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By generalizing the traditional concept of heat dQ and work dW to alsoinclude their time-dependent irreversible components d_{i}Q and d_{i}W allowsus to express them in terms of the instantaneous internal temperature T(t) andpressure P(t), whereas the conventional form uses the constant values T_{0} andP_{0} of the medium. This results in an extremely useful formulation ofnon-equilibrium thermodynamics so that the first law turns into the Gibbsfundamental relation and the Clausius inequality becomes an equality\ointdQ(t)/T(t)\equiv0 in all cases, a quite remarkable but unexpected result.We determine the irreversible components d_{i}Q\equivd_{i}W and discuss howthey can be determined to obtain the generalized dW(t) and dQ(t).

Generalized Non-equilibrium Heat and Work and the Fate of the Clausius Inequality