Clausius’ Equivalence of Transformations

Clausius wrote about the equivalence of two different types of transformations, each involving heat and work. He took as an assumption that work could be converted to heat and vice-versa, that heat could be converted to work.

Specifically, he equated a process that converts heat to work with a separate process that converts heat from a high temperature to a low temperature. The conversion of heat to work is one transformation. The conversion of heat from a high temperature to a low temperature is a second transformation.

Clausius asserts that these two processes are equivalent transformations. His assertion of the equivalence of these two transformations is the essence of his definition of the 2nd law of thermodynamics.

Heat Engine

A generic heat engine performs both transformations. It will

1. transform some input heat to work
2. transfer some input heat from the hot source to the cold source

Diagram showing heat flows for a generic heat engine

Heat flows for a Carnot cycle

Transformation 1 — conversion of Heat to Work

The transformation of some of the input heat to work will produce the first equivalence

• Equivalence 1 = (Qh-Qc)/Th=W/Th

Note that this quantity uses the total heat converted to work in the numerator, divided by the temperature of the engine or hot bath, Th.

Transformation 2 — transfer of heat from a high temperature to a low temperature

The second transformation has to do with the transfer of some of the input heat from a high temperature to a low temperature. The heat that is transferred from a high temperature to a low temperature is the heat that does not do any work. By contrast, the heat that does contribute to work is consumed in the heat engine and is not transferred from a high temperature to a low temperature but is completely converted to work.

The amount of heat transferred from a high temperature bath to a low temperature bath is Qc. Qc is the amount of heat transferred from the hot source to the cold source and added into the cold reservoir. This amount of heat Qc does not do any work. This second equivalence is then

• Equivalence 2 = -Qc (1/Th — 1/Tc)

Equality of Transformations

Per Clausius, the two transformations are equal to each other

• Transformation 1 = Transformation 2
• W/Th = -Qc (1/Th — 1/Tc)

Clausius’s statement requires that the impact of the first transformation be exactly offset by the impact due to the second transformation, where the impact is defined as heat transformed divided by the temperature of at which the transformation occurs.

This relation can be brought into a more familiar form by noting that the work, W, is

• Qh -Qc = W

Hence, substituting this relation for work W into the above equivalence equation leads to

• (Qh -Qc)/Th = -Qc (1/Th- 1/Tc)

And, finally,

• Qh/Th = Qc/Tc

Equality of Transformations and Entropy Conservation

Asserting the equivalence of transformations across the engine cycle

• Equivalence 1 = (Qh-Qc)/Th=W/Th
• Equivalence 2 = -Qc (1/Th — 1/Tc)

led to the assertion that a new quantity — the entropy — was conserved across the cycle

• Qh/Th = Qc/Tc

This final relation shows that the entropy loss due to moving Qh out of the hot bath at Th is compensated for by the entropy gain due to adding Qc to the cold bath, at Tc. Entropy is conserved across the cycle.

When will the transformations be equal? When will entropy be conserved?

The net entropy across an engine cycle will be zero when there are no frictional forces present. When there are no frictional forces present, the entropy increase in one part of the cycle is exactly compensated for by an entropy decrease elsewhere in the cycle. When there are no frictional forces present, a cycle is reversible. These are all equivalent statements.

• Entropy conservation
• Lack of frictional forces
• Equality of transformations

Entropy is conserved for a cycle that is free of frictional or non-conserved forces. Such friction free cycles are reversible.

Entropy Analysis — Engine, Environment, Universe

Clausius’s statement applies to the parts as well as to the whole.

Let’s continue with the example from above. It contains three systems :

• system1 is the engine
• system 2 is the environment and contains the two baths, one hot, one cold
• system 3 is the universe and contains systems 1 and 2

Can it be shown that entropy is conserved for each system separately?

In other words, can it be shown that the transformations that impact the environment are equal to the transformations that impact the engine?

Yes. The analysis follows the same lines as above but more carefully labels the contributions as they exit or enter each system in turn. The quantities are all the same but differ only by a sign as the entropy crosses a system boundary.

From Heat flow to Entropy flow

First, consider the two heat flows from the hot bath, Qc and W. The hot bath emits Qh. Qh consists of Qc and W.

Recall that W=Qh-Qc; hence, Qh=W+Qc.

Qc flows from the hot bath to the cold bath and does no work. W also flows from the hot bath to the engine and is 100% converted to work in the engine.

