Entropy and the Second Law
3. Reversible and Irreversible Processes
Learning outcomes
- I can define reversible processes.
- I can describe irreversible processes.
- I can compare their efficiencies.
- I can identify sources of irreversibility.
- I can explain why perfectly reversible processes are idealizations.
Reversible and Irreversible Processes
Thermodynamic processes can be classified as reversible or irreversible depending on whether the process can be reversed while restoring both the system and its surroundings exactly to their original states.
A reversible process is an idealized process that can be reversed by an infinitesimally small change without producing a net change in the system and surroundings.
An irreversible process produces a net increase in total entropy and cannot be completely reversed without leaving changes elsewhere.
The distinction is central to the Second Law of Thermodynamics and establishes the maximum possible efficiency of engines and other energy-conversion devices.
Reversible Processes
A perfectly reversible process is a theoretical process that can proceed:
forward or backward
through a sequence of states that remain extremely close to equilibrium.
If the process is reversed:
system returns to original state
and:
surroundings return to original state
with no net change anywhere.
For an ideal reversible process:
ΔS_total = 0
No entropy is produced.
Reversibility Does Not Simply Mean "Can Go Backward"
This distinction is important.
Suppose you compress a gas and later allow it to expand.
The gas may return to its original:
- pressure
- temperature
- volume
But this does not necessarily mean the original process was thermodynamically reversible.
For true reversibility:
both the system and surroundings must be restored exactly to their original conditions
If friction, heat dissipation or other entropy-producing effects occurred, the process was irreversible.
Conditions for a Reversible Process
A perfectly reversible process requires idealized conditions such as:
- no friction
- no turbulence
- no electrical resistance
- no uncontrolled expansion
- no mixing
- no diffusion
- no finite temperature differences during heat transfer
- no finite pressure differences during expansion or compression
The system must remain extremely close to:
thermodynamic equilibrium
throughout the process.
Quasi-Static Processes
A process that occurs extremely slowly is called:
quasi-static
The system passes through a continuous sequence of states that are very close to equilibrium.
For example, imagine a gas inside a cylinder with a piston.
If tiny weights are removed from the piston one at a time, the gas expands extremely slowly.
If each weight is infinitesimally small, adding it back could approximately reverse the process.
This approaches a reversible expansion.
Quasi-Static Does Not Automatically Mean Reversible
A slow process can still be irreversible.
For example, imagine moving a piston extremely slowly while substantial friction acts between the piston and cylinder.
The process is:
slow
but friction converts mechanical energy into internal energy.
Entropy is produced.
Therefore:
quasi-static + friction ≠ reversible
A reversible process must avoid all sources of entropy production.
Reversible Heat Transfer
Heat transfer can approach reversibility only when the temperature difference is extremely small.
Suppose one object is at:
300.000 K
and another is at:
300.001 K
Heat transfer between them could occur through an extremely small temperature difference.
As:
ΔT → 0
the process approaches reversible heat transfer.
Irreversible Heat Transfer
Now consider:
hot object = 500 K
cold object = 300 K
Heat naturally transfers:
500 K → 300 K
This occurs across a finite temperature difference.
The process produces entropy and is:
irreversible
The larger the temperature difference, the farther the system is from equilibrium.
Entropy and Reversibility
For a complete isolated system:
Reversible process
ΔS_total = 0
Irreversible process
ΔS_total > 0
This gives us an important way to distinguish the two.
Reversible processes do not generate entropy.
Irreversible processes:
produce entropy
Entropy Production
We can think of total entropy change as including entropy generated by irreversibility.
Sources of entropy production include:
- friction
- viscosity
- turbulence
- electrical resistance
- diffusion
- mixing
- chemical reactions
- heat transfer across finite temperature differences
- expansion through finite pressure differences
Greater irreversibility generally means:
greater entropy production
Friction
Friction is one of the most familiar sources of irreversibility.
Suppose a block slides across a rough floor.
