Heat Flow and Spontaneous Processes

A spontaneous process is a process that can proceed naturally under a particular set of conditions without requiring continuous external work to drive it.

One of the most familiar spontaneous processes is:

heat transfer from a hotter object to a colder object

If two objects at different temperatures are placed in thermal contact:

hot → cold

Thermal energy continues to transfer until the objects reach thermal equilibrium.

This direction is governed by the Second Law of Thermodynamics and is associated with an increase in the total entropy of the system and surroundings.

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What Is Heat?

Heat is energy transferred between systems because of a temperature difference.

Heat is therefore not something an object simply "contains."

An object contains:

internal energy

Heat describes:

energy being transferred because of a temperature difference

The symbol commonly used for heat transfer is:

Q

and its SI unit is:

joule (J)

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Temperature and Heat Flow

Temperature is related to the microscopic energy distribution of particles in a system.

When objects at different temperatures interact, energy tends to transfer:

higher temperature → lower temperature

For example:

hot coffee → cooler room

warm hand → ice cube

hot stove → cooking pot

warm building → cold outdoor air

The direction can be predicted simply by comparing temperatures.

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Predicting Heat-Flow Direction

Suppose:

Object A = 400 K

Object B = 300 K

Thermal energy flows:

A → B

because:

T_A > T_B

If:

T_A = T_B

there is:

no net heat transfer

The objects are in thermal equilibrium.

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Thermal Equilibrium

Thermal equilibrium occurs when objects in thermal contact have the same temperature and there is no net transfer of thermal energy between them.

At equilibrium:

T₁ = T₂

and:

Q_net = 0

Individual particles still exchange energy microscopically.

However, there is:

no net macroscopic energy transfer

in either direction.

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Reaching Equilibrium

Consider a hot metal block placed in cooler water.

Initially:

T_metal > T_water

Therefore:

metal → water

thermal energy transfers.

As this happens:

• metal cools
• water warms
• temperature difference decreases
• heat-transfer rate generally decreases

Eventually:

T_metal = T_water

At this point:

thermal equilibrium has been reached

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Spontaneous Does Not Mean Instantaneous

A spontaneous process does not necessarily happen quickly.

For example:

• hot coffee cooling is spontaneous
• iron rusting can be spontaneous
• gases mixing is spontaneous

Yet these processes occur at very different rates.

Therefore:

spontaneous ≠ fast

Spontaneity tells us whether a process has a natural thermodynamic direction under the stated conditions, not how rapidly it happens.

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The Second Law and Heat Flow

The Second Law of Thermodynamics explains why heat naturally flows from hot to cold.

For an isolated system:

ΔS_total ≥ 0

For a spontaneous irreversible process:

ΔS_total > 0

where:

ΔS_total = ΔS_system + ΔS_surroundings

Heat transfer from hot to cold produces an overall increase in entropy.

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Entropy Change During Heat Transfer

For an ideal reversible transfer of heat at constant absolute temperature:

ΔS = Q_rev/T

where:

• ΔS = entropy change
• Q_rev = reversible heat transfer
• T = absolute temperature in kelvin

This relationship helps us understand why the direction:

hot → cold

is spontaneous.

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Hot Object Losing Energy

Suppose an amount of energy Q leaves a hot object at temperature T_h.

Its entropy change is:

ΔS_hot = −Q/T_h

The negative sign indicates that the hot object:

loses entropy

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Cold Object Receiving Energy

The same amount of energy enters a colder object at T_c.

