Second Law of Thermodynamics
The first law of thermodynamics establishes that energy is conserved: it can be transformed from one form to another, but it cannot be created or destroyed.
This quantitative framework, discussed in our article on the First Law and Joule’s Experiment, is essential for understanding energy balances in any physical or industrial system.
The second law instead introduces this fundamental limitation. It defines the direction in which processes naturally occur, sets limits on the efficiency of energy conversion, and explains why perpetual motion machines of the second kind are impossible.
Entropy and the Direction of Natural Processes
The Second Law of Thermodynamics can be expressed through several equivalent statements. Kelvin emphasized the limits of converting heat into work, Clausius focused on the direction of heat transfer, and Carnot defined the maximum theoretical efficiency of any heat engine. Although these formulations differ in appearance, they describe the same physical principle: natural processes have a preferred direction, and no cyclic device can completely convert heat into work.
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Continue to Access the PDFThis principle leads to the definition of entropy, a property that captures these inherent limitations.
Unlike energy—which is conserved according to the First Law—entropy tends to increase in real processes. This increase is the signature of irreversibility.
Every spontaneous transformation involves finite temperature differences, friction, mixing, turbulence, viscosity, or non-equilibrium effects. These mechanisms generate entropy and make the process impossible to reverse without external intervention.
Typical examples include heat flowing spontaneously from a hotter body to a colder one, a gas expanding to fill a larger volume, or mechanical energy degrading into low-grade heat.
For a reversible process, entropy is defined as: dS=δQ/TdS
where
- dS is the infinitesimal entropy change,
- δQrev is the reversible heat exchanged,
- T is the absolute temperature.
In irreversible processes—essentially all real processes—the total entropy change of the system and surroundings satisfies: ΔStotal>0 and, in differential form: dS>δQ/TdS.
The entropy of an isolated system can remain constant (reversible limit) or increase, but never decrease. This is a universal observation: systems naturally evolve toward states of higher probability, which correspond to higher entropy.
This principle sets a fundamental limitation on energy use.
Even though energy is conserved, only part of it remains available to perform useful work; the rest is inevitably degraded into forms that cannot be reconverted into mechanical energy.
Universal Form of the Second Law
For any real process, reversible or irreversible, the Second Law can be written in its most general differential form: dS≥δQ/T
The equality applies to reversible transformations, while the strict inequality applies to irreversible ones. This compact expression captures the essence of the Second Law: every real process generates entropy, and only an ideal reversible process achieves the lower bound.
Clausius Inequality
Among the equivalent formulations of the Second Law, the most general mathematical form is the Clausius Inequality: ∮δQ/T ≤ 0
It states that in any cyclic process the total entropy change cannot be negative.
Equality holds only for reversible cycles; real cycles always satisfy the strict inequality.
This result directly rules out any machine that would convert all the heat from a single reservoir into work.
Perpetual Motion of the Second Kind
A perpetual motion machine of the second kind is a hypothetical device that would convert heat from a single reservoir entirely into work, operating indefinitely without rejecting heat to a colder sink.
At first sight, such a device does not violate the First Law: the heat absorbed would equal the work produced. The violation appears when entropy is considered.
Withdrawing heat from a reservoir decreases its entropy; if no heat is discharged elsewhere, there is no compensating increase. The total entropy would decrease, contradicting the Second Law.
For this reason, the Kelvin–Planck statement declares:
No cyclic device can transform all the heat absorbed from a single reservoir entirely into work.
Every real heat engine must therefore operate between two reservoirs—a hot source and a cold sink—and must reject part of the absorbed heat to satisfy the entropy balance. A machine that tries to avoid this is not only impractical: it is fundamentally impossible.
Conclusion
The Second Law of Thermodynamics goes beyond energy conservation. It defines the direction of natural processes and sets the limits of efficiency in every fundamental transformation. While the First Law tells us that energy is never lost, the Second Law reminds us that not all energy remains usable.
Ing. Ivet Miranda
⬆️ Back to TopSecond Law of Thermodynamics – Quiz
Which statement correctly describes a Perpetual Motion Machine of the Second Kind?
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- Work in Thermodynamics. PV Diagrams
FAQ
Why is the Second Law important in industrial plants?
Because it sets the maximum achievable efficiency of energy conversion systems. Steam turbines, compressors, and refrigeration units are all constrained by entropy increase. For example, no turbine can convert all heat into work — part of the energy must always be rejected as low-grade heat.
What are the two classical statements of the Second Law?
Clausius statement: Heat cannot spontaneously flow from a colder body to a hotter one.
Kelvin–Planck statement: No cyclic machine can convert all the heat absorbed from a single reservoir entirely into work.
How does entropy affect process design?
Entropy analysis helps engineers identify irreversibilities (pressure drops, heat losses, friction) and improve efficiency. For instance, in heat exchangers, a minimum temperature difference must always exist; otherwise, heat transfer would stop.
What is the difference between a reversible and an irreversible process?
A reversible process is an idealized transformation that occurs infinitely slowly, with the system always in equilibrium with its surroundings. Entropy change can be exactly calculated as dS=δQrev/T.
In reality, almost all processes are irreversible: friction, heat transfer across finite temperature differences, mixing, and rapid expansions all increase entropy and make the process one-way.
Why does irreversibility matter in industrial systems?
Irreversibility sets the practical efficiency limits of real equipment. For example:
A turbine cannot reach the Carnot efficiency because of friction and pressure drops.
Heat exchangers always require a temperature difference to transfer heat, which produces entropy.
Compressors and pumps consume extra power due to mechanical losses.
These irreversibilities explain why no industrial system can convert energy with 100% efficiency.