Initially, it has:
organized kinetic energy
Friction transforms this into internal energy distributed among:
- the block
- the floor
- surrounding air
The energy is conserved.
However, the organized motion has become dispersed microscopic motion.
This produces entropy.
Why Friction Is Irreversible
Imagine trying to reverse the process.
The block and floor would need to spontaneously cool while their microscopic molecular motions coordinated themselves to launch the block backward.
This would represent a large spontaneous decrease in entropy.
It is overwhelmingly improbable.
Therefore:
friction makes processes irreversible
Free Expansion
Consider a gas confined to one side of an insulated container.
A partition is suddenly removed.
The gas expands into the empty region.
This is called:
free expansion
The gas spontaneously spreads throughout the larger volume.
The process is irreversible.
Why Free Expansion Is Irreversible
Before expansion:
gas confined to smaller volume
After expansion:
gas occupies larger volume
The final state has vastly more accessible microscopic arrangements.
Therefore:
entropy increases
The gas does not spontaneously return to one side of the container.
To compress it again requires external intervention and changes in the surroundings.
Diffusion
Suppose perfume is released in one part of a room.
The molecules spread throughout the air.
This is:
diffusion
The reverse process would require all the perfume molecules to spontaneously gather back into one small region.
Because the dispersed state corresponds to vastly more possible microscopic arrangements:
diffusion is irreversible
Mixing
Mixing is another major source of irreversibility.
Imagine two different gases separated by a partition.
Remove the partition:
gas A + gas B → mixed gases
The mixed state has many more possible microscopic arrangements.
The gases do not spontaneously separate again.
Therefore:
mixing produces entropy
and is generally irreversible.
Electrical Resistance
Electrical current flowing through a resistor produces:
Joule heating
Electrical energy becomes thermal energy distributed among microscopic particle motions.
The power dissipated is:
P = I²R
or:
P = VI
This energy dispersal produces entropy.
Therefore:
electrical resistance is a source of irreversibility
Viscosity
Viscosity is internal friction within a fluid.
When layers of fluid move past one another, mechanical energy can be transformed into internal energy.
Examples include:
- oil flowing through pipes
- air moving over aircraft
- water flowing through narrow channels
Viscosity therefore produces:
irreversibility
Turbulence
Turbulent fluid motion contains irregular swirling structures called:
eddies
Large-scale organized motion gradually transfers energy into smaller-scale motion and eventually into random molecular motion.
Therefore turbulence is strongly:
irreversible
Reducing turbulence can improve the efficiency of some engineering systems.
Inelastic Collisions
Consider two objects that collide and stick together.
Before:
organized kinetic energy
After:
some kinetic energy has become:
- internal energy
- sound
- deformation
The total energy is conserved, but the process cannot spontaneously reverse itself.
Therefore an inelastic collision is:
irreversible
Combustion
Burning fuel is another strongly irreversible process.
Chemical energy becomes:
- thermal energy
- radiation
- expanding gases
- chemical products
The products do not spontaneously collect the dispersed energy and reform the original fuel and oxygen.
Therefore:
combustion is highly irreversible
Reversible Compression and Expansion
Imagine an ideal gas in a frictionless piston.
If the external pressure differs from the gas pressure by only an infinitesimal amount, the piston can move extremely slowly.
For expansion:
P_gas ≈ P_external
A tiny decrease in external pressure causes expansion.
A tiny increase can reverse it.
This approaches:
reversible expansion
Irreversible Expansion
Suppose the gas pressure is much greater than the external pressure.
The piston accelerates rapidly outward.
The system is far from mechanical equilibrium.
This expansion is:
irreversible
A finite pressure difference is therefore another important source of irreversibility.
Pressure Differences
For a reversible mechanical process:
ΔP → 0
between the system and surroundings.
For an irreversible process:
finite ΔP
can drive rapid expansion or compression.
