WHAT IS CLAUSIUS THEOREM IN THERMODYNAMICS ?

We were discussing “Carnot cycle and its efficiency” as well as “efficiency of reversible heat engine” and “COP of refrigerator and heat pump” in our previous posts. We have also seen the concept of “corollary of Carnot’s theorem” and simultaneously “equivalence of Kelvin Planck statement and Clausius statement” in our recent posts.

Today we will see here the concept of Clausius’ theorem with the help of this post.

Let us see here now Clausius’ theorem

According to Clausius’ theorem a reversible path or process or line could be replaced by two reversible adiabatic processes or lines and one reversible isothermal process or line.

Let us see the following figure, a system is at equilibrium state i.e. at point i and system reaches to another equilibrium state i.e. at point f by following the reversible path i-f.

According to Clausius’ theorem reversible process i.e. i-f could be replaced by two reversible adiabatic processes i.e. i-a and b-f and one reversible isothermal process i.e. a-b.

Process i-f: Reversible process

Processes i-a and b-f: Reversible adiabatic processes

Process a-b: Reversible isothermal process

The replacement will be done in such a way that the area below the reversible path i-f must be equal to the area below i-a-b-f. Mathematically we will say that

Area below i-f = Area below i-a-b-f

Let us apply the "first law of thermodynamics" for following path or process

Process: i-f

Q _if = W_if + U_f-U_i

Process: i-a-b-f

Q _iabf = W_iabf + U_f-U_i

As we have seen above that the area below the reversible path i-f must be equal to the area below i-a-b-f or W_if = W_iabf

Therefore, we will have

Q _if = Q _iabf

Q _if = Q _ia + Q _ab + Q _bf

As we know that i-a and b-f are reversible adiabatic processes and therefore Q _ia = Q _bf = 0

Q _if = Q _ab

From above expression, we can say that heat transferred during the reversible process i-f will be equal to the heat transferred during the reversible isothermal process a-b.

Therefore we can say that any reversible process or path could be replaced by a reversible zigzag path with same end states and this reversible zigzag path will have two reversible adiabatic paths and one reversible isothermal path.

This replacement will be done in such a way that heat transferred during the original reversible process will be equal to the heat transferred during the reversible isothermal process.

We were discussing above single reversible process, now we will analyze here one complete reversible cycle as shown in figure. Let the reversible cycle is divided into large number of strips and these strips will indicate the reversible adiabatic lines as shown in figure. These strips are closed on bottom and tops with the help of reversible isothermal lines.

We can see here that original reversible cycle will be divided here into numbers of small-small Carnot’s cycle as shown in figure.

Let us focus over here the Carnot’s cycle abcd, dQ₁ heat is absorbed reversibly at temperature T₁ and dQ₂ heat is rejected reversibly at temperature T₂.

dQ₁/T₁= dQ₂/T₂

Let us consider the sign convention and we will take heat absorption as positive and heat rejection as negative.

dQ₁/T₁+ dQ₂/T₂= 0

In similar way for Carnot’s cycle efgh, dQ₃ heat is absorbed reversibly at temperature T₃ and dQ₄ heat is rejected reversibly at temperature T₄.

dQ₃/T₃+ dQ₄/T₄= 0

Similarly for complete reversible cycle, we will have following equation

The cyclic integral of dQ/T for a reversible process will be zero and that is the mathematical expression of Clausius theorem. We must note it here that above equation will be only valid for a reversible cycle.

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We will see another topic "Clausius inequality and explain its significance" in our next post in the category of thermal engineering.