Entropy Change Calculator – ΔS for Reactions, Heating, Gases and Mixing
This entropy change calculator finds ΔS four ways: the standard reaction entropy ΔS° from tabulated S° values, the entropy gained by heating, melting and boiling, ΔS for ideal-gas expansions and compressions, and the entropy of mixing. Every result ends in the same second-law ledger: ΔS_univ = ΔS_sys + ΔS_surr, which is positive for anything that happens on its own.
What entropy measures
Rudolf Clausius named entropy in 1865 as heat transferred reversibly divided by temperature, dS = dq_rev/T. Ludwig Boltzmann later tied it to counting: S = k·ln W, where W is the number of microscopic arrangements consistent with what you see. "Disorder" is a loose metaphor; "how widely energy and matter are spread" is closer. Doubling the volume a gas can roam doubles the positions open to each molecule, which works out to ΔS = R·ln 2 = 5.763 J/(mol·K).
Absolute entropies and the third law
Formation enthalpies are measured relative to the elements, so every element has ΔfH° = 0. Entropy is different: the third law fixes S = 0 for a perfect crystal at 0 K, so S° values are absolute and an element's S° is never zero. O₂(g) carries 205.138 J/(mol·K). The one exception is aqueous ions, which are measured relative to H⁺(aq) = 0; small, highly charged ions such as Ca²⁺ and CO₃²⁻ order the water around them and end up with negative S°.
Predicting the sign of ΔS°
Gases carry most of the entropy, so the sign of ΔS° usually follows Δn(gas): ammonia synthesis turns 4 mol of gas into 2 and has ΔS° = −198.76 J/(mol·K). When no gas changes, smaller effects decide: solid → liquid → solution, more atoms per molecule, and the water that ions order or release. That is why calcite precipitating from Ca²⁺(aq) and CO₃²⁻(aq) raises the entropy by 202.9 J/(mol·K).
The second law in practice
A reaction that releases heat raises the surroundings' entropy by ΔS_surr = −ΔH/T. Adding the two gives ΔS_univ = −ΔG/T, so the entropy verdict and the Gibbs verdict always agree. Ammonia synthesis is favoured below about 464 K because its heat outweighs the lost gas; limestone only decomposes above about 1110 K, where the gained CO₂ outweighs the heat it costs. To work in free energies directly, use the Gibbs Free Energy Calculator.
Heating, melting and boiling
Warming a substance adds m·c·ln(T₂/T₁); each phase change adds ΔH_trans/T_trans, always with temperatures in kelvin. On a T–S diagram the area under a reversible path equals the heat, which makes the large plateaus easy to see: boiling supplies 68 % of the entropy gained between −20 °C ice and 120 °C steam. Most liquids gain about 88 J/(mol·K) on boiling (Trouton's rule); water gains 109 J/(mol·K) because hydrogen bonds order the liquid. Vapor pressures and ΔH_vap fits belong in the Clausius-Clapeyron Calculator.
Irreversible changes
The system's ΔS depends only on its end states, so the tool finds it along a reversible route even when the real process is irreversible: supercooled water freezing at −10 °C, a gas rushing into a vacuum, or hot copper dropped into cold water. What depends on the path is the entropy generated, the part that lands in ΔS_univ. A free expansion and a reversible isothermal expansion give the gas the same +5.763 J/K; only the second leaves the universe unchanged.
Mixing and the Gibbs paradox
When ideal gases mix, each one expands into the total volume, giving ΔS_mix = −R·Σ nᵢ·ln xᵢ, which peaks at R ln 2 for a 50:50 binary mixture. Mixing a gas with more of the same gas creates nothing, which is the resolution of Gibbs's paradox: identical molecules have no "which one is where" to count. The same −Σ p·ln p form measures information, as the Shannon Entropy Calculator shows in bits.