$Claussius$ $Integral$ states that $\oint \dfrac{δQ}{T}\le0$ .
Now this $integral$ states that in a $PVT$ graph,the path of the line integral doesn't matter ,the only thing that matters is that the path has to be closed.Now why is this the case ?
How do I intuitively understand $Claussius$ $Inequality$ ?
Clausius' inequality is not about how your "system" $\mathcal S$ behaves under inspection but rather how any environment $\mathcal E$ surrounding "the" system $\mathcal S$ must behave so that when the state of $\mathcal S$ is changed and moved around any thermodynamic path (maybe irreversibly), starting from one equilibrium state but always returning to the same. Note that $\delta Q$ is the heat leaving the environment $\mathcal E$ at temperature $T$ that is again of the environment.
When equality $=$ holds the integral is a form of conservation law, as it shows that the quantity represented by the infinitesimal $\frac{\delta Q}{T}$ when absorbed (+) is equally compensated by the amount emitted (-). When strict inequality holds it shows that the "system" because of internal irreversibilities (say, friction and such) must produce a positive amount of "heat" internally to be absorbed by $\mathcal E$ so that $\mathcal S$ may be returned to its original equilibrium state.
All state functions satisfy $$\oint dF = 0$$ Take $F$ to be volume $V$: if we subject the system to a series of processes that end up on the same state where we started, then $V$ at the end is equal to $V$ at the beginning, which means that the integral of $dV$ over any closed path is zero.
The Clausius equality, $$\oint \frac{dQ_\text{rev}}{T}=0$$ proves that the quantity $dS\equiv dQ/T$ is the differential of a state function, thus we recognize the existence of entropy and its relationship to temperature and heat. Notice the important condition of reversibility, required to obtain this equality.
The inequality applies if the condition of reversibility does not hold during the path of the process: the more the process deviates from equilibrium, the bigger the inequality. This is the second law.
Only entropy has this unique dependence on reversibility. The integral of $dV$ (or any other state function) over any closed path is exactly zero regardless of whether the process was conducted reversibly or not. But entropy is a special property that is associated with all inequalities that appear in thermodynamics.