Its entropy change is:

ΔS_cold = +Q/T_c

The cold object therefore:

gains entropy

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Total Entropy Change

The total entropy change is:

ΔS_total = −Q/T_h + Q/T_c

or:

ΔS_total = Q(1/T_c − 1/T_h)

Since:

T_h > T_c

then:

1/T_c > 1/T_h

Therefore:

ΔS_total > 0

This is why spontaneous heat flow occurs:

hot → cold

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Worked Example: Spontaneous Heat Transfer

Suppose:

Q = 1200 J

moves from an object at:

T_h = 600 K

to an object at:

T_c = 300 K

Hot object:

ΔS_hot = −1200/600

ΔS_hot = −2.0 J/K

Cold object:

ΔS_cold = +1200/300

ΔS_cold = +4.0 J/K

Total:

ΔS_total = −2.0 + 4.0

ΔS_total = +2.0 J/K

Since:

ΔS_total > 0

the direction is thermodynamically spontaneous.

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What About the Reverse Direction?

Suppose the same 1200 J spontaneously moved:

300 K → 600 K

Then:

Cold object:

ΔS_cold = −1200/300

ΔS_cold = −4.0 J/K

Hot object:

ΔS_hot = +1200/600

ΔS_hot = +2.0 J/K

Therefore:

ΔS_total = −2.0 J/K

This would decrease the total entropy of the isolated system.

Therefore, it is not a spontaneous process.

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Heat Can Be Moved from Cold to Hot

Heat transfer from cold to hot is not impossible.

It simply requires:

external work

Examples include:

• refrigerators
• freezers
• air conditioners
• heat pumps

These devices use electrical or mechanical work to force thermal energy in the non-spontaneous direction.

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A Refrigerator

A refrigerator removes thermal energy from its cold interior.

It transfers that energy to the warmer room.

The compressor requires:

electrical energy

For the refrigerator:

Q_hot = Q_cold + W

where:

• Q_cold = thermal energy removed from the interior
• W = work input
• Q_hot = thermal energy released to the room

Therefore:

Q_hot > Q_cold

The refrigerator does not violate the Second Law.

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Spontaneous Processes and Entropy

A spontaneous process generally proceeds in the direction that increases the total entropy of an isolated system.

Examples include:

• hot objects cooling
• cold objects warming in hotter surroundings
• gases expanding
• gases mixing
• diffusion
• friction
• many chemical reactions
• melting under suitable conditions

The key criterion for an isolated system is:

ΔS_total > 0

for an irreversible spontaneous process.

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Entropy and Probability

Why does entropy tend to increase?

Entropy is related to the number of microscopic arrangements available to a system:

S = k_B ln Ω

where:

• S = entropy
• k_B = Boltzmann constant
• Ω = number of accessible microstates

Higher-entropy macrostates correspond to vastly more microscopic arrangements.

Therefore they are:

overwhelmingly more probable

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Energy Spreading

A useful way to understand spontaneous heat flow is:

energy tends to become more widely distributed

Imagine energy concentrated in one hot object while another object is cold.

There are relatively fewer microscopic ways to maintain this unequal distribution.

After energy has spread between the objects:

more microscopic arrangements become accessible

Therefore:

entropy increases

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Hot and Cold Are Unequal Energy Distributions

Consider two identical blocks.

Initially:

Block A = hot

Block B = cold

Energy is unevenly distributed.

After contact:

Block A = warm

Block B = warm

Energy is more evenly distributed.

The equilibrium condition corresponds to a vastly more probable macrostate.

Therefore, the system spontaneously approaches:

thermal equilibrium

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Equilibrium and Maximum Entropy

For an isolated system under fixed constraints, thermodynamic equilibrium corresponds to the overwhelmingly most probable macrostate and maximum entropy.

At equilibrium:

• temperatures are equal
• pressures may equalize
• concentrations may become uniform
• no net macroscopic flows remain

The system has no spontaneous tendency to move away from equilibrium.

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Thermal Equilibrium Is Dynamic Microscopically

Equilibrium does not mean that particles stop moving.

Particles continue to:

• move
• collide
• exchange energy

But statistically:

energy transfer one way ≈ energy transfer the other way

Therefore:

net transfer = 0

Equilibrium is a dynamic microscopic state with no net macroscopic change.

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Zeroth Law Connection

The Zeroth Law of Thermodynamics states:

If A is in thermal equilibrium with B, and B is in thermal equilibrium with C, then A is in thermal equilibrium with C.

