Entropy and the Second Law
1. Entropy
Learning outcomes
- I can define entropy.
- I can relate entropy to disorder and probability.
- I can explain why entropy tends to increase.
- I can compare high- and low-entropy systems.
- I can identify processes that increase entropy.
Entropy
Entropy is a thermodynamic quantity related to how energy and matter can be distributed among the possible microscopic arrangements of a system.
It is commonly introduced as a measure of disorder, but a more accurate description is:
higher entropy → more possible microscopic arrangements and more widely dispersed energy
lower entropy → fewer possible microscopic arrangements and more concentrated or constrained energy
Entropy is represented by:
S
and its SI unit is:
J/K
Entropy is fundamental to understanding why many natural processes occur spontaneously in one direction but not the reverse.
Macroscopic and Microscopic Descriptions
Thermodynamics describes systems using large-scale quantities such as:
- temperature
- pressure
- volume
- internal energy
- entropy
These are macroscopic properties.
At the microscopic level, however, matter consists of enormous numbers of particles.
Each particle can have:
- position
- velocity
- energy
- direction of motion
A particular microscopic arrangement of the particles is called a:
microstate
Microstates
A microstate is one specific way the particles and energy of a system can be arranged while still producing the same overall macroscopic state.
For example, imagine four particles distributed between two boxes.
One possible arrangement is:
all four particles in the left box
Another is:
two particles in each box
There are many more individual ways to arrange four particles with two on each side than to place all four on one side.
Therefore:
evenly distributed states usually have more microstates
and therefore:
higher entropy
Entropy and Probability
The relationship between entropy and the number of possible microstates is expressed by:
S = k_B ln Ω
where:
- S = entropy
- k_B = Boltzmann constant
- Ω = number of accessible microstates
The Boltzmann constant is:
k_B ≈ 1.38 × 10⁻²³ J/K
This equation provides a statistical interpretation of entropy.
Boltzmann's Equation
Consider:
S = k_B ln Ω
If Ω increases:
S increases
Therefore:
more accessible microstates → greater entropy
This explains why entropy is closely connected to:
probability
A macrostate that can be produced by an enormous number of microscopic arrangements is much more probable than one that can occur in only a few ways.
A Simple Example
Imagine four gas particles.
Situation A
All four particles happen to be in one small corner.
There are relatively few arrangements corresponding to this condition.
Therefore:
low probability → low entropy
Situation B
The particles are spread throughout the container.
There are vastly more arrangements corresponding to this condition.
Therefore:
high probability → high entropy
Why Gas Spreads Out
Suppose a gas initially occupies only the left half of a container.
A partition is removed.
The gas spontaneously spreads throughout the entire container.
Why?
There are vastly more microscopic arrangements in which the molecules are spread throughout the container than arrangements in which every molecule remains confined to one side.
Therefore:
spread-out state → overwhelmingly more probable
This is why the gas naturally expands.
Could the Gas Return to One Side?
Quantum and classical microscopic laws do not simply say that gas molecules are forbidden from all moving into one half of the container simultaneously.
But for a macroscopic number of molecules, the probability is unimaginably small.
A typical sample may contain around:
10²³ particles
The number of possible microscopic arrangements is enormous.
Therefore, spontaneous concentration of all those particles into one corner is effectively never observed.
Entropy gives a statistical explanation for the direction of the process.
Entropy and Disorder
Entropy is often described as:
a measure of disorder
This can be useful as a first approximation.
For example:
ordered crystal → relatively low entropy
gas → relatively high entropy
However, "disorder" can sometimes be misleading.
Entropy is more fundamentally related to:
- number of accessible microstates
- probability
- dispersal of energy
So instead of simply remembering:
entropy = disorder
it is better to think:
entropy measures how many microscopic arrangements are compatible with the system's macroscopic condition
Low-Entropy Systems
A low-entropy system has relatively few accessible microscopic arrangements.
