Clausius deduced his famous "Entropy always increases" from two
premises:

1. Any irreversible process can be closed by a reversible process to
become a cycle.

2. Closed integral of dQ/T is negative or zero for any cycle (Q is the
heat absorbed, T is the temperature).

The falsehood of the first premise is almost obvious. I am not able to
prove RIGOROUSLY the falsehood of the second, but its truth is not
proved either (Clausius' proof is invalid). The problem with the
second premise is mentioned he

http://www.chem.umd.edu/~devoe/thermo/3steps.pdf

Unfortunately the analysis is too technical to be discussed via
email.

Pentcho Valev