Entropy is a state variable, i.e. the data about the initial and final states would suffice to define change in entropy . However, while applying the formula dS = integral of dQ/T, I have distinctly been told that dQ is to be calculated by assuming a hypothetical reversible process, irrespective of whether we are given a reversible or an irreversible process in the problem. Although heat depends on the thermodynamic path, S doesn't. So why should it matter whether the process is irreversible, whether Q is calculated for an irreversible process, as S doesn't depend on all this? (I know that the math won't work out and we'll end up with different values, but my question is a conceptual one.)
For both the reversible and the irreversible paths of a system, the T in the equation is supposed to be the value at the interface between the system and surroundings through which dQ flows (see Fermi's Thermodynamics text). This integral is different between the reversible path and the irreversible path. The various integrals of dQ/T for all the possible irreversible paths are different from one another, but the integrals of dQ/T for all the reversible paths are the same, and greater than those for all the irreversible paths.