Examples can include:
- particles confined to a small region
- highly ordered crystals
- energy concentrated in one region
- gases compressed into small volumes
- systems with large temperature differences
Low entropy often corresponds to:
greater constraint
or:
greater concentration of energy or matter
High-Entropy Systems
A high-entropy system has many accessible microscopic arrangements.
Examples include:
- gas spread throughout a container
- mixed substances
- energy distributed throughout a system
- thermal equilibrium
- melted or vaporized materials
High entropy often corresponds to:
greater dispersal of matter and energy
Comparing Solid, Liquid and Gas
For the same substance under comparable conditions, a useful general trend is:
solid → liquid → gas
with entropy generally increasing.
Solid
Particles are strongly constrained around fixed positions.
Relatively fewer microstates are available.
Liquid
Particles can move past one another.
More arrangements are possible.
Gas
Particles can move throughout a much larger volume.
A vastly greater number of arrangements are available.
Therefore, generally:
S_solid < S_liquid < S_gas
Melting and Entropy
During melting:
solid → liquid
Particles become less constrained and can occupy more possible arrangements.
Therefore:
entropy increases
For the substance:
ΔS > 0
during melting.
Freezing and Entropy
During freezing:
liquid → solid
The particles become more constrained.
The entropy of the substance therefore:
decreases
However, this does not violate the Second Law of Thermodynamics because heat is released to the surroundings.
The total entropy change must include both:
system + surroundings
Vaporization and Entropy
During vaporization:
liquid → gas
particles become much more widely distributed and gain access to many more microscopic arrangements.
Therefore:
entropy increases substantially
Vaporization is usually associated with a large positive entropy change for the substance.
Condensation and Entropy
During condensation:
gas → liquid
particles become more constrained.
Therefore, the entropy of the condensing substance:
decreases
Again, heat is transferred to the surroundings, so the entropy change of the entire system plus surroundings must be considered.
Mixing and Entropy
Suppose two different gases are initially separated.
Remove the partition.
The gases mix spontaneously.
The mixed state has many more possible arrangements than the separated state.
Therefore:
mixing generally increases entropy
This helps explain why gases naturally mix but do not spontaneously separate again.
Diffusion and Entropy
Diffusion is another example.
Suppose perfume molecules are concentrated in one region of a room.
Over time they spread throughout the air.
Initially:
molecules concentrated → fewer arrangements
Later:
molecules dispersed → more arrangements
Therefore:
diffusion increases entropy
This connects entropy with processes already familiar from particle theory.
Heat Transfer and Entropy
Consider two objects:
hot object + cold object
When they are placed in thermal contact:
energy transfers from hot → cold
until thermal equilibrium is reached.
The total energy is conserved.
But its distribution changes.
Initially, energy is concentrated more strongly in the hotter object.
Eventually, energy becomes more evenly distributed.
The final equilibrium state corresponds to:
greater total entropy
Entropy and Thermal Equilibrium
At thermal equilibrium:
T_hot = T_cold
There is no net heat transfer.
For an isolated system, equilibrium corresponds to the overwhelmingly most probable macroscopic state and, under the usual constraints, the state of maximum entropy.
Therefore, entropy helps explain why:
heat flows naturally from hot objects to cold objects
rather than the reverse.
The Second Law of Thermodynamics
The tendency of entropy to increase is expressed by the:
Second Law of Thermodynamics
For an isolated system:
ΔS_total ≥ 0
This means:
- entropy increases in an irreversible spontaneous process
- entropy remains constant in an ideal reversible process
- total entropy does not spontaneously decrease in an isolated system
This law gives thermodynamic processes a preferred direction.
Why Entropy Tends to Increase
Entropy increases because high-entropy macrostates correspond to vastly more possible microstates.
Imagine throwing many coins.
There is only one way to obtain:
all heads
But there are enormous numbers of arrangements giving approximately:
half heads and half tails
Therefore, the approximately equal mixture is much more probable.
A thermodynamic system behaves similarly.
