Thermodynamic entropy — UPSC CSE 2009 Physics PYQ

Define entropy. How is it related to disorder? Hence, derive the Boltzmann relation S = k \log \Omega, where \Omega is the probability and k is the Boltzmann constant. Show that for any type of process, involving a closed system \Delta S \ge \frac{\Delta Q}{T} where the equality sign applies for internally reversible processes and the inequality for internally irreversible processes.