Entropy and Free Energy
Explore spontaneity, entropy, Gibbs free energy, and the second and third laws of thermodynamics.
Chemistry reference tablesEntropy and Microstates
Entropy (S) measures the dispersal of energy and matter in a system. Formally, it is related to the number of microstates (W) — the number of different ways particles and energy can be arranged while producing the same macroscopic state: S = k ln W, where k is Boltzmann’s constant.
Higher entropy means more microstates are accessible. Gases have much higher entropy than liquids, which in turn have higher entropy than solids, because particles in the gas phase have far more possible positions and energy distributions. Dissolving a solid in a solvent generally increases entropy because the solute particles become dispersed throughout the solution.
The third law of thermodynamics establishes an absolute reference point: the entropy of a perfect crystal at 0 K is exactly zero (W = 1). This allows chemists to tabulate standard molar entropies (S°) as absolute values, unlike enthalpy where only changes are measured.
Go deeperEntropy is counting, not messiness
The “disorder” metaphor eventually misleads; the microstate definition never does. A shuffled deck is not messier than a sorted one in any physical sense: it is simply one particular arrangement out of the astronomical number of arrangements that all look “shuffled,” while only a handful look “sorted.” Systems drift toward shuffled outcomes for the same reason a fair coin flipped a thousand times lands near 500 heads: not because any force pushes them there, but because that is where almost all the possibilities live.
This is why gas released in a corner fills the room. No repulsion drives it outward; the spread-out arrangements simply outnumber the corner-huddled ones so overwhelmingly that spreading is a statistical certainty.
Predicting Entropy Changes
You can predict the sign of ΔS for a physical or chemical process using straightforward rules:
- ΔS > 0 (entropy increases): phase changes from more ordered to less ordered (solid → liquid → gas); dissolving a solid or liquid; increasing temperature; reactions that produce more moles of gas than they consume.
- ΔS < 0 (entropy decreases): condensation or freezing; crystallization from solution; reactions that produce fewer moles of gas.
The number of moles of gas is the strongest predictor because gases have enormously more microstates than condensed phases. For example, the decomposition of CaCO3(s) into CaO(s) + CO2(g) has ΔS > 0 because one mole of gas is produced from zero moles of gas in the reactants.
Standard molar entropies generally increase with molar mass and molecular complexity, reflecting the greater number of ways energy can be distributed among more atoms and bonds.
Go deeperTest yourself: three entropy signs
Predict the sign of ΔS for each, then check below.
- 2 H2(g) + O2(g) → 2 H2O(g)
- Sugar dissolving in water
- Water freezing
Answers: The first is negative: three moles of gas become two, and gas-mole count dominates every other consideration. The second is positive: an ordered crystal disperses among the solvent molecules. The third is negative: liquid to solid is toward order, which is allowed to happen spontaneously below 0 °C because the heat released raises the surroundings’ entropy by more than the system loses (the second law watches the total, as the second-law section below makes precise).
The Second Law of Thermodynamics
The second law states that the total entropy of the universe always increases for any spontaneous process:
ΔSuniverse = ΔSsystem + ΔSsurroundings > 0
A process can decrease the entropy of the system (as in freezing water) as long as the surroundings gain even more entropy. At constant pressure and temperature, the entropy change of the surroundings is: ΔSsurroundings = −ΔHsystem / T. An exothermic reaction releases heat, warming the surroundings and increasing their entropy; an endothermic reaction absorbs heat, decreasing the surroundings’ entropy.
This explains why some endothermic processes are spontaneous (the large positive ΔSsystem outweighs the negative ΔSsurroundings) and why some exothermic processes are nonspontaneous (the entropy decrease of the system is too large to overcome).
Go deeperWhy this matters: life does not cheat the second law
A growing organism assembles atoms into fantastically ordered structures, and a refrigerator makes cold spots in a warm kitchen: both look like entropy running backwards. Neither is. The second law audits the universe, system plus surroundings, and both examples pay their entropy bill elsewhere: the fridge dumps more heat out its back coils than the order it creates inside, and living things ride a river of low-entropy sunlight, exporting waste heat that raises the universe’s entropy more than their own construction lowers it.