It naturally evolves toward macroscopic states corresponding to the largest numbers of microscopic arrangements.
Probability, Not a Mysterious Force
Entropy does not increase because there is an "entropy force" deliberately making systems disordered.
Instead:
high-entropy states are overwhelmingly more probable
For systems containing enormous numbers of particles, the probability difference becomes extraordinary.
This statistical tendency appears to us as a reliable law of nature.
Entropy Change
Entropy change is represented by:
ΔS
where:
ΔS = S_final − S_initial
If:
ΔS > 0
entropy increases.
If:
ΔS < 0
entropy decreases.
If:
ΔS = 0
there is no net entropy change.
Entropy and Reversible Heat Transfer
For an ideal reversible transfer of heat at constant absolute temperature:
ΔS = Q_rev/T
where:
- ΔS = entropy change
- Q_rev = reversibly transferred thermal energy
- T = absolute temperature in kelvin
The unit is:
J/K
This equation connects entropy with thermal energy transfer.
Worked Example: Entropy Change
Suppose:
600 J
of thermal energy is reversibly transferred into a system at:
300 K
Use:
ΔS = Q/T
ΔS = 600/300
ΔS = 2.0 J/K
The entropy of the system increases by:
2.0 J/K
Worked Example: Energy Leaving a System
Suppose a system loses:
400 J
of heat reversibly at:
200 K
For the system:
Q = −400 J
Therefore:
ΔS = Q/T
ΔS = −400/200
ΔS = −2.0 J/K
The entropy of the system decreases.
This is possible because the surroundings receive the energy.
The Second Law concerns the:
total entropy of system + surroundings
Why Heat Flows from Hot to Cold
Suppose an amount of heat Q moves from a hot object at T_h to a cold object at T_c.
The hot object loses entropy:
ΔS_hot = −Q/T_h
The cold object gains entropy:
ΔS_cold = +Q/T_c
Therefore:
ΔS_total = Q/T_c − Q/T_h
Since:
T_h > T_c
then:
1/T_c > 1/T_h
so:
ΔS_total > 0
Therefore, spontaneous heat transfer from hot to cold increases total entropy.
Worked Example: Hot to Cold
Suppose:
100 J
of heat moves from an object at:
400 K
to an object at:
300 K
Hot object:
ΔS_hot = −100/400
ΔS_hot = −0.25 J/K
Cold object:
ΔS_cold = +100/300
ΔS_cold ≈ +0.333 J/K
Total:
ΔS_total = 0.333 − 0.25
ΔS_total ≈ +0.083 J/K
Total entropy increases.
Therefore, the process is consistent with the Second Law.
Why the Reverse Does Not Happen Naturally
Imagine 100 J spontaneously moving from the:
300 K object → 400 K object
Then:
Cold object:
ΔS = −100/300 ≈ −0.333 J/K
Hot object:
ΔS = +100/400 = +0.25 J/K
Total:
ΔS_total ≈ −0.083 J/K
This would decrease the total entropy of an isolated system.
Therefore, it does not occur spontaneously.
Refrigerators and Entropy
A refrigerator appears to move heat in the "wrong" direction:
cold region → warm surroundings
Does this violate the Second Law?
No.
A refrigerator requires:
external work
Electrical energy operates the compressor.
The refrigerator transfers heat from the cold interior while releasing even more energy to the warmer surroundings.
When the entire system is considered:
total entropy still increases
Local Entropy Can Decrease
The Second Law does not say that entropy must increase everywhere.
Entropy can decrease in one location.
Examples include:
- water freezing
- crystals forming
- refrigerators cooling food
- organisms building complex structures
But these processes transfer energy and entropy to their surroundings.
The important quantity is:
total entropy change
for the complete system and surroundings.
Entropy and Living Organisms
Living organisms create highly organized structures.
This might seem to contradict entropy increase.
It does not.
Organisms are:
open systems
They exchange both matter and energy with their surroundings.