This principle allows us to define and measure:

temperature

It also explains why a thermometer can determine the temperature of another object after reaching thermal equilibrium with it.

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Diffusion as a Spontaneous Process

Suppose perfume is released in one part of a room.

Initially:

high concentration in one region

Eventually:

molecules spread throughout the room

The dispersed state has vastly more possible microscopic arrangements.

Therefore:

diffusion is spontaneous

and:

entropy increases

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Concentration Equilibrium

Diffusion continues until there is no large-scale concentration difference.

At equilibrium:

concentration becomes approximately uniform

Molecules still move randomly.

However:

net diffusion = 0

This is analogous to thermal equilibrium.

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Gas Expansion

Suppose gas is confined to half of an insulated container.

A partition is removed.

The gas spontaneously expands into the available space.

The larger volume provides many more possible particle arrangements.

Therefore:

Ω increases

so:

S increases

The gas approaches equilibrium throughout the container.

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Why Gas Does Not Spontaneously Return

Nothing in the microscopic motion of an individual gas molecule says it cannot move back into the original half.

But a macroscopic gas contains an enormous number of particles.

The probability that all of them spontaneously gather into the original half is extraordinarily small.

Therefore the expansion is effectively:

irreversible

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Mixing

Suppose two gases are separated by a partition.

Remove the partition.

They spontaneously mix.

Before:

separated gases

After:

mixed gases

The mixed state generally has many more accessible microstates.

Therefore:

entropy increases

The gases do not spontaneously unmix.

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Melting as a Spontaneous Process

Suppose ice is placed in a room above its melting temperature.

Thermal energy flows:

warmer surroundings → colder ice

The ice melts.

Under these conditions:

melting is spontaneous

However, whether a phase change is spontaneous depends on conditions such as:

• temperature
• pressure

Water does not always spontaneously melt.

At sufficiently low temperatures:

freezing may instead be spontaneous.

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Direction Depends on Conditions

"Spontaneous" does not mean that one direction is always preferred under every condition.

For example:

Above the melting point:

ice → water

can be spontaneous.

Below the melting point:

water → ice

can be spontaneous.

Thermodynamic conditions determine the preferred direction.

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Friction as a Spontaneous Process

Suppose a box slides across a rough floor.

Friction slows it down.

Organized kinetic energy becomes:

internal energy

distributed among enormous numbers of microscopic particles.

This process occurs naturally.

The reverse does not.

The floor does not spontaneously cool while launching the box forward.

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Why Friction Is Irreversible

Before friction:

energy concentrated in coordinated motion

After friction:

energy distributed among random microscopic motion

There are vastly more ways to distribute energy microscopically than to organize it into one coordinated motion.

Therefore:

entropy increases

and the process is irreversible.

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Irreversible Processes

A process is irreversible if the system and surroundings cannot both be restored exactly to their original states without producing additional changes elsewhere.

Common irreversible processes include:

• spontaneous heat transfer
• friction
• diffusion
• mixing
• free expansion
• combustion
• electrical resistance
• inelastic collisions

For an irreversible process:

ΔS_total > 0

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Why Spontaneous Heat Transfer Is Irreversible

Suppose a hot object cools while warming a colder object.

The final state has greater total entropy.

To return everything exactly to the initial state, thermal energy would need to move:

cold → hot

This does not occur spontaneously.

External work would be required.

That work would produce additional changes in the surroundings.

Therefore, the original process cannot be completely reversed.

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Reversible Heat Transfer

An ideal reversible process would transfer heat across an infinitesimally small temperature difference:

ΔT → 0

The process would occur extremely slowly.

In the ideal reversible limit:

ΔS_total = 0

But real heat transfer requires a finite temperature difference to proceed at a useful rate.

Therefore:

real heat transfer is irreversible

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Why Perfect Reversal Is Impossible in Practice

Imagine hot coffee cooling in a room.