Local order is always for sale; the second law only insists the total price be paid. That accounting is exactly the ΔSuniverse equation in this section.
Calculating Standard Entropy Changes
The standard entropy change for a reaction is calculated from tabulated S° values using a products-minus-reactants approach:
ΔS°rxn = ∑nS°(products) − ∑nS°(reactants)
Each S° value is multiplied by the stoichiometric coefficient (n) of that substance in the balanced equation. Unlike ΔH°f values, standard molar entropies are never zero for elements in their standard states — every substance at temperatures above 0 K has a positive entropy.
Typical S° values at 298 K: H2(g) = 130.7 J/(mol·K), O2(g) = 205.2, H2O(l) = 69.9, H2O(g) = 188.8. Note the dramatic difference between liquid and gaseous water, reflecting the much greater disorder in the vapor phase. Units are J/(mol·K), not kJ, so be careful to convert when combining with ΔH values (which are in kJ).
Gibbs Free Energy
Gibbs free energy (G) unifies enthalpy and entropy into a single criterion for spontaneity at constant temperature and pressure:
ΔG = ΔH − TΔS
- ΔG < 0: the process is spontaneous (thermodynamically favorable).
- ΔG > 0: the process is nonspontaneous (the reverse is spontaneous).
- ΔG = 0: the system is at equilibrium.
Four scenarios arise from the signs of ΔH and ΔS: (1) ΔH < 0 and ΔS > 0 — always spontaneous; (2) ΔH > 0 and ΔS < 0 — never spontaneous; (3) ΔH < 0 and ΔS < 0 — spontaneous at low T; (4) ΔH > 0 and ΔS > 0 — spontaneous at high T. The crossover temperature where ΔG = 0 is T = ΔH/ΔS.
ΔG°rxn can also be computed from standard free energies of formation: ΔG° = ∑nΔG°f(products) − ∑nΔG°f(reactants).
Go deeperThe melting point is where delta G changes sign
Ice melting has ΔH > 0 (costs heat) and ΔS > 0 (gains freedom): the two terms of ΔG = ΔH − TΔS pull in opposite directions, and temperature referees. Below 0 °C the enthalpy term wins, ΔG > 0, and water freezes; above it the TΔS term wins, ΔG < 0, and ice melts. At exactly 0 °C, ΔG = 0: neither direction is favored, and ice and water coexist.
Read that again as a definition: a melting point is the temperature where ΔG for the phase change crosses zero. Every transition temperature in chemistry is this same tug-of-war, settled: T = ΔH/ΔS.
Free Energy and the Equilibrium Constant
The standard free energy change is directly related to the equilibrium constant:
ΔG° = −RT ln K
where R = 8.314 J/(mol·K) and T is in kelvins. This equation reveals that:
- K > 1 (ΔG° < 0): products predominate at equilibrium.
- K < 1 (ΔG° > 0): reactants predominate at equilibrium.
- K = 1 (ΔG° = 0): neither side is favored.
A large positive E°cell in electrochemistry corresponds to a large negative ΔG° and a large K, linking all three thermodynamic indicators of reaction favorability.
Go deeperPredicting K without running the reaction
ΔG° = −RT ln K is a bridge between two worlds: tabulated formation data on one side, equilibrium behavior on the other. It means a chemist can compute the equilibrium constant of a reaction that has never been run, from table lookups alone. The exponential makes modest energies decisive: at 25 °C, ΔG° = −20 kJ/mol (a small number by bond-energy standards) already gives K ≈ 3000, and every additional −5.7 kJ/mol multiplies K by ten.
This is how reaction feasibility gets screened across industries: paper thermodynamics first, flasks only for the candidates that survive.