They maintain internal organization by:
- consuming energy
- carrying out chemical reactions
- releasing thermal energy
- producing waste
The total entropy of the organism plus environment can still increase.
Irreversible Processes
Many everyday processes are:
irreversible
Examples include:
- friction
- diffusion
- gas expansion
- mixing
- heat transfer across a temperature difference
- combustion
- breaking objects
These processes naturally proceed in one direction and increase total entropy.
Friction and Entropy
When an object slides across a rough surface, organized kinetic energy is transformed into internal energy.
Initially:
energy concentrated in coordinated motion
After friction:
energy distributed among random microscopic motions
The energy has not disappeared.
It has become more dispersed.
Therefore:
friction increases entropy
Gas Expansion
Suppose a gas expands freely into an empty region.
Before:
particles occupy smaller volume
After:
particles occupy larger volume
The larger volume allows vastly more possible particle positions.
Therefore:
entropy increases
This is known as:
free expansion
Mixing
When different substances mix:
separated state → mixed state
there are generally more possible microscopic arrangements in the mixed state.
Therefore:
entropy increases
Examples include:
- gases mixing
- dye spreading through water
- perfume spreading through air
Combustion
Combustion generally converts concentrated chemical energy into:
- thermal energy
- radiation
- moving molecules
- gaseous products
Energy becomes more widely dispersed.
Therefore combustion generally produces an overall:
increase in entropy
when the complete system and surroundings are considered.
Low Entropy vs High Entropy
| Low Entropy | High Entropy |
|---|---|
| Fewer accessible microstates | More accessible microstates |
| Less probable macrostate | More probable macrostate |
| Matter more constrained | Matter more dispersed |
| Energy more concentrated | Energy more dispersed |
| Large gradients may exist | Closer to equilibrium |
| Example: compressed gas | Example: expanded gas |
| Example: ordered crystal | Example: gas |
| Example: hot/cold separation | Example: thermal equilibrium |
Entropy and Energy Quality
The First Law says:
energy is conserved
But conservation of energy does not mean all energy remains equally useful.
For example, mechanical energy can be transformed through friction into thermal energy spread throughout the environment.
The total amount of energy remains the same.
However, that dispersed thermal energy is generally harder to convert completely back into useful mechanical work.
Entropy therefore helps describe the:
quality and availability of energy
First Law vs Second Law
First Law
Deals with:
conservation of energy
Energy cannot be created or destroyed.
Second Law
Deals with:
direction and energy dispersal
Total entropy of an isolated system does not decrease.
Together:
First Law → how much energy exists
Second Law → which energy transformations occur spontaneously and how useful energy becomes dispersed
Entropy and Heat Engines
A heat engine takes thermal energy from a hot reservoir and converts some of it into:
useful work
However, not all the thermal energy can be converted into work.
Some energy must be transferred to a colder reservoir.
This limitation is a consequence of the:
Second Law of Thermodynamics
No real heat engine can be 100% efficient.
Entropy and the Arrow of Time
Many fundamental microscopic equations do not strongly distinguish between forward and backward time.
Yet everyday processes clearly have a preferred direction.
For example:
ice melts in a warm room
but we do not see:
warm water spontaneously form an ice cube while heating the room
Similarly:
perfume spreads throughout a room
but does not spontaneously gather back into the bottle.
Entropy provides a thermodynamic:
arrow of time
The direction toward higher total entropy corresponds to the direction in which macroscopic processes naturally occur.
Broken Glass
Imagine dropping a glass.
It shatters.
The reverse process—fragments spontaneously assembling into the original glass—is not normally observed.
Energy conservation alone does not forbid us from imagining a reversed motion.
But the shattered state plus dispersed thermal and acoustic energy corresponds to vastly more possible microscopic configurations.
The reverse process is therefore extraordinarily improbable.
This illustrates the statistical nature of entropy.
Entropy and Information
Entropy also appears in information theory.
Although thermodynamic entropy and information entropy arise in different contexts, both involve:
numbers of possible states
A system with many possible microscopic configurations requires more information to specify exactly.
