Question

How do you mathematically derive the "2nd Law of Thermodynamics"?

Answer 1

The second law of thermodynamics states that, the state of entropy of entire universe as an isolated system will always increase over time. The second law also states that the changes in the entropy in the universe can never be negative.

To understand why entropy increases and decreases, it is important that two changes in entropy have to considered at all times. The entropy change of universe is equivalent to the sum of the changes in entropy of the system and surroundings.

In an isothermal reversible expansion, the heat q absorbed by system from the surroundings is,

Since the heat absorbed by the system is the amount lost by the surroundings,

Therfore,

If the process is irreversible however, the entropy change is

If we put the two equations for together for both processes, we get the second law of thermodynamics,

where equals zero for a truly reversible process and is greater than zero for an irreversible process. In reality truly reversible process never happen so it is safe to say all thermodynamic processes are irreversible in direction they occur.