# Negative value of $dq/dT$ in a polytropic process

Consider a reversible polytropic process :$$PV^x=K$$ where $$K$$ is a constant. We can easily derive , By differentiating this and using the gas laws, that for such a process: $$PdV=\frac {(nRdT)}{(1-x)}$$.

Consider the First law for a reversible process: $$dq=du+P_{gas}dV$$ (since $$P_{ext}=P_{gas}).$$ Substituting

$$PdV=\frac{(nRdT)}{(1-x)}$$ $$\text {and}$$ $$dU=nC_vdT$$

and dividing by $$dT$$ , we get $$\frac{dq}{dT}= nR\left (C_v+\dfrac{1}{1-x}\right )$$ Further, since $$C_v=\dfrac{R}{\gamma -1}$$ we can clearly see that for $$x$$ in $$[1,\gamma]$$, we will get a negative value for dq/dT

Physically, This seems to imply that whenever $$dq>0, dT<0$$. That is, adding heat to the system decreases the temperature. Likewise, If $$dq<0, dT>0$$. Thus, extracting heat raises temperature. Physically, This seems to make no sense at all. However, I dont see any fault with the mathematics either.I have seen this general formula in many textbooks, and even questions where dq/dT comes out to be negative: But there is never an explanation.

How exactly is $$dq/dT < 0$$ making sense/plausible? My current "guess" is that it isn't possible to carry out polytropic processes where $$x<\gamma$$ reversibly.

If $$x=\gamma$$, it means that the gas is expanding and cooling because you're not adding any heat. If x = 1, it means that the gas is expanding, and you are adding just enough heat to keep the temperature constant. If x is in between these values, it means that your are adding heat but not enough to keep the temperature constant, so it is cooling.