Second, update the heat flow diagram to reflect entropy flows. The diagram below shows the entropy transfers due to moving heat Qc from a high temperature to a low temperature (transformation 1). It also shows the entropy transfer converting an amount of heat W to work (transformation 2) with 100% efficiency. The signs in the diagram reflect the impact on the environment eg

• -Qc/Th indicates that entropy has left the hot bath
• -W/Th indicates that heat W has left the hot bath
• +Qc/Tc indicates that heat Qc has entered the cold bath

Entropy flows — signs are with respect to environment as system of focus eg hot and cold baths

Transformation 1 — conversion of heat to work

The heat-to-work transformation (top right, above) captures the flow of entropy out of the environment and into the engine for the amount of heat,W=Qh-Qc, that does work.

• -W/Th = impact of transformation 1 on environment e.g. entropy is leaving the hot bath
• +W/Th = impact of transformation 1 on engine e.g. entropy is entering the engine

The entropy that is flowing out of the environment and into the engine is the same entropy, differing only by the sign associated with the quantity.

Note — a transformation of type 1, conversion of heat to work, decreases the entropy of the environment and increases the entropy of the engine. It’s the same entropy, flowing across system boundaries; as expected, the impact on the environment is the negative of the impact on the engine.

Also note that all of the entropy, W/Th, available to do work does do work. No amount of energy is lost in the engine or in transit to the engine to friction.

Transformation 2 — transfer of heat from a high temperature to a low temperature

The second transformation (top center, above) captures the flow of entropy out of the hot bath and into the cold bath for the amount of heat Qc, which does no work. This transformation consists of two parts — one part at a high temperature and one part at a low temperature. The high temperature part has

• -Qc/Th = impact of transformation 2 on hot bath e.g. entropy is leaving the hot bath
• +Qc/Th = impact of transformation 2 on engine e.g. entropy is entering the engine

And the low temperature part has

• +Qc/Tc = impact of transformation 2 on cold bath e.g. entropy is entering the cold bath
• -Qc/Tc = impact of transformation 2 on engine e.g. entropy is leaving the engine

Hence, the change in entropy due to Qc going from a hot source to a cold source is then

• -Qc (1/Th — 1/Tc) = impact of transformation on environment >0
• +Qc (1/Th — 1/Tc) = impact of transformation on engine < 0

Note — transformation of type 2, transfer of heat from a high temperature to a low temperature, increases the entropy of the environment and reduces the entropy of the engine. It’s the same entropy, flowing across system boundaries; as expected, the impact on the environment is the negative of the impact on the engine.

Net Impact on Environment

The net impact of both transformations on the environment is then

• -W/Th = impact of transformation 1
• -Qc(1/Th-1/Tc) = impact of transformation 2

Note that the sign of transformation 1 is negative, indicating a loss of entropy for the environment.

Using the fact that W=Qh-Qc and adding the two impacts together yields the net impact on the environment

• -W/Th -Qc(1/Th-1/Tc)
• -(Qh-Qc)/Th -Qc(1/Th-1/Tc)
• -Qh/Th +Qc/Th — Qc/Th +Qc/Tc
• -Qh/Th + Qc/Tc = Net change in entropy on environment

Net Impact on the engine

The net impact of both transformations on the engine is then just

• +W/Th = impact of transformation 1
• +Qc(1/Th-1/Tc) = impact of transformation 2

Note the sign of the second transformation is positive, since it reflects the transit of Qc into and out of the engine.

Using the fact that W=Qh-Qc and adding the two impacts together yields the net impact on the engine

• +W/Th +Qc(1/Th-1/Tc)
• +(Qh-Qc)/Th +Qc(1/Th-1/Tc)
• +Qh/Th -Qc/Th + Qc/Th -Qc/Tc
• +Qh/Th — Qc/Tc = Net change in entropy on the engine

Net impact on Universe

Self-evidently, the impact on the engine is the negative of the impact on the environment.

Hence, the net impact on the universe is zero

• change in entropy of engine + change in entropy of environment = 0

In the case where all forces are conserved, and, hence, the cycle is reversible, entropy is conserved for every system. The net change in entropy on the universe is the sum of the change in entropy on all systems in the universe, which in this case, sums to zero.

Clausius’s Equivalence of transformations

Clausius’s transformations capture the entropy change due to two main processes

1. Conversion of heat to work or work to heat
2. Transfer of heat from a high temperature to a low temperature

Clausius’s statement of the 2nd law is that the sum of all type 1 and type 2 transformations across the cycle of a (reversible) engine is zero.

The following are equivalent statements

• transformations of type 1 and 2 are equal
• entropy is conserved
• process is reversible
• only conserved forces apply

If the Carnot cycle is broken up into type 1 and type 2 transformations and if these transformations are added up across the cycle, paying careful attention to defining the impacts as being on the system or on the environment, the net impact on the system or, alternatively, on the environment, is zero.

Can the Carnot cycle be analyzed in terms of Clausius’ transformations? Yes — check out this blog to see how.