The second law of thermodynamics is developed for systems undergoing cycles while exchanging heat with two reservoirs. One corollary of the second law is known as the Clausius inequality, which states that the cyclic integral of is always less than or equal to zero. That is,

where
       delta;Q = differential heat transfer at the system
              boundary during a cycle
      T = absolute temperature at the
            boundary
      = integration over the entire cycle

The Clausius inequality is valid for all cycles, reversible or irreversible.

To demonstrate the Clausius inequality, consider two heat engines that work between the same two reservoirs. Heat engine A is an reversible heat engine and heat engine B is an irreversible one. Both of these two engines absorb heat Q_H from a heat source, which has a temperature of T_H and reject heat to a heat sink at temperature T_L. Applying the first law to both of the engines gives,

      W_A,rev = Q_H - Q_A,L

      W_B,irr = Q_H - Q_B,L

Since a reversible engine has the highest efficiency of any heat engines working between the same source and sink, a reversible heat engine produces more work than an irreversible heat engine for the same heat input Q_H.

      W_A,rev = Q_H - Q_A,L > W_B,irr = Q_H - Q_B,L

      Q_A,L < Q_B,L

Since the temperature T_H is constant during the heat transfer Q_H, and the temperature T_L is constant during the heat transfer Q_A,L and Q_B,L, the cyclic integral of for both heat engines is

Comparing the above two equations with Q_A,L < Q_B,L yields,

       (1)

A reversible heat engine holds the following relation:

      
Therefore, the cyclic integral of for a reversible engine is zero.

Combining the above equation with equation (1) gives,

To summarize the above analysis, the Clausius inequality states

       for internally reversible cycles
       for irreversible cycles

From the discussion above, , the Clausius inequality states, for an internally reversible cycle

The cyclic integral of is zero. Therefore, is a property. Clausius in 1865 named this new thermodynamic property entropy. The definition is

The entropy change for process1-2 can be determined as

The Entropy Change between
two Specified States are the
Same Regardless the
Process Is Reversible or Not

Note that the above definition actually defines the change of entropy, instead of entropy itself. Because entropy is a property, the entropy change between two specified states is the same whether the process is reversible or not.

The previous discussion only means that entropy is a useful property in the second law analysis of an engineering device. But what is entropy? Entropy can be viewed as a measure of molecular disorder. In other words, entropy is the amount of disorder in a system. A pure substance at absolute zero temperature is in perfect order, and its entropy is zero. This is the third law of thermodynamics.

If the substance is not a pure crystal like substance, its entropy at absolute zero is not zero.

When the molecular disorder is produced, the ability of doing work is reduced.

Cycle 1-2-1

Consider a cycle as described in the diagram at the left. Peocess1-2 is an arbitrary (reversible or irreversible) process, Process 2-1 is an internally reversible process. From the Clausius inequality,

      
or

where the equality holds for internally reversible process and inequality holds for irreversible process.

For an isolated system, the heat transfer at the system boundary is zero. The above equation becomes

      ΔS_isolated 0

This equation is known as the increase of entropy principle. It states

Entropy is an extensive property, and thus the total entropy of a system is equal to the sum of the entropies of all parts of the system. The combination of a system and its surroundings is an isolated system. For this case, the increase of entropy principle is

      ΔS_isolated = ΔS_sys + ΔS_surr 0

The entropy change of a system can be negative. But the entropy change of an isolated system formed by this system and its surroundings can not be negative.

The second law of thermodynamics states that processes can occur only in a certain direction. But which direction? The increase of entropy principle answers this question: a process must proceed in the direction that complies with the increase of entropy principle. A process that violates this principle is impossible.