This deep mathematical connection has become important in:
- statistical physics
- computing
- quantum information
- black-hole physics
Entropy and Probability
A crucial idea is:
entropy increase is statistical
For very small systems, temporary fluctuations toward lower entropy can occur.
For macroscopic systems containing around 10²³ particles, however, large spontaneous entropy decreases are so improbable that they are effectively never observed.
Thus the Second Law emerges from:
overwhelming statistical probability
Processes That Generally Increase Entropy
Important examples include:
- heating a colder object
- melting
- vaporization
- sublimation
- gas expansion
- diffusion
- mixing
- friction
- combustion
- irreversible deformation
- heat transfer from hot to cold
These processes generally increase either the entropy of the system or the total entropy of the system and surroundings.
Processes That Can Decrease a System's Entropy
Examples include:
- freezing
- condensation
- compression
- crystallization
- cooling
However:
system entropy decreasing does not mean total entropy decreases
The surroundings must also be considered.
Worked Example: Comparing States
Consider the same amount of water as:
ice
liquid water
water vapor
The general entropy ranking is:
S_ice < S_liquid < S_vapor
Why?
Ice molecules are highly constrained.
Liquid molecules have greater freedom.
Gas molecules have far more possible positions and motions.
Therefore:
number of accessible microstates increases
from solid to liquid to gas.
Worked Example: Gas Expansion
A gas is initially trapped in one half of an insulated container.
A partition is removed.
The gas spreads throughout the entire container.
Energy may remain essentially unchanged during ideal free expansion, but the gas now has access to:
many more possible positions
Therefore:
Ω increases
and from:
S = k_B ln Ω
entropy increases.
This demonstrates that entropy is not simply another word for energy.
Worked Example: Probability
Suppose one macrostate has:
Ω₁ = 100
possible microstates.
Another has:
Ω₂ = 10,000
possible microstates.
Using:
S = k_B ln Ω
the second state has greater entropy because:
Ω₂ > Ω₁
It is also much more statistically likely.
Thus:
more microstates → greater probability → greater entropy
Entropy Does Not Mean "Everything Becomes Messy"
The word "disorder" is useful for simple examples, but it should not be taken too literally.
A messy bedroom is not a precise thermodynamic measurement.
Thermodynamic entropy refers to:
the number and probability of microscopic states available to a physical system
The scientific meaning is much more precise than everyday ideas of tidiness.
Common Mistakes
Mistake: Defining entropy only as disorder.
Disorder is a useful analogy, but entropy is more precisely related to accessible microstates and energy dispersal.
Mistake: Saying entropy always increases everywhere.
Local entropy can decrease. The Second Law concerns the total entropy of an isolated system.
Mistake: Saying entropy increase violates conservation of energy.
Energy remains conserved. Entropy describes how that energy is distributed.
Mistake: Saying high entropy means high energy.
Entropy and energy are different physical quantities.
Mistake: Saying freezing violates the Second Law because entropy decreases.
The entropy of the water decreases, but heat is released and the surroundings gain entropy.
Mistake: Saying living organisms violate the Second Law because they become organized.
Living organisms are open systems that exchange energy and matter with their surroundings.
Mistake: Saying entropy increase is caused by a special force.
High-entropy states are overwhelmingly more statistically probable.
Mistake: Saying a high-entropy state is impossible to reverse.
Microscopically, reversals may not be strictly impossible, but macroscopic entropy decreases are overwhelmingly improbable without external changes.
Mistake: Using Celsius in ΔS = Q/T.
Temperature must be measured in kelvin.
Key Terms
Entropy (S) — thermodynamic quantity related to the number of accessible microscopic arrangements and the dispersal of energy.
Microstate — one particular microscopic arrangement of a system's particles and energy.
Macrostate — large-scale state specified by quantities such as temperature, pressure and volume.
Multiplicity (Ω) — number of microscopic states corresponding to a particular macrostate.
