Chemistry: Atoms First 2e · Thermodynamics

Entropy

10 min read

Constants cited (k_B = 1.38 × 10^-23 J/K; ΔH_vap water 40.7 kJ/mol at 373.15 K; ΔH_fus ice 6.01 kJ/mol at 273.15 K; standard molar entropies of N₂, H₂, NH₃) are standard reference values; calculations are original worked examples.

On this page 9 sections

In 30 seconds
Why this matters
The college version
Eli explains
Worked example
Key takeaway
Check yourself
Study tools
Sources & references

In 30 seconds

The previous topic established that spontaneity cannot be explained by energy alone — a second tendency, the dispersal of energy and matter, also drives change. Entropy (S) is the thermodynamic quantity that measures this dispersal: the number of ways the energy and particles of a system can be arranged while keeping the same macroscopic state. The more microstates available, the higher the entropy.

Entropy is a : its change depends only on the initial and final states, not the path between them. For a at constant temperature, the entropy change is

$Δ S = q rev T$

where q_rev is the heat absorbed reversibly and T is the absolute temperature in kelvins. This topic covers what entropy means at the molecular level, how to predict whether a process increases or decreases entropy, and how to calculate entropy changes for phase changes and chemical reactions using standard molar entropies.

Why this matters

The arrow of time: Entropy is the only thermodynamic quantity that has a built-in direction — it is why time seems to flow one way and why broken cups never reassemble.
Predicting spontaneity: The total entropy change of the universe (system + ) decides whether any process can occur; the second law (Topic 3) makes this the master criterion.
Calculating reaction feasibility: Standard molar entropy values let chemists compute ΔS^°_rxn for any balanced reaction and combine it with enthalpy to judge feasibility.
Energy technology: Heat engines, refrigerators, and power plants are limited by entropy; "efficiency" is really an entropy argument.
Exam value: Predicting entropy sign and calculating ΔS^°_rxn are guaranteed exam skills.

The college version

Core Concepts

Entropy as the number of arrangements

Imagine distributing a fixed amount of energy among a set of particles. Each distinct way of doing so is a ; entropy is proportional to the logarithm of the number of microstates W:

$S = k B ln W$

where k_B = 1.38 × 10^-23 J/K is Boltzmann's constant. More microstates → higher entropy. A gas in one corner of a room has few microstates (low entropy); the same gas spread evenly has vastly more (high entropy). Systems naturally drift toward the most probable arrangement — the one with the most microstates — which is why gases expand and heat flows from hot to cold. This statistical view makes entropy concrete: it is not "disorder" as a vague idea but a countable multiplicity of arrangements.

Entropy is a state function

Entropy depends only on the current state of the system — its temperature, pressure, phase, and composition — not on how it got there. This is what allows the definition ΔS = q_rev/T: we compute the heat that would be exchanged along a reversible path, even if the real process is irreversible. A cup of hot water cooling to room temperature has the same ΔS whether it cools in 5 minutes or 5 hours.

Predicting the sign of ΔS

Without calculating, you can usually predict whether entropy increases (+) or decreases (-) from the change in dispersal:

Phase changes: gas > liquid > solid. Melting and vaporizing increase entropy; freezing and condensing decrease it.
Gas moles in reactions: reactions that produce more gas molecules than they consume increase entropy (more particles, more arrangements). Watch moles of gas, not total moles: N₂(g) + 3H₂(g) → 2NH₃(g) decreases entropy (4 mol gas → 2 mol gas).
Dissolving: solids dissolving in liquids usually increase entropy (ions/molecules spread through the solvent), though some hydration effects can make it a small net change.
Temperature: heating a substance increases entropy (more energy to distribute); cooling decreases it.
Volume: allowing a gas to expand into a larger volume increases entropy.

Entropy change for phase changes

At a phase-change temperature, the process is reversible and isothermal, so the entropy change is simply the heat of transition divided by the transition temperature (in kelvins):

$Δ S transition = Δ H transition T transition$

For example, water vaporizes at its normal boiling point:

$Δ S vap = Δ H vap T b$

Melting, vaporizing, and sublimation all give positive ΔS; the reverse processes give the negative of the same value.

Standard molar entropies and ΔS^°_rxn

The standard molar entropy S^° is the entropy of one mole of a substance in its standard state at 298 K (and 1 bar), typically in J/(mol·K). Unlike standard enthalpies of formation, standard molar entropies are absolute — they are not referenced to an arbitrary zero because the third law (Topic 3) sets S = 0 at 0 K for a perfect crystal.

