You can prove that entropy is a system property with the use of Clausius inequality for a reversible cycle. Hence, system entropy is supposed to be independent of the path taken.

However, for an irreversible process apparently there is an extra increase in entropy $dS_{irrev}$ due to the process being irreversible. However, if the system undergoes an irreversible cycle, but ends up in the same state, shouldn't the entropy be the same, since it's a system property?

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    $\begingroup$ The entropy of the system will be the same, if it ends up in the same state it started in. However, if the process is irreversible, the entropy of the environment will have increased. $\endgroup$ – noah May 3 '17 at 16:54
  • $\begingroup$ If the entropy is not the same at the end of the irreversible process then it is not the same sate. Entropy is a state function as anything else in the sense of same state if and only if same entropy, same pressure, same temperature, volume, etc. $\endgroup$ – hyportnex May 3 '17 at 17:03

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