Temperature Dependence of Spontaneity
Because ΔG = ΔH − TΔS, the temperature dependence of spontaneity is set by the signs of ΔH and ΔS. Four combinations exist:
- ΔH < 0, ΔS > 0: spontaneous at all temperatures (ΔG < 0 always) — e.g., the incomplete combustion 2 C(s) + O2(g) → 2 CO(g).
- ΔH > 0, ΔS < 0: never spontaneous (ΔG > 0 at all T).
- ΔH < 0, ΔS < 0: spontaneous at low temperature, where the favorable enthalpy term dominates (|ΔH| > |TΔS|).
- ΔH > 0, ΔS > 0: spontaneous at high temperature, where the entropy term dominates (TΔS > ΔH).
For the two temperature-dependent quadrants, the crossover temperature where spontaneity flips is found by setting ΔG = 0:
T = ΔH° / ΔS°
Watch the units: ΔH° is usually tabulated in kJ while ΔS° is in J/K — convert one before dividing (e.g., T = 100,000 J ÷ 250 J/K = 400 K).
The same ΔG = 0 condition describes a phase transition at equilibrium: at the normal boiling point, liquid and vapor coexist, so Tbp ≈ ΔH°vap / ΔS°vap. Using tabulated data for water (ΔH°vap = +44.01 kJ/mol, ΔS°vap = +118.8 J/(mol·K)) gives T ≈ 370.5 K = 97.3 °C — within 3 K of the true 373.15 K, the small error coming from the assumption that ΔH° and ΔS° do not vary with temperature. The same method estimates the boiling point of formic acid (≈ 388 K = 115 °C) from its formation data.
Two cautions: the crossover temperature marks where ΔG° changes sign, not where the reaction becomes fast (kinetics is a separate question); and the constant-ΔH°/ΔS° assumption degrades over wide temperature ranges — treat computed thresholds as estimates.
Free Energy Under Nonstandard Conditions
Under nonstandard conditions (concentrations other than 1 M, pressures other than 1 bar), the free energy change depends on the reaction quotient Q:
ΔG = ΔG° + RT ln Q
When Q < K, ΔG is negative and the reaction proceeds forward to produce more products. When Q > K, ΔG is positive and the reaction proceeds in reverse. At equilibrium, Q = K and ΔG = 0.
This equation explains why a reaction with a positive ΔG° can still proceed forward if the concentrations are far from equilibrium (Q << K). Conversely, a reaction with negative ΔG° can be driven backward if products are already in great excess (Q >> K).
The distinction between ΔG° (fixed value for a reaction at a given temperature) and ΔG (varies with composition) is essential: ΔG° tells you which direction is favored from standard conditions, while ΔG tells you the direction from the current state of the system.
Go deeperCommon mistake: confusing delta G with delta G standard
ΔG° is a fixed property of the reaction at a given temperature: everything at standard conditions, one number, table-worthy. ΔG is a live reading that depends on the actual mixture through Q, and it changes as the reaction proceeds. The error that follows: claiming “ΔG° = 0 at equilibrium.” It is ΔG that hits zero at equilibrium; ΔG° is only zero for the special reaction whose K happens to equal 1.
Keep them straight with their questions: ΔG° answers “where does equilibrium lie?” (via K); ΔG answers “which way does this particular mixture move right now?” (via Q). One is the map, the other is your position on it.
Thermodynamics Decision Framework: ΔS, ΔG, and K Together
Students perform best when thermodynamics is treated as a decision chain:
- Predict sign of ΔS qualitatively from phase and particle changes.
- Use ΔG = ΔH − TΔS for spontaneity at the stated temperature.
- Connect to equilibrium with ΔG° = −RT lnK.
- For nonstandard conditions, evaluate ΔG = ΔG° + RT lnQ.
This links macroscopic direction, temperature dependence, and equilibrium composition in one model. Quick checks: if K ≫ 1 then ΔG° should be negative; if Q < K then ΔG should be negative (forward-favored); if Q > K then ΔG should be positive (reverse-favored).
