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A polytropic process is a process that obeys the relation $pV^n=C.$ However, when I try to solve a problem involving this relationship with, for example, $n = 1.5$ my use of units in my equations breaks down. At some point, I find myself having to solve for the constant $C$ to evaluate the integral which in the case of $n = 1.5$ gives the constant $C$ with the units $\mathrm{\frac{kg\cdot m^{3.5}}{s^2}}$ and does not match up with the expected units for energy. (Assuming units of $\mathrm{m^3}$ for $V$ and units of $\mathrm{\frac{kg}{m\cdot s^2}}$ for $p$.)

Usually I drop the units entirely at this point and arrive at the correct answer regardless, but this does not feel like a rigorous solution. How would I avoid this contradiction in my dimensional analysis, and why does it exist?

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    $\begingroup$ $C$ takes on the necessary units - if $n$ is not 1, $C$ will not be in energy units. $\endgroup$
    – Jon Custer
    Sep 1, 2021 at 1:41
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    $\begingroup$ You have a hidden assumption regarding what units should apply to C. That hidden assumption is incorrect. $\endgroup$ Sep 1, 2021 at 2:18

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The product $pV$ (where $p$ is pressure and $V$ is volume, each with their standard units in your favorite system) has units of energy. As noted in the comments, the product $pV^n$ when $n\neq 1$ does not. Nor should you expect it to. We're no longer dealing with energy but with an invariant in certain adiabatic processes, for example.

I don't recommend ever dropping units entirely, as this will catch you up someday. Instead, consider replacing the constant $C$ with $p_0V_0^n$, where $p_0$ and $V_0$ describe a reference state (which could be a final state) measured in the same units.

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