Boltzmann constant (k_B) — constant relating entropy to the number of microstates, approximately 1.38 × 10⁻²³ J/K.
Boltzmann equation — S = k_B ln Ω.
Entropy change (ΔS) — difference between final and initial entropy: ΔS = S_final − S_initial.
Second Law of Thermodynamics — principle stating that the total entropy of an isolated system cannot decrease.
Thermal equilibrium — condition in which systems have the same temperature and there is no net heat transfer.
Reversible process — idealized process that can be reversed without producing a net increase in total entropy.
Irreversible process — real process that produces an increase in total entropy.
Energy dispersal — spreading of energy among more particles, states or locations.
Spontaneous process — process capable of occurring naturally under the given conditions without continuous external intervention.
Free expansion — expansion of a gas into an available empty volume without doing external work.
Arrow of time — observed direction of macroscopic processes toward increasing total entropy.
Key Takeaways
- Entropy is represented by S and measured in J/K.
- Entropy is related to the number of microscopic arrangements available to a system.
- A particular microscopic arrangement is called a microstate.
- Boltzmann's equation is S = k_B ln Ω.
- More accessible microstates generally mean greater entropy.
- High-entropy macrostates are overwhelmingly more probable than low-entropy macrostates.
- "Disorder" can be a useful introduction to entropy but is not its complete scientific definition.
- Energy dispersal provides another useful way of understanding entropy.
- Gases generally have greater entropy than liquids, and liquids generally have greater entropy than solids of the same substance.
- Melting generally increases the entropy of a substance.
- Vaporization generally produces an even larger entropy increase.
- Gas expansion increases entropy because more spatial arrangements become available.
- Diffusion and mixing generally increase entropy.
- Friction increases entropy by dispersing organized mechanical energy into microscopic thermal motion.
- Heat naturally flows from hot objects to colder objects because this increases total entropy.
- Thermal equilibrium corresponds to a highly probable, high-entropy state under the given constraints.
- The Second Law states that ΔS_total ≥ 0 for an isolated system.
- Entropy may decrease locally as long as the total entropy of the complete system and surroundings does not decrease.
- Refrigerators do not violate the Second Law because they require external work and increase total entropy overall.
- Living organisms do not violate the Second Law because they exchange matter and energy with their environment.
- Entropy helps explain why many processes have a preferred direction.
- The First Law describes conservation of energy; the Second Law describes the direction and limitations of energy transformations.
- Increasing entropy makes energy more dispersed and often less available for useful work.
- Entropy explains why no heat engine can convert all absorbed thermal energy into useful work.
- The tendency toward increasing entropy provides a statistical arrow of time.
- For microscopic systems, entropy fluctuations can occur, but large entropy decreases in macroscopic systems are extraordinarily improbable.
- Entropy connects thermodynamics with probability, statistical mechanics, information theory and modern physics.
Check Your Understanding
- Define entropy.
- What is a microstate?
- What is the difference between a microstate and a macrostate?
- State Boltzmann's entropy equation.
- What does Ω represent in S = k_B ln Ω?
- Why does a system with more accessible microstates have greater entropy?
- Explain why entropy is related to probability.
- Why is describing entropy simply as "disorder" incomplete?
- Compare the entropy of a solid, liquid and gas of the same substance.
- Explain why melting generally increases entropy.
- Why does gas expansion into an empty container increase entropy?
- Explain why diffusion tends to increase entropy.
- State the Second Law of Thermodynamics in terms of entropy.
- Why does heat naturally flow from a hotter object to a colder object?
- Explain why freezing does not violate the Second Law.
- How does friction increase entropy?
- Why can a refrigerator decrease the entropy inside it without violating the Second Law?
- Explain why living organisms do not violate the Second Law by creating organized structures.
- Compare a low-entropy system with a high-entropy system in terms of microstates, probability and energy dispersal.
- Explain how entropy provides an "arrow of time" for macroscopic processes.