Therefore:
finite pressure difference → entropy production
Temperature Differences
Similarly, for reversible heat transfer:
ΔT → 0
For ordinary irreversible heat transfer:
finite ΔT
Therefore:
finite temperature difference → entropy production
This gives engineers an important strategy:
reducing unnecessary temperature differences can reduce irreversibility.
Reversible vs Irreversible
| Reversible | Irreversible |
|---|---|
| Idealized | Realistic |
| Infinitely slow limit | Can occur at finite rate |
| Very close to equilibrium | Often away from equilibrium |
| No friction | Friction may occur |
| No turbulence | Turbulence may occur |
| Infinitesimal temperature differences | Finite temperature differences |
| Infinitesimal pressure differences | Finite pressure differences |
| No entropy production | Produces entropy |
| ΔS_total = 0 | ΔS_total > 0 |
| Maximum theoretical efficiency | Lower efficiency |
Reversibility and Efficiency
Reversible processes are important because they establish the:
maximum theoretical efficiency
possible between specified conditions.
An irreversible process loses some ability to perform useful work because energy becomes more dispersed.
The energy itself is not destroyed.
Instead:
useful energy becomes less available for work
Energy Quality
Consider 100 J of organized mechanical energy.
In principle, much of it might be converted into another useful form.
Now suppose friction converts it into 100 J of low-temperature thermal energy distributed throughout the environment.
The total energy remains:
100 J
But recovering all 100 J as useful mechanical work is impossible in a cyclic process.
Thus irreversibility reduces:
energy quality
or, more precisely, the system's ability to perform useful work.
Heat Engines
A heat engine receives thermal energy from a hot reservoir.
It converts some into:
work
and rejects the rest to a colder reservoir.
For any engine:
Q_hot = W + Q_cold
Efficiency is:
η = W/Q_hot
Irreversibility reduces the amount of useful work that can be obtained.
The Reversible Heat Engine
An ideal reversible heat engine provides the maximum possible efficiency between two reservoir temperatures.
The theoretical maximum is given by the:
Carnot efficiency
η_max = 1 − T_c/T_h
where:
- T_c = cold reservoir temperature
- T_h = hot reservoir temperature
Temperatures must be in:
kelvin
Worked Example: Ideal Efficiency
Suppose:
T_h = 800 K
and:
T_c = 400 K
Then:
η_max = 1 − 400/800
η_max = 1 − 0.50
η_max = 0.50
Therefore:
maximum theoretical efficiency = 50%
No engine operating between these temperatures can exceed this efficiency.
Comparing Real and Ideal Engines
Suppose an ideal reversible engine has a maximum efficiency of:
50%
A real engine might achieve:
35%
Why is the real efficiency lower?
Possible sources of irreversibility include:
- friction
- combustion
- turbulence
- heat loss
- finite temperature differences
- exhaust losses
- mechanical resistance
Therefore:
real efficiency < reversible maximum
Worked Example: Real Engine
An engine absorbs:
2000 J
of thermal energy and produces:
700 J
of useful work.
Efficiency:
η = W/Q_hot
η = 700/2000
η = 0.35
Therefore:
η = 35%
If the theoretical reversible maximum were 50%, the difference reflects unavoidable and avoidable irreversibilities in the real system.
Carnot's Insight
Sadi Carnot showed that an ideal reversible engine operating between two temperatures has the greatest possible efficiency.
An important consequence is:
all reversible engines operating between the same two reservoir temperatures have the same maximum efficiency
The particular engine design does not change this fundamental limit.
Only:
T_h and T_c
determine the Carnot efficiency.
Why Real Engines Are Less Efficient
A real engine operates with:
- friction
- finite temperature differences
- finite pressure differences
- rapid fluid flow
- turbulence
- combustion
- electrical resistance
- heat leakage
All these processes produce entropy.
Therefore:
real engine → irreversible → lower efficiency
Improving Efficiency
Engineers cannot eliminate the Second Law.
However, they can reduce unnecessary irreversibility.
