Why can the equation of adiabatic process $P_1 V_1^\gamma=P_2 V_2^\gamma$ be written as

$$\frac{\Delta P}{P}=-\gamma \frac{\Delta V}{V},$$

where $P_2-\Delta P=P_1$ and $V_2+\Delta V=V_1$ as in this website


  • 3
    $\begingroup$ Use MathJax please. $\endgroup$
    – user139621
    Jan 22, 2018 at 13:12

1 Answer 1


It's only an approximation for small changes in pressure and volume. We can write your original equation as $$PV^{\gamma}=C$$ If we take the differential of this relationship using the product rule, we obtain:$$(dP)V^{\gamma}+\gamma PV^{\gamma-1}dV=0$$Dividing this by $PV^{\gamma}$ yields: $$\frac{dP}{P}+\gamma \frac{dV}{V}=0$$or, equivalently, $$\frac{dP}{P}=-\gamma \frac{dV}{V}$$


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