For a reaction, the standard entropy change is:

$Δ S ° rxn = \sum n S ° (products) - \sum m S ° (reactants)$

where n and m are the stoichiometric coefficients. Units matter: entropies are in J/(mol·K), so ΔS^°_rxn comes out in J/K — usually a small number of kJ scale only after division by 1000, so watch the units when combining with ΔH (kJ).

Entropy of the surroundings

A system's entropy change is only half the story; the surroundings also gain or lose entropy. When a process releases heat (ΔH < 0), the surroundings absorb it and their entropy increases:

$Δ S surr = - Δ H sys T$

at constant temperature and pressure. This relationship is the bridge to the second law in Topic 3, where the sum ΔS_sys + ΔS_surr decides spontaneity.

Common Confusions

Do Not Confuse	With	Difference
Entropy	Enthalpy	Enthalpy is heat at constant pressure (kJ); entropy is dispersal/arrangements (J/K). Exothermic ≠ high entropy.
More disorder	More microstates	"Disorder" is a loose metaphor; entropy is precisely k_B lnW — a count of arrangements.
ΔS_rxn sign	ΔH_rxn sign	A reaction can be exothermic yet have ΔS < 0 (NH₃ synthesis); the two are independent properties.
Moles of gas	Total moles	Only gas moles matter for the entropy rule: 4 mol gas → 2 mol gas decreases entropy even though 4 → 2 overall.
Standard molar entropy S^°	Standard enthalpy of formation ΔH^°_f	S^° is absolute (zero at 0 K per the third law); ΔH^°_f is relative to an arbitrary zero (elements = 0).
J vs kJ	—	S^° values are J/(mol·K); ΔH values are usually kJ/mol. Convert (÷1000) before combining — a classic unit trap.
Reversible	Irreversible	Reversible is an idealized equilibrium path used for calculation; real processes are irreversible and produce net entropy.

Eli explains

The same idea, in plain words

Explain it like I’m 10

Imagine your LEGO bricks. You can build one tall tower (that's ONE arrangement — low entropy), or you can scatter the bricks all over the floor in millions of different ways (high entropy). Nature always drifts toward the way with the most arrangements — the scattered floor. A solid ice cube has few arrangements for its molecules; liquid water has many more; steam has the most. Entropy counts how many different ways the pieces can be arranged.

Worked example

Worked example 1 — entropy of vaporization of water. The enthalpy of vaporization of water is ΔH_vap = 40.7 kJ/mol at its normal boiling point T_b = 373.15 K. Find ΔS_vap.

Use the phase-change relation:

$Δ S vap = Δ H vap T b$

Convert the enthalpy to J/mol first (dimensional analysis):

$40.7 kJ mol \times 1000 J 1 kJ = 4.07 \times 104 J/mol$

$Δ S vap = 4.07 \times 104 J/mol 373.15 K = 109 J/(mol\cdotK)$

Liquid → gas increases entropy by about 109 J/(mol·K) per mole — a large gain, consistent with the huge increase in molecular freedom. The reverse (condensation) has ΔS = -109 J/(mol·K).

Worked example 2 — sign of ΔS by inspection. Predict the sign of ΔS for each process:

2H₂(g) + O₂(g) → 2H₂O(l): 3 mol gas → 0 mol gas; fewer particles and a gas→liquid change → ΔS < 0.
CaCO₃(s) → CaO(s) + CO₂(g): gas is produced → ΔS > 0.
NaCl(s) → Na⁺(aq) + Cl^-(aq): solid disperses into ions → ΔS > 0.
Freezing water at −5 °C: liquid → solid → ΔS < 0.

The gas-moles rule handles the first two instantly; phase and dissolution reasoning cover the rest.

Worked example 3 — ΔS^°_rxn from standard molar entropies. Calculate ΔS^°_rxn for the synthesis of ammonia at 298 K:

$N 2 (g) + 3 H 2 (g) \to 2 NH 3 (g)$

Standard molar entropies (J/(mol·K)): S^°(N₂) = 191.5, S^°(H₂) = 130.6, S^°(NH₃) = 192.8.