Strategies include:
- reducing friction
- improving lubrication
- reducing turbulence
- improving insulation
- optimizing combustion
- reducing electrical resistance
- recovering waste heat
- reducing unnecessary temperature differences
- improving aerodynamic design
This can move a real device closer to its theoretical maximum.
Refrigerators and Reversibility
Refrigerators use work to transfer thermal energy:
cold region → warm surroundings
Real refrigerators contain sources of irreversibility such as:
- compressor friction
- electrical resistance
- pressure losses
- heat transfer across finite temperature differences
A perfectly reversible refrigerator would provide the:
maximum theoretical performance
for given hot and cold temperatures.
Real refrigerators always perform below that ideal limit.
Power Stations
Power stations contain many irreversible processes.
Examples include:
- combustion
- heat transfer
- steam flow
- turbine friction
- generator resistance
- cooling
Reducing these irreversibilities increases the amount of electrical energy that can be produced from a given amount of fuel.
This has both:
economic
and:
environmental
benefits.
Cars
A car engine contains many sources of irreversibility:
- combustion
- friction
- hot exhaust
- cooling systems
- air resistance
- tire deformation
- electrical resistance
Reducing these losses improves fuel economy.
However, no design can completely eliminate all thermodynamic irreversibility.
Regenerative Braking
Traditional friction brakes transform kinetic energy into thermal energy.
This is highly irreversible.
Electric and hybrid vehicles can instead use:
regenerative braking
The electric motor operates as a generator and converts some kinetic energy into electrical energy stored in the battery.
This does not make the process perfectly reversible.
But it reduces the amount of useful energy that becomes dispersed as heat.
Therefore it improves overall efficiency.
Why Perfect Reversibility Requires Infinite Slowness
Suppose heat must move from one object to another.
To transfer heat at a useful rate, there must normally be a:
temperature difference
But a finite temperature difference causes irreversibility.
To approach reversible heat transfer:
ΔT → 0
As the temperature difference approaches zero, the heat-transfer rate also tends to become extremely small.
Similarly, reversible mechanical expansion requires:
ΔP → 0
which produces extremely slow motion.
Thus a perfectly reversible process would require an ideal limit of:
infinitely slow operation
This makes perfect reversibility impractical.
Maximum Efficiency vs Useful Power
This creates an important engineering trade-off.
A system operating extremely slowly may approach reversible efficiency.
But:
work/time → very small
so its:
power output → very small
Real machines must produce useful power in finite time.
Therefore engineers usually balance:
efficiency
against:
power, size, cost and practicality
Perfectly Reversible Processes Are Idealizations
Perfect reversibility requires impossible or impractical conditions such as:
- zero friction
- zero viscosity
- zero turbulence
- zero electrical resistance
- infinitesimal temperature differences
- infinitesimal pressure differences
- infinitely slow changes
- perfect insulation where required
Real physical systems cannot satisfy all these conditions exactly.
Therefore:
perfect reversible processes do not occur in practice
They are theoretical models.
Why Study an Impossible Ideal?
Reversible processes are extremely useful even though they are idealizations.
They provide a:
theoretical benchmark
If an ideal reversible engine could achieve 60% efficiency while a real engine achieves 30%, engineers know there is substantial room for improvement.
If the real engine already achieves 58%, further improvements are fundamentally limited.
Therefore reversible processes tell us:
the best that physics allows
Reversibility and the Second Law
The Second Law gives the fundamental distinction:
reversible: ΔS_total = 0
irreversible: ΔS_total > 0
An irreversible process produces entropy.
That entropy production represents a reduction in the maximum amount of useful work that can be obtained.
This is why reversibility and efficiency are closely related.
Everyday Example: Cooling Coffee
Hot coffee cools in a room because:
T_coffee > T_room
Heat transfers across a finite temperature difference.
Therefore the process is:
irreversible
The coffee will not spontaneously absorb the dispersed thermal energy and return to its original temperature while making the room colder.
Everyday Example: Braking
A bicycle uses friction brakes.