To perfectly reverse the process, every effect would need to be undone:

• coffee returned to its exact original temperature
• room returned to its exact original state
• air motions reversed
• thermal energy distribution restored
• no additional energy dispersed elsewhere

Realistically, this cannot be achieved.

The system may be restored approximately, but not without:

additional entropy production

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Reversible vs Spontaneous Irreversible Processes

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Driving Forces

Spontaneous processes are often driven by differences or gradients.

Examples include:

temperature difference → heat flow

concentration difference → diffusion

pressure difference → fluid flow or expansion

electrical potential difference → electric current

These gradients represent departures from equilibrium.

As the spontaneous process proceeds:

gradient decreases

until equilibrium is approached.

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Temperature Gradient

A temperature gradient is a change in temperature across distance.

Large temperature gradient:

stronger driving force for heat transfer

As heat transfers:

temperature difference decreases

Eventually:

ΔT = 0

and thermal equilibrium is reached.

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Conduction

During conduction, thermal energy transfers through particle interactions.

For example, heat one end of a metal rod.

Energy transfers:

hot end → cold end

until the temperature becomes more uniform.

This is a spontaneous process driven by a:

temperature gradient

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Convection

Convection transfers thermal energy through bulk movement of fluids.

Warmer fluid often becomes less dense and rises while cooler fluid sinks.

This produces:

convection currents

These currents help redistribute thermal energy and move the system toward equilibrium.

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Radiation

Thermal radiation also transfers energy.

A warmer object generally emits more thermal radiation than a colder object.

Two objects exchange radiation in both directions, but if their temperatures differ:

net energy transfer is from hotter → colder

At equal temperature:

net radiative transfer = 0

even though both objects continue emitting and absorbing radiation.

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Everyday Example: Ice in a Drink

Place ice in a warm drink.

Thermal energy flows:

drink → ice

The drink cools.

The ice warms and may melt.

Eventually, the system approaches a common temperature.

The reverse—melted water spontaneously freezing while making the drink hotter—is not spontaneous under the same conditions.

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Everyday Example: Hot Food

Hot food left on a table cools because:

T_food > T_room

Energy transfers through:

• conduction
• convection
• radiation

until the food approaches room temperature.

This is spontaneous heat transfer toward equilibrium.

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Everyday Example: Opening an Oven

When a hot oven door is opened:

hot oven → cooler kitchen

Thermal energy spreads through convection and radiation.

The kitchen does not spontaneously transfer enough thermal energy back into the oven to restore its original hot state.

The process is:

irreversible

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Everyday Example: Cold Drink

A cold drink in a warm room warms up.

Heat flows:

room → drink

Notice that "heat flows hot to cold" does not mean the object we are interested in must always lose energy.

A cold object gains energy because the surroundings are hotter.

Always compare:

temperatures

to determine heat-flow direction.

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Everyday Example: House in Winter

Inside:

295 K

Outside:

270 K

Thermal energy naturally transfers:

inside → outside

Heating systems must continually replace this energy.

Insulation slows the spontaneous heat transfer but does not reverse its natural direction.

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Everyday Example: Air Conditioning

During hot weather:

outside > inside

Thermal energy naturally tends to enter a cooled building.

An air conditioner uses external work to:

• remove energy from indoors
• transfer it outside

Thus air conditioning actively opposes the spontaneous direction of heat flow.

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Heat Flow and Efficiency

Large temperature differences allow faster heat transfer but also produce greater irreversibility.

To approach reversible heat transfer:

ΔT → 0

But then the rate of heat transfer becomes very small.

This creates an important engineering trade-off:

fast energy transfer vs thermodynamic efficiency

Real devices must balance both.

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Equilibrium Is Not "No Energy"

A system at equilibrium can still contain enormous amounts of internal energy.

