Entropy Calculator: ΔS for Reactions & Phase Changes

Entropy has two equivalent definitions that this calculator draws on in its two modes. The thermodynamic definition is differential: dS = dQ_rev / T, so entropy change is the heat exchanged along a reversible path divided by the absolute temperature at which it flows. Because the divisor is T, the same joule of heat produces a larger entropy change when delivered cold than when delivered hot, which is why heat spontaneously moves from hot bodies to cold ones. The statistical definition, S = k_B * ln(W), counts microstates consistent with the macroscopic state, with k_B fixed exactly at 1.380649 times ten to the minus 23 J/K by the 2019 SI redefinition. The reaction mode uses tabulated absolute entropies, deltaS_rxn = sum(n * S products) minus sum(n * S reactants), in J per mol per K. Unlike formation enthalpies, standard molar entropies are absolute rather than relative, because the third law fixes a perfect crystal at 0 K as the zero point. Values are therefore positive and fall in recognizable bands: about 2.4 for diamond and 5.7 for graphite, 41 to 70 for typical liquids including 69.9 for liquid water, and 130 to 260 for gases, with 130.7 for hydrogen and 213.8 for carbon dioxide. A useful heuristic is that the sign of deltaS_rxn usually follows the change in moles of gas. Two caveats: the Q/T form is valid only for a reversible path, so an irreversible process needs a reversible path constructed between the same endpoints, and spontaneity is governed by total entropy including the surroundings.

Sadi Carnot laid the groundwork in 1824 with his analysis of an idealized heat engine, showing that efficiency depends only on the two reservoir temperatures. Rudolf Clausius formalized what Carnot had glimpsed, and in 1865 introduced the name entropy, choosing it from the Greek for transformation so that it would resemble the word energy. His statement that the energy of the universe is constant while its entropy tends to a maximum became the canonical phrasing of the second law. Ludwig Boltzmann supplied the microscopic interpretation through the 1870s, arguing that entropy counts the molecular arrangements compatible with a given macroscopic state; Max Planck later wrote the relation in the compact form S = k log W, and it is engraved on Boltzmann tombstone in Vienna. Absolute entropies became measurable after Walther Nernst proposed his heat theorem in 1906, work recognized with the 1920 Nobel Prize in Chemistry; Planck sharpened it to the statement that the entropy of a perfect crystal vanishes at absolute zero. Gilbert Lewis and Merle Randall, in their 1923 textbook on thermodynamics, named this the third law and set out the calorimetric procedure of integrating measured heat capacity divided by temperature from near 0 K upward, which is how the standard molar entropies tabulated by NIST and CODATA are still obtained.