EntropyCourse level: AP Chemistry

Entropy and the Second Law

What you'll be able to do: Predict the sign of an entropy change and calculate ΔS from standard molar entropies.

Best after: Energy, Heat and the First Law

Open question bankReturn to unit

Introduction

Enthalpy alone cannot explain why ice melts on a warm day or why gases mix without being pushed. The missing ingredient is entropy, a measure of how many ways the energy and particles can be arranged.

These are recommended, not required. You can start this lesson at any time.

Learning objectives

  • Predict the sign of ΔS from a balanced equation
  • Calculate ΔS from standard molar entropy values
  • State the second and third laws of thermodynamics
  • Distinguish system entropy from universe entropy

What entropy counts

Entropy is a measure of how many microscopic arrangements of position and energy give the same overall state. More available arrangements means higher entropy.

Describing it as disorder is a rough shorthand that fails for cases such as protein folding, so counting arrangements is the safer picture.

Predicting the sign

For the same substance, melting, vaporization and heating at fixed phase and ordinary constraints generally increase entropy. Dissolution and increases in gas moles often suggest an increase, but competing effects require a closer comparison. A large gas-mole change is often the leading qualitative clue, but tabulated entropies settle cases with competing effects.

Qualitative comparisons work best when other features are controlled. Vaporization of the same substance gives many more accessible configurations; compression of a gas at fixed temperature reduces them. Dissolving a salt releases lattice ions into the solvent, but ordering of surrounding water can offset that gain, so dissolution entropy is not always positive. When gas moles are unchanged, molecular complexity, phase and mixing can still matter. A memorized particle-count rule is not a replacement for the standard-entropy calculation.

Gas-mole changes are a useful first clue to ΔS; check data when other phase, mixing or molecular effects compete.

Standard molar entropies

Tabulated as S in J/mol K, these are absolute values rather than changes, because the third law sets the entropy of a perfect crystal at 0 K to exactly zero. Every substance above absolute zero therefore has a positive S, elements included.

Distinguish S for a whole sample, in J/K, from molar entropy Sₘ, in J/(mol·K). Standard molar values are scaled by stoichiometric amounts before subtracting. For N₂ + 3 H₂ → 2 NH₃, 2(192.8) − [191.6 + 3(130.7)] = −198.1 J/K for the stoichiometric batch, or −198.1 J/(mol reaction·K). Unlike formation enthalpies, elemental molar entropies at 298 K are not zero. A perfect ordered crystal at 0 K supplies the third-law reference, not an arbitrary zero assigned separately to each element.

ΔS(rxn) = S n S(products) - S n S(reactants)

Entropies are in J/mol K while enthalpies are in kJ/mol. Convert before combining them in a Gibbs calculation.

The second law

A process is spontaneous when the total entropy of the universe increases. The system alone is allowed to become more ordered, as when water freezes, provided the surroundings gain more entropy than the system loses.

ΔS(universe) = ΔS(system) + ΔS(surroundings) > 0 for a spontaneous change

Entropy of the surroundings

Heat released by an exothermic reaction spreads into the surroundings and raises their entropy, and it does so more effectively at low temperature. This is why exothermic reactions are so often spontaneous, and it is the idea the Gibbs equation packages up.

For a large thermal reservoir at T, ΔSsurr = qsurr/T. With constant-pressure reaction heat and only pressure-volume work, this becomes −ΔHsys/T. If ΔHsys = −10.0 kJ per reaction amount, ΔSsys = −20.0 J/K and T = 298 K, ΔSsurr = +10000/298 = +33.6 J/K, so ΔSuniv = −20.0 + 33.6 = +13.6 J/K. The system loses entropy but the total increases. Use reservoir temperature in kelvin and match joules with joules. At equilibrium the total entropy has no first-order increase for a permitted infinitesimal change; molecular exchange need not stop.

ΔS(surroundings) = -ΔH(system) / T

Mixing can increase entropy without a reaction

Two different ideal gases mixed at the same temperature occupy more accessible arrangements than when separated. Their mixing entropy is positive even if neither gas reacts and there is no net change in gas moles.

Entropy is not merely a count of gaseous particles; the number of ways to distribute distinct particles also matters.

Entropy of mixing two ideal gases. Read this visual alongside the section; the text alternative describes its scientific content and assumptions.
Distinguish mixing entropy from a change in phase or mole count.

Key ideas

Definition
Entropy (S)

Entropy for a sample has units J/K; standard molar entropy has units J/(mol·K). It reflects accessible microscopic states and energy distributions.