Kinetic energy becomes:
thermal energy in brake pads + wheel + surroundings
The bicycle stops.
The brakes do not spontaneously cool and accelerate the bicycle back to its original speed.
Therefore:
friction braking is irreversible
Everyday Example: Mixing Milk and Coffee
Add milk to coffee.
The liquids mix.
The mixed state corresponds to vastly more microscopic arrangements than the separated state.
The milk does not spontaneously separate itself from the coffee.
Therefore:
mixing is irreversible
Everyday Example: Inflating and Releasing a Balloon
Compressed air inside a balloon is released.
The air rapidly expands into the room.
This involves:
- pressure differences
- turbulence
- mixing
The process is highly:
irreversible
Returning the air to the balloon requires external work.
Everyday Example: Ice Melting
Ice placed in a warm room absorbs thermal energy across a finite temperature difference.
It melts.
Under the same environmental conditions, the water does not spontaneously freeze while transferring energy back exactly as before.
Therefore, the real melting process is:
irreversible
Sources of Irreversibility
The major sources worth recognizing are:
Friction
Converts organized motion into dispersed internal energy.
Heat transfer across finite temperature differences
Produces entropy as energy moves from hot to cold.
Unrestrained expansion
Gas expands through a finite pressure difference.
Mixing and diffusion
Particles become distributed among more possible arrangements.
Viscosity
Mechanical energy is dissipated within flowing fluids.
Turbulence
Organized fluid motion becomes irregular and eventually thermalized.
Electrical resistance
Electrical energy becomes thermal energy.
Chemical reactions
Processes such as combustion produce entropy.
Inelastic deformation
Mechanical energy becomes internal energy during deformation.
Reducing Irreversibility
Engineers try to reduce irreversibility by:
friction → better bearings and lubrication
heat loss → better insulation
turbulence → improved fluid-flow design
electrical resistance → appropriate conductors
waste heat → heat recovery
large temperature differences → staged heat exchange
poor combustion → improved combustion control
The goal is not perfect reversibility.
The goal is:
greater practical efficiency
Reversible vs Irreversible Energy Transfer
Consider two methods of lowering a heavy object.
Method A
Drop it.
Gravitational potential energy becomes:
- kinetic energy
- sound
- deformation
- thermal energy
Highly irreversible.
Method B
Lower it slowly using an ideal generator.
Much of the gravitational potential energy can be converted into:
electrical energy
This process can approach greater reversibility and recover more useful energy.
The example shows why reducing irreversibility improves useful energy recovery.
Energy Is Not Destroyed
When we call a process inefficient, it is important not to say:
energy disappeared
The First Law still applies.
Energy is conserved.
The problem is that some energy becomes dispersed in forms from which it is difficult or impossible to recover completely as useful work.
Therefore:
irreversibility reduces useful energy availability, not total energy
Common Mistakes
Mistake: Saying reversible means a process can simply happen backward.
True thermodynamic reversibility requires both the system and surroundings to return exactly to their initial states.
Mistake: Saying slow processes are automatically reversible.
A slow process can still contain friction, resistance or other irreversibilities.
Mistake: Saying irreversible means physically impossible to reverse.
A system can often be returned to its original state, but additional changes occur in the surroundings.
Mistake: Saying friction destroys energy.
Friction conserves energy but disperses it into internal energy and produces entropy.
Mistake: Saying reversible processes increase entropy.
An ideal reversible process produces no net entropy: ΔS_total = 0.
Mistake: Saying real processes can be perfectly reversible.
Perfect reversibility is an idealized limiting case.
Mistake: Saying reversible engines are 100% efficient.
A reversible engine has the maximum efficiency possible between its temperatures, but that efficiency is still below 100% when T_c > 0 K.
Mistake: Saying high efficiency means no waste heat exists.
Heat engines must reject some energy to a colder reservoir.
Mistake: Saying energy loss and energy destruction are the same thing.
"Lost" energy usually means energy that is no longer readily available for useful work.