Key Equations
Learning Objectives
After studying this topic, you should be able to:
- Predict the sign of ΔS for physical and chemical processes from changes in moles of gas, phase, and molecular disorder
- Apply the second-law ΔS_universe > 0 spontaneity criterion and the third-law S(perfect crystal, 0 K) = 0 reference baseline to chemical processes
- Calculate the standard entropy change (ΔS°rxn) for a reaction from standard molar entropies
- Calculate ΔG° from ΔH° and ΔS° and determine whether a reaction is thermodynamically favorable
- Calculate ΔG° from K (or K from ΔG°) using ΔG° = −RT ln K, and predict spontaneity at standard conditions from K's magnitude
- Predict the temperature dependence of thermodynamic favorability from the signs of ΔH° and ΔS°
- Calculate ΔG under nonstandard conditions from ΔG° + RT ln Q, and use the sign of ΔG to predict reaction direction
How-To Procedure
How to Determine Whether a Reaction Is Spontaneous
- Identify ΔH (enthalpy change) and ΔS (entropy change) for the reaction, either from given data or by calculating from standard values.
- Make sure both values use the same energy units (convert ΔS from J/K to kJ/K if ΔH is in kJ, or vice versa).
- Convert the temperature to kelvins: T(K) = T(°C) + 273.15.
- Calculate ΔG = ΔH − TΔS.
- Interpret the result: ΔG < 0 means spontaneous; ΔG > 0 means nonspontaneous; ΔG = 0 means the system is at equilibrium.
- If both ΔH and ΔS have the same sign, note that spontaneity is temperature-dependent and find the crossover temperature where ΔG = 0 using T = ΔH / ΔS.
Worked Example
Determining Spontaneity from ΔH and ΔS
For the reaction 2 H2O2(l) → 2 H2O(l) + O2(g), ΔH° = −196.1 kJ and ΔS° = +125.7 J/K. Calculate ΔG° at 25 °C and determine if the reaction is spontaneous.
- Convert temperature to Kelvin: T = 25 + 273.15 = 298.15 K.
- Convert ΔS to kJ/K to match ΔH units: ΔS° = 0.1257 kJ/K.
- Apply the Gibbs equation: ΔG° = ΔH° − TΔS° = (−196.1) − (298.15)(0.1257) = −196.1 − 37.5 = −233.6 kJ.
- Since ΔG° < 0, the reaction is spontaneous under standard conditions. Both ΔH < 0 and ΔS > 0, so this reaction is spontaneous at all temperatures.
ΔG° = −233.6 kJ. The decomposition of hydrogen peroxide is spontaneous at 25 °C (and at all temperatures, since it is exothermic with a positive entropy change).
Test Your Understanding
Ice melts spontaneously at room temperature even though the process is endothermic (ΔH > 0). How can an endothermic process be spontaneous? What drives it forward?
Self-Study Questions
What is a spontaneous process?
What is entropy and what does it measure?
How do you predict the sign of ΔS for a reaction based on the states and number of moles of reactants and products?
Hint: More gas-phase molecules and greater disorder generally mean positive ΔS.
What is the second law of thermodynamics?
What is the third law of thermodynamics and what does it establish about absolute entropy?
What is Gibbs free energy and what does its sign tell you about spontaneity?
What is the equation ΔG = ΔH − TΔS and what does each term represent?
Under what combinations of ΔH and ΔS is a reaction always spontaneous, never spontaneous, or temperature-dependent?
How is ΔG° related to the equilibrium constant K?
What is the difference between ΔG and ΔG°?
Content Sources
Concept sections adapted from open educational resources under Creative Commons licensing:
- OpenStax Chemistry 2e, Ch 16.1: Spontaneity (CC BY 4.0)
- OpenStax Chemistry 2e, Ch 16.2: Entropy (CC BY 4.0)
- OpenStax Chemistry 2e, Ch 16.3: The Second and Third Laws of Thermodynamics (CC BY 4.0)
- OpenStax Chemistry 2e, Ch 16.4: Free Energy (CC BY 4.0)