If we consider a [[Carnot Cycle|Carnot heat engine]] we find that the following relation is true: $\oint\partial Q=Q_H−Q_L>0$ Because of the relation between $Q_H$, $Q_L$, $T_H$, and $T_L$​ for [[Reversibility|reversible]] [[Cycle|cycles]], it is clear that: $\oint \dfrac{\partial Q}{T}=\dfrac{Q_H}{T_H}−\dfrac{Q_L}{T_L}=0$ For an irreversible [[Heat Engine|heat engine]]: $Q_{L,irr}>Q_{L,rev}​$ As such: $\dfrac{Q_H}{T_H}<\dfrac{Q_{L,irr}}{T_L}​​$ Therefore: $\oint\dfrac{\partial Q}{T}=\dfrac{Q_H}{T_H}−\dfrac{Q_{L,irr}}{T_L}<0$ A similar proof exists for [[Refrigeration Cycle|refrigeration cycles]], therefore leading to the Inequality of Clausius: $\oint\dfrac{\partial Q}{T}=0\to\text{reversible}$ $\oint\dfrac{\partial Q}{T}<0\to\text{irreversible}$ $\oint\dfrac{\partial Q}{T}>0\to\text{impossible}$