Clausius Statement: Proof, Example, and Applications

Clausius Statement states that “It is practically impossible for any system to work in such a way that the sole result would be a heat transfer from a cooler to a hotter body”.

• Heat is a form of energy that naturally travels from a hotter region to a colder region.
• If the surroundings or system or both accomplish the heat transfer, a cooler body can transfer heat to a hotter body.
• Refrigerators and air conditioners are devices that are designed to transfer heat from a cold place to its warmer surroundings.
• However, in both circumstances, they require power.
• Clausius's statement states that an air conditioner will not cool a room without power.
• We cannot construct a refrigerator or AC that does not require any work as input.

Key Terms: Second law of thermodynamics, Work, Heat, Clausius Theorem, Thermodynamics, Heat engine, Carnot cycle, Entropy, Enthalpy

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Who was Rudolf Clausius?

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A German mathematician and scientist, Rudolf Julius Emanuel Clausius was one of the most important pioneers of thermodynamics science.

• Rudolf Clausius proposed a few fundamental concepts, one of which is the theory of heat, which is a restatement of the Carnot cycle.
• He wrote a paper on "The Moving Force of Heat" in 1850.
• He introduced entropy in 1865 and the virial theorem in 1870.

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Clausius Statement of Second Law of Thermodynamics

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In the year 1850, Clausius published one of the earliest statements on the Second Law of Thermodynamics.

According to Clausius's statement,

"It is impossible to construct a mechanism that runs on a cycle and produces no other effect than heat transfer from a cold body to a hot one". 

Heat cannot flow spontaneously from a cold system to a hot system unless the system is subjected to external work.

• This is exactly what refrigerators and heat pumps do.
• In a refrigerator, heat travels from cold to hot only when driven by external work.
• Refrigerators are driven by electric motors that require work from the surroundings to function.

According to Clausius, the algebraic sum of each transformation that occurs throughout a cyclic process must be less than zero and, in extreme situations, equal to nothing.

\(\oint \frac {\delta Q}{T}=0\)

Where

• δQ is energy input into the system due to heating
• T represents the absolute temperature of the body whenever that energy is absorbed, and is discovered to be accurate for any cyclic and reversible process.

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Clausius Statement Proof

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Consider the diagram shown below

The diagram represents a system, having gas in the piston.

• To add heat Q_sys to the system at a local, variable temperature T, we use a Carnot heat engine/pump.
• Work W_sys is done on the system during the process.
• Q_sys and W_sys can both be positive or negative.
• The Carnot engine's cold reservoir is at temperature T₀.

According to the Kelvin-Planck statement of the second law, we cannot have extracted net work at the end of a complete cycle of the system (or else we would have converted heat from the cold reservoir into work). Looking at the diagram to understand how the signs of the various works are defined, indicates 

W_carnot + W_sys > 0

Because of energy conservation and since the system and engine have returned to their initial states, any net work done must result in heat transfer to the reservoir.

Q_res ≤ 0

Looking at the Carnot engine now, we can see that if we add heat dQ_sys at temperature T at some point throughout the cycle, heat dQ_res is extracted from the reservoir, and

dQ_res = (T₀/T) dQ_sys

The total heat extracted is less than zero, therefore we have

\(Q_{res}=\oint dQ_{res} =\oint \frac {T_0}{T} \:dQ_{sys} = T_0 \oint \frac {dQ_{sys}}{T}\leq0\)

This proves the inequality

\(\oint \frac {dQ}{T} \leq0\)

Clearly, if the system is driven through a reversible cycle, all quantities will simply change signs. However, if Q was less than zero before, it would be greater after the reversed cycle, indicating a net extraction of work and violating the Kelvin-Planck statement. Therefore Q must be exactly zero for a reversible system, and

\(\oint \frac {dQ_{rev}}{T}=0\)

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Clausius Statement Example

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Heating a cup of coffee is a basic example.

• The coffee starts heating at a lower temperature than the surroundings.
• It will gain heat spontaneously from warmer surroundings until it reaches thermal equilibrium with the environment, at this point no more heat transfer will occur.

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History of Clausius Statement

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In the early nineteenth century, scientists investigated the concept of energy and how it was transported and converted.

• This led to the development of the first law of thermodynamics, which states that energy is conserved.
• However, it was discovered that not all energy is useful and capable of being turned into work.
• Natural processes involving heat and work have limitations and irreversible properties.
• In the 1850s, German scientist Rudolf Clausius developed the concept of entropy to measure the tendency of energy to decay into useless forms.
• Clausius proposed his famous statement of the second law based on this concept of entropy. 
• Because natural processes tend to produce entropy and disorder, Clausius' statement explains why perpetual motion machines and technologies that violate the second law are impossible.
• Simultaneously, William Thomson (Lord Kelvin) and others developed analogous statements of the second law, focusing on the limitations of thermal machines and devices.
• However, Clausius' statement in terms of heat flow and temperature was seen as the most fundamental expression of the second law, containing its insight into the irreversible nature of nature.​​

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Clausius Statement Application

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Natural observations show that spontaneous heat transfer happens only in one direction: from a hot to a cold place.

• A refrigerator is an engineering application of Clausius’ statement of the second law.
• Heat should be transferred from the inside to the surrounding region using a compressor to keep the interior of a refrigerator cold. 
• Thomson and Clausius used the Carnot cycle to formulate the second law.
• This led to the development of the concept of entropy as a new thermodynamic property.

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Things to Remember

• Heat is a form of energy.
• Heat energy always flows from a hotter body to a colder body.
• Clausius published one of the earliest statements on the Second Law of Thermodynamics in 1850.
• According to Clausius's statement, it is impossible to transfer heat from a cooler body to a hotter body.
• Refrigerators are driven by electric motors that require work from their surroundings to function.
• Refrigerators and air conditioners are equipment that is designed to transfer heat from a cold environment to its heated surroundings.
• According to the Kelvin-Planck statement of the second law, we cannot have extracted net work at the end of a complete cycle of the system.