$Δ S ° rxn = \sum n S ° (products) - \sum m S ° (reactants)$

$Δ S ° rxn = 2(192.8) - [1(191.5) + 3(130.6)]$

$Δ S ° rxn = 385.6 - [191.5 + 391.8] = 385.6 - 583.3 = -197.7 J/K$

The reaction has ΔS^°_rxn = -197.7 J/K, a substantial decrease — consistent with 4 mol of gas collapsing into 2 mol. This negative entropy change is why ammonia synthesis needs high pressure and a catalyst: thermodynamics favors the reverse (decomposition) unless the surroundings' entropy gain from the exothermic reaction compensates (Topic 3).

Key takeaways

Entropy S measures the number of microstates: S = k_B lnW; more arrangements → higher entropy.
ΔS = q_rev/T for a reversible isothermal process — entropy is a state function.
Phase order: S_gas ≫ S_liquid > S_solid; melting/vaporizing increase S.
Gas moles rule: more moles of gas in products than reactants → ΔS_rxn > 0. Count gas moles only.
Heating increases entropy; cooling decreases it; gas expansion increases it.
Phase-change entropy: ΔS = ΔH_transition/T_transition (T in kelvins).
Standard molar entropy S^° (J/(mol·K)) is absolute (third law zero), unlike ΔH^°_f.
Reaction entropy: ΔS^°_rxn = ∑n S^°(products) - ∑m S^°(reactants).
Surroundings entropy: ΔS_surr = -ΔH_sys/T — exothermic reactions boost surroundings entropy.

Check yourself

6 review questions from the chapter. Try each one, then open the answer.

State the Boltzmann relation between entropy and microstates, and explain what "more microstates" means physically.
Show answer
S = k_B lnW with k_B = 1.38 × 10^-23 J/K. More microstates means more distinct ways to arrange energy/particles consistent with the same macroscopic state — and a more probable state.
Predict the sign of ΔS for: (a) CO₂(s) → CO₂(g); (b) 2NO(g) + O₂(g) → 2NO₂(g); (c) sugar dissolving in water.
Show answer
(a) Positive (sublimation: solid → gas). (b) Negative (3 mol gas → 2 mol gas). (c) Positive (solute disperses into solution).
The enthalpy of fusion of ice is 6.01 kJ/mol at 273.15 K. Calculate ΔS_fus of water.
Show answer
ΔS_fus = ΔH_fus/T = (6.01 × 10³ J/mol)/(273.15 K) = 22.0 J/(mol·K).
Why are standard molar entropies absolute values while standard enthalpies of formation are relative?
Show answer
The third law gives a real zero of entropy (perfect crystal at 0 K), so S^° values are absolute; enthalpies are measured relative to elements in their standard states, an arbitrary zero.
In N₂(g) + 3H₂(g) → 2NH₃(g), ΔS^°_rxn is negative even though the reaction is famously used industrially. Explain this apparent contradiction.
Show answer
The system entropy decreases (4 mol gas → 2 mol gas), but the reaction is exothermic, so the surroundings gain entropy (-ΔH/T); under the right conditions the surroundings' gain exceeds the system's loss, and the total can still favor the forward reaction.
A reaction is exothermic. What happens to the entropy of the surroundings, and what equation describes it?
Show answer
The surroundings gain entropy: ΔS_surr = -ΔH_sys/T — heat released by the system becomes dispersed thermal energy in the surroundings.

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Study tools & related lessonsKey vocabulary · Related

Key vocabulary

Entropy (S): Measure of the number of ways energy/matter can be arranged (microstates).
Microstate: One specific arrangement of energy among particles.
State function: Property depending only on current state, not the path taken.
Reversible process: An idealized path through equilibrium states; the system and surroundings can be restored exactly.
Standard molar entropy (S°): Entropy of one mole of a substance in its standard state at 298 K, J/(mol·K).
Boltzmann's constant (kB): 1.38 × 10^-23 J/K; connects microstates to entropy.
Heat of transition (Δ Htrans): Enthalpy absorbed/released at a phase change (fusion, vaporization).
Surroundings: Everything outside the system that can exchange energy with it.

Sources & references

openstax.org — Chemistry Atoms First 2e

This lesson was adapted from the open educational references above; their licenses and attributions are preserved. See Copyright & Licensing.

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In 30 seconds

Why this matters

The college version

Core Concepts

Entropy as the number of arrangements

Entropy is a state function

Predicting the sign of ΔS

Entropy change for phase changes

Standard molar entropies and ΔS°rxn

Entropy of the surroundings

Common Confusions

The same idea, in plain words

Worked example

Key takeaways

Check yourself

Keep learning

Key vocabulary

Related lessons

Sources & references

Standard molar entropies and ΔS^°_rxn