=const$ can never occur at constant temperature except for n=1? –  Green Noob Aug 18 '12 at 15:43
For an ideal gas, yes. The ideal gas law tells you that $PV=\text{const}$ for a constant-temperature process. For a non-ideal gas, you can still control the state of the gas by adjusting the volume and temperature, but the process for which $PV=\text{const}$, for example, would not be precisely isothermal. –  David Z Aug 18 '12 at 15:46
Thanks for replying. In your answer, you also mentioned that $PV^n=const$ is not likely to occur naturally other than in the special cases mentioned. Does this also mean that there are no engineering applications? –  Green Noob Aug 19 '12 at 7:20

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.

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That was my first thought as well, but the adiabatic exponent $\gamma$ isn't an integer. –  David Z Aug 18 '12 at 5:41
Sorry I did not get your question. The adiabatic index is related to the degree of freedom of the molecules. Gamma=2 means the motion is stranded in a plane. en.wikipedia.org/wiki/Adiabatic_index –  Shaktyai Aug 18 '12 at 5:48
Yes, I know, $\gamma = 1 + 2/n$ where $n$ is the number of DOF. Constrained planar motion is an exceptional case, though; in any normal system $n \ge 3$. And still, the formula does not allow for arbitrary integers, only 2. –  David Z Aug 18 '12 at 6:09

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 $P=f(V)$

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