# Contradiction of Units in Polytropic Process

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?

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

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.