# Derivation of polytropic process equation

I know that a polytropic process equation is often the result of an empirical fit of a P-V curve that allows one to derive some nice analytical equations for the process. Is there actually a derivation for the polytropic process equation?

Indeed, the general polytropic equation $pV^n=C$ is meant as a fit to describe a wide variety of situations - but the general situations don't have to exactly obey any polytropic equation with any $n$, anyway (because the right $n$ depends on $p$ and/or $V$), so there is nothing to derive.

A major, general enough exception is the adiabatic equation $pV^\gamma = C$. It only differs by using the name $\gamma$ for $n$ where $\gamma = C_P/C_V$, the ratio of heat capacities at constant pressure and volume, respectively. The adiabatic process is one in which no heat is transferred between the gas and the environment, so all the work being used or consumed is used to change the internal thermal energy of the gas. Nevertheless, $\gamma$ can have many different values such as $5/3$ etc. Microscopically, the value is linked to a function of the degrees of freedom of a single (average) molecule.

In that case, the derivation of the adiabatic equation - based on $\alpha p \,dV = dU$ - is here:

From the first law of thermodynamics; $$dU+dW=dQ$$ As; $$C_V = \frac {dU}{dT},$$ $$C=\frac {dQ}{dT},$$ $$dW=PdV,$$ We have; $$C_VdT + PdV = CdT$$ Multiplying throughout by $$R$$ ; $$C_V.RdT + R.PdV = R.CdT$$ By the ideal gas law; $$PV=RT$$ $$→PdV+VdP=RdT$$ Hence; $$C_V(PdV+VdP) + R.PdV=R.CdT$$ $$→PdV(C_V+R) + C_V.VdP = R.CdT$$ But $$C_P - C_V = R$$. Hence; $$C_P.PdV + C_V.VdP = R.CdT$$ $$→C_V.VdP = R.CdT - C_P.PdV$$ $$=(C_P-C_V).CdT - C_P.PdV$$ Dividing throughout by $$C_V$$ , noting that $$\frac {C_P}{C_V} = \gamma$$ and rearranging; $$VdP = -[(1-\gamma)CdT - \gamma .PdV]$$ $$=-[(1-\gamma)dQ - \gamma .dW]$$ $$=-\left[(1-\gamma)\frac {dQ}{dW} - \gamma \right].dW$$ $$-\left[(1-\gamma)\frac {dQ}{dW} - \gamma \right].PdV$$ Noting that for a polytropic process $$\frac {dQ}{dW} = \text {constant} = K$$ and rearranging; $$\frac {dP}{P} + [(1-\gamma)K - \gamma ].\frac {dV}{V} = 0$$ Integrating both sides , we have; $$\ln P + [(1-\gamma)K - \gamma ].\ln V = C' = \ln C$$ $$~PV^{(1-\gamma)K - \gamma } = C$$ In a simplified way , we can write; $$PV^n = \text {constant}$$