Equilibrium means there is:

no net driving force for macroscopic change

For thermal equilibrium:

no net heat flow

It does not mean:

• particles stop moving
• energy disappears
• temperature becomes zero

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Equilibrium and Entropy

For an isolated system under fixed constraints, equilibrium corresponds to the overwhelmingly most probable macrostate.

It therefore corresponds to:

maximum entropy under those constraints

Once equilibrium is reached, spontaneous macroscopic changes no longer occur because there is no more probable accessible macrostate toward which the system can evolve under those conditions.

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Spontaneous Does Not Mean Energy Is Created

When a process occurs spontaneously, it does not create energy.

The:

First Law of Thermodynamics

still applies.

Energy is conserved.

The Second Law determines the:

preferred direction

in which energy becomes distributed.

Therefore:

First Law → energy conservation

Second Law → direction of spontaneous change

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Worked Example: Predicting Direction

Object A:

T_A = 450 K

Object B:

T_B = 350 K

Since:

450 K > 350 K

the spontaneous direction is:

A → B

Energy transfers until the temperatures approach equilibrium.

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Worked Example: Equal Temperatures

Two objects are both:

320 K

They are placed in thermal contact.

Since:

T₁ = T₂

there is:

no net heat transfer

They are already in thermal equilibrium.

Microscopic energy exchanges may continue, but there is no net transfer.

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Worked Example: Entropy Test

Suppose a proposed process gives:

ΔS_system = −3 J/K

and:

ΔS_surroundings = +5 J/K

Then:

ΔS_total = −3 + 5

ΔS_total = +2 J/K

Since:

ΔS_total > 0

the process is thermodynamically allowed as a spontaneous irreversible process under the stated conditions.

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Worked Example: Non-Spontaneous Proposal

Suppose:

ΔS_system = +2 J/K

and:

ΔS_surroundings = −5 J/K

Then:

ΔS_total = 2 − 5

ΔS_total = −3 J/K

An isolated system cannot spontaneously undergo this overall change.

External intervention or another compensating process would be required.

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Worked Example: Reversible Limit

Suppose:

ΔS_system = +4.0 J/K

and:

ΔS_surroundings = −4.0 J/K

Then:

ΔS_total = 0

This corresponds to the ideal:

reversible limit

No net entropy is produced.

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Spontaneous vs Reversible

A common misconception is that "spontaneous" means "reversible."

In thermodynamics, ordinary spontaneous processes are generally:

irreversible

For example:

hot → cold

is spontaneous.

But:

cold → hot

does not spontaneously occur to restore the original state.

The spontaneous process creates entropy.

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Common Mistakes

Mistake: Saying heat is stored inside an object.
Objects store internal energy. Heat is energy transferred because of a temperature difference.

Mistake: Saying heat always flows upward.
Heat spontaneously flows from higher temperature to lower temperature, regardless of direction.

Mistake: Saying cold flows from one object to another.
It is thermal energy that transfers.

Mistake: Saying a spontaneous process must happen quickly.
Spontaneous describes thermodynamic direction, not reaction or process rate.

Mistake: Saying entropy must increase in every object.
One object's entropy can decrease while the total entropy increases.

Mistake: Saying heat can never move from cold to hot.
It can when external work is supplied, as in a refrigerator.

Mistake: Saying equilibrium means particles stop moving.
Microscopic motion continues; there is simply no net macroscopic transfer.

Mistake: Saying equilibrium means zero energy.
A system at equilibrium can contain large amounts of internal energy.

Mistake: Saying a spontaneous process can be perfectly reversed simply by making the system return to its original state.
The surroundings must also be restored without additional entropy production.

Mistake: Using Celsius in entropy calculations involving Q/T.
Use absolute temperature in kelvin.

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Key Terms

Heat — energy transferred between systems because of a temperature difference.

Spontaneous process — process capable of proceeding naturally in a particular direction under the given conditions.

Thermal equilibrium — condition in which systems have the same temperature and no net heat transfer occurs.

Temperature gradient — variation in temperature across a distance.

