Clausius Inequality: To Determine Whether A Cycle Is Possible

The Clausius inequality is a consequence of the second law of thermodynamics. Clausius inequality is related to the entropy of thermodynamics system. It is also known as "Clausius Theorem". While entropy is defined as a measure of the randomness or disorder of a thermodynamic system.

Clausius Inequality

It states that- "In a cyclic operation, the sum of the 'dQ/T' term over a complete cycle (or cyclic integration of the 'dQ/T' term) is less than or equal to zero, depending on whether the process or a cycle is reversible or irreversible". Thus;

Where,

dQ is the amount of heat added to the

system at any point

T is the temperature at this point

The above equation is known as the Clausius inequality and it suggests to us the criteria for reversibility and irreversibility of the thermodynamic cycle;

➛The Clausius inequality is valid for all the cycles whether it is reversible or irreversible or the refrigeration cycle.

➛ The Clausius Inequality applies to any real engine cycle and implies a negative change in entropy on the cycle.

➛ "Clausius Inequality Is Valid For Refrigerator And Heat Pump Only".

Let's look at the derivation of the formula for the Clausius inequality;

According to Carnot's theorem, the efficiency of a reversible heat engine is greater than the efficiency of an irreversible heat engine when they are both operating between the same thermal reservoirs. Thus;

➝ η(reversible) ≻ η(irreversible)

--------------------(1)

Let us consider reversible and irreversible heat engines operating between two identical thermal reservoirs (high temperature reservoir, Tн and low temperature reservoir, Tc). Let the heat absorbed from the hot reservoir be Qн and the transfer of heat to the cold reservoir be Qc

Here,

• For both reversible and irreversible heat engines, { QнR ＝ Qнエ＝ Qн }

• But the heat rejected to the cold reservoir is different, { QcR < Qエ }

➩ { Qcエ＝ QcR ＋ Q'}

The efficiency of a reversible heat engine (or Carnot engine) operating in a cycle;

--------------------(a)

The efficiency of a irreversible heat engine operating in a cycle;

---------------------(b)

From both the above results;

Thus, we get;

The above equation is known as the "CLAUSIUS INEQUALITY". It tells us whether a cycle (or a process) is reversible or irreversible.

If the term on the left-hand side is greater than zero then the cycle is not possible otherwise possible.

Case(a):-

For A Reversible Cycle;

Thus;

Case(b):-

For An Irreversible Cycle;

In case of irreversible cycle, the amount of heat rejected to the cold reservoir is greater than the amount of heat rejected to the cold reservoir in case of reversible cycle. That means, for an irreversible process;