Key Terms
Reversible process — ideal process that can be reversed while restoring both system and surroundings exactly to their initial states.
Irreversible process — process that generates entropy and cannot be completely reversed without additional changes.
Quasi-static process — process occurring sufficiently slowly that the system remains close to equilibrium.
Thermodynamic equilibrium — condition in which there are no net driving forces for macroscopic change.
Entropy production — generation of entropy during an irreversible process.
Friction — resistive interaction that converts organized mechanical energy into internal energy.
Viscosity — internal friction within a fluid.
Turbulence — irregular fluid motion involving eddies and energy dissipation.
Free expansion — spontaneous expansion of a gas into an available volume without controlled opposing pressure.
Finite temperature difference — non-zero temperature difference that drives irreversible heat transfer.
Finite pressure difference — pressure difference that drives irreversible mechanical motion or fluid flow.
Carnot engine — ideal reversible heat engine establishing the maximum possible efficiency between two temperatures.
Carnot efficiency — maximum theoretical heat-engine efficiency, η_max = 1 − T_c/T_h.
Energy quality — informal description of how readily energy can be converted into useful work.
Idealization — simplified theoretical model representing a limiting case rather than a perfectly realizable physical system.
Key Takeaways
- A reversible process can ideally be reversed while restoring both the system and surroundings exactly.
- An irreversible process produces a net increase in total entropy.
- For a reversible process, ΔS_total = 0.
- For an irreversible process, ΔS_total > 0.
- Most real processes are irreversible.
- A quasi-static process occurs extremely slowly and remains close to equilibrium.
- Slow operation alone does not guarantee reversibility.
- Friction is a major source of irreversibility.
- Heat transfer across a finite temperature difference is irreversible.
- Expansion through a finite pressure difference is irreversible.
- Diffusion and mixing are irreversible.
- Electrical resistance, viscosity, turbulence and combustion also produce irreversibility.
- Reversible processes provide the maximum theoretical efficiency.
- Irreversibility reduces the amount of energy available for useful work.
- Energy is still conserved during irreversible processes.
- Reversible heat engines establish the maximum efficiency possible between two temperatures.
- Carnot efficiency is η_max = 1 − T_c/T_h.
- A reversible heat engine is not necessarily 100% efficient.
- Real engines have lower efficiencies than corresponding reversible engines.
- Engineers improve efficiency by reducing sources of irreversibility.
- Perfect reversible processes require ideal conditions such as zero friction and infinitesimal temperature and pressure differences.
- Approaching perfect reversibility generally requires extremely slow operation.
- Extremely slow operation produces very little practical power, creating an efficiency-power trade-off.
- Perfectly reversible processes are therefore theoretical idealizations rather than real processes.
- Despite being idealizations, reversible processes are valuable because they establish the best performance permitted by thermodynamics.
Check Your Understanding
- Define a reversible process.
- Define an irreversible process.
- What condition must be satisfied for both the system and surroundings in a reversible process?
- What is the total entropy change for an ideal reversible process?
- What is the total entropy change for an irreversible process?
- What is meant by a quasi-static process?
- Explain why a slow process is not necessarily reversible.
- Why does friction make a process irreversible?
- Explain why heat transfer across a finite temperature difference is irreversible.
- Why does free expansion of a gas increase entropy?
- Explain why mixing two gases is an irreversible process.
- Identify four different sources of irreversibility.
- How does electrical resistance produce irreversibility?
- Explain why turbulence can reduce the efficiency of a machine.
- Compare the efficiency of an ideal reversible engine with a real irreversible engine operating between the same temperatures.
- An ideal engine operates between 700 K and 350 K. Calculate its maximum theoretical efficiency.
- Why does reducing friction improve the efficiency of a real machine?
- Explain why perfectly reversible processes would have to occur extremely slowly.
- Why are perfectly reversible processes useful even though they cannot be achieved exactly?
- Explain the statement: "Irreversibility does not destroy energy, but it reduces the amount of energy available for useful work."