Entropy (S) — thermodynamic quantity related to accessible microscopic arrangements and energy dispersal.

Entropy change (ΔS) — difference between final and initial entropy.

Second Law of Thermodynamics — principle requiring total entropy of an isolated system not to decrease.

Irreversible process — process that produces entropy and cannot be completely reversed without additional changes.

Reversible process — ideal process producing no net entropy and capable of exact reversal in the ideal limit.

Microstate — particular microscopic arrangement of a system.

Equilibrium — state with no net macroscopic driving force for change.

Conduction — thermal energy transfer through particle interactions without bulk movement of matter.

Convection — thermal energy transfer involving bulk motion of a fluid.

Thermal radiation — transfer of energy by electromagnetic radiation.

Heat pump — device using external work to transfer thermal energy from a colder region to a warmer region.

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Key Takeaways

• Heat is energy transferred because of a temperature difference.
• Thermal energy spontaneously transfers from higher temperature to lower temperature.
• The direction of heat flow is determined by temperature, not by which object contains more total energy.
• Heat transfer continues until thermal equilibrium is reached.
• At thermal equilibrium, temperatures are equal and there is no net heat transfer.
• Microscopic energy exchanges continue even at equilibrium.
• A spontaneous process can proceed naturally under the given conditions.
• Spontaneous does not mean instantaneous or fast.
• The Second Law determines the preferred direction of spontaneous thermodynamic processes.
• For a spontaneous irreversible process, ΔS_total > 0.
• Heat transfer from hot to cold increases total entropy.
• Heat transfer from cold to hot requires external work or another compensating change.
• Refrigerators, air conditioners and heat pumps move thermal energy from colder regions to warmer regions using work.
• Energy tends to become more widely dispersed during spontaneous processes.
• Higher-entropy macrostates correspond to vastly more possible microscopic arrangements.
• Diffusion, mixing, gas expansion and friction are examples of spontaneous irreversible processes.
• Equilibrium corresponds to the overwhelmingly most probable macrostate under the given constraints.
• For an isolated system under fixed constraints, equilibrium corresponds to maximum entropy.
• Spontaneous processes generally move systems toward equilibrium.
• Real spontaneous heat transfer occurs across a finite temperature difference and is irreversible.
• A reversible heat-transfer process is an ideal limiting case with an infinitesimally small temperature difference.
• A spontaneous process cannot be completely reversed without additional changes because it produces entropy.
• The First Law explains energy conservation.
• The Second Law explains the natural direction of energy transfer.
• Everyday phenomena such as cooling coffee, melting ice, heating cold drinks and warming buildings can all be understood using spontaneous heat transfer and equilibrium.

Check Your Understanding

1. Define heat.
2. What is a spontaneous process?
3. Predict the direction of heat flow between objects at 450 K and 280 K.
4. What happens to heat transfer as two objects approach the same temperature?
5. Define thermal equilibrium.
6. Does particle motion stop at thermal equilibrium? Explain.
7. State the Second Law in terms of entropy.
8. Why does spontaneous heat transfer from hot to cold increase total entropy?
9. Explain why heat does not spontaneously flow from a colder object to a hotter object.
10. Can heat be transferred from cold to hot? Explain how.
11. Why does a refrigerator not violate the Second Law?
12. Explain why a spontaneous process does not necessarily occur quickly.
13. How does diffusion illustrate a spontaneous process?
14. Why is free expansion of a gas spontaneous?
15. Explain how equilibrium is related to entropy and probability.
16. Why is spontaneous heat transfer irreversible?
17. A process has ΔS_system = −4 J/K and ΔS_surroundings = +7 J/K. Determine ΔS_total and state whether the entropy change is consistent with a spontaneous irreversible process.
18. Explain why a cold drink warms spontaneously in a warm room.
19. Compare a real spontaneous heat-transfer process with an ideal reversible heat-transfer process.
20. Explain why the First Law alone cannot predict whether heat will flow from hot to cold or from cold to hot.