Rule
Gas moles rule

An increase in gas moles often suggests positive reaction entropy; calculate from data when other effects compete.

Rule
Third law

A perfect crystal at 0 K has zero entropy, which makes absolute entropies measurable.

Rule
Second law

For a spontaneous change the entropy of the universe increases.

Key concept
Elements are not zero

Unlike ΔHf, standard molar entropies of elements are positive.

Equation
Reaction entropy

ΔS(rxn) = Σ n S(products) - Σ n S(reactants)

  • S = standard molar entropy in J/mol K
  • n = stoichiometric coefficient
Equation
Entropy change of the surroundings

ΔS(surroundings) = -ΔH(system)/T

  • T = absolute temperature in K

Worked examples

Worked example 1

Predict the sign of ΔS for CaCO₃(s) → CaO(s) + CO₂(g).

Try it first: Count the moles of gas on each side.

    0 of 2 steps revealed.

    Worked example 2

    Calculate ΔS for N₂(g) + 3 H₂(g) → 2 NH₃(g) given S: N₂ = 191.6, H₂ = 130.7, NH₃ = 192.8 J/mol K.

      0 of 3 steps revealed.

      Worked example 3

      Water freezes at -10 C, yet the system becomes more ordered. Explain how this is consistent with the second law.

        0 of 4 steps revealed.

        Worked example 4

        A partition between equal amounts of N₂ and O₂ is removed in an insulated container. Predict the sign of ΔSmix.

        Try it first: Think about the accessible positions of each type of molecule.

          0 of 3 steps revealed.

          Common mistakes

          Common mistakes and better checks
          MistakeWhy it's wrongCheck instead
          Assuming any process that orders the system violates the second law.The law applies to the universe, not the system alone.Include the surroundings term before judging spontaneity.
          Combining S in J/mol K with ΔH in kJ/mol without converting.The entropy term ends up 1000 times too large.Convert entropy to kJ/mol K, or enthalpy to J/mol, before combining.
          Giving elements a standard entropy of zero.Only a perfect crystal at 0 K has zero entropy.Look up the tabulated positive value for the element.
          Predicting ΔS from the total number of particles rather than gas particles.Gases carry far more entropy per mole than solids or liquids.Count gas moles as an initial clue, then consider phases, mixing and tabulated entropy data.

          Scroll horizontally if needed to read all columns.

          After the lesson · Question bank

          Independent practice

          Work through new problems on your own. The lesson's quick checks and worked examples stay above if you need to review a step.

          31 topic questions across 3 difficulty levels.

          Loading practice questions…

          Practice this skillBrowse all topic questionsBack to unit

          What you should now know

          Entropy describes the multiplicity of microscopic states and energy distributions. Phase, temperature, mixing and gas-mole changes guide qualitative predictions, while standard molar entropy data give a quantitative products-minus-reactants sum. Elemental entropies at ordinary temperatures are not assigned zero. Spontaneity requires positive total entropy change of system plus surroundings; the system alone may lose entropy.

          • Entropy counts available microscopic arrangements
          • Rising gas moles often suggests positive ΔS; use data for a definitive comparison
          • Standard molar entropies are absolute and positive, elements included
          • ΔS = products minus reactants, scaled by coefficients
          • Spontaneity requires the entropy of the universe to increase
          Review prerequisiteReturn to unit

          Your next step

          Lesson: Not marked completed

          Marking a lesson completed records that you finished reading it. Practice establishes mastery separately. Saved on this device.

          Try a short set of different questions to check what you can do independently.

          Practice this skill

          Watch and Learn

          An optional video to support this lesson.

          Loading video resources…

          Supplementary learning resources

          More resources for this topic →

          External resources belong to their respective creators. Chem Help is not affiliated with or endorsed by these providers. Links open the original provider; their terms apply.

          Sources and further reading

          • Chemistry 2e, Section 16.2: Entropy
            Paul Flowers, Klaus Theopold, Richard Langley, William R. Robinson · OpenStax, Rice University · Chapter 16.2
            View source

            Lesson-specific rights and provenance need review.

            The current online Chemistry 2e edition lists CC BY-NC-SA 4.0. The source version used for this lesson and its applicable rights remain unverified.

            Recorded use: reference. This describes the stored citation, not verified authorship or reuse permission.

            Recorded attribution / license claim (applicability unverified): Chemistry 2e, OpenStax, Rice University, licensed CC BY 4.0.

            Source rights and review status