Mixing Entropy Simulator — Ideal Gas Three-Component Mixing
Compute the mixing entropy Delta S_mix = -n R Sum x_i ln x_i for a three-component ideal gas in real time from total moles n, mole fractions x_1, x_2, x_3 and temperature T. The tool also reports Delta G_mix = -T Delta S_mix, the equimolar maximum Delta S_max = n R ln 3 and the ratio Delta S / Delta S_max, and visualises the mixture as particles plus a ternary diagram in barycentric coordinates.
Parameters
Total moles n
mol
Mole fraction x_1
Mole fraction x_2
Temperature T
K
Defaults: n = 1.00 mol, x_1 = 0.50, x_2 = 0.30 (so x_3 = 0.20), T = 298 K. The third mole fraction is x_3 = 1 - x_1 - x_2. If x_3 becomes negative a warning is shown and the mixing calculation is skipped. Uses R = 8.314 J/(mol K).
x_3 = 1 - x_1 - x_2 is negative. Adjust the sliders so that x_1 + x_2 < 1.
Results
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Delta S_mix
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Delta G_mix
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Max Delta S (equimolar)
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Delta S / Delta S_max
Three-component mixture (particles)
Red = species 1 (x_1 fraction) / green = species 2 (x_2 fraction) / blue = species 3 (x_3 fraction). Particle counts are proportional to n and x_i, normalised to a total of about 300 particles.
Ternary diagram (barycentric coordinates)
Each vertex is a pure component (x_i = 1), the centre is the equimolar point (x_1 = x_2 = x_3 = 1/3, maximum entropy). The yellow circle marks the current (x_1, x_2, x_3). Background shading shows the iso-entropy contours.
Theory & Key Formulas
The mixing entropy of a three-component ideal gas is given by the sum of mole-fraction log terms:
$n$ is the total moles, $x_i$ is the mole fraction of species $i$ ($\sum_i x_i = 1$) and $R = 8.314$ J/(mol K). Because ideal gases have no intermolecular forces, $\Delta H_{\text{mix}} = 0$ and the free-energy change reduces to:
Microscopically, Boltzmann's $S = k_B \ln W$ with Stirling's approximation gives $-N k_B \sum x_i \ln x_i$, which matches the macroscopic formula exactly.
What is the Mixing Entropy Simulator
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My textbook says "mixing two gases increases entropy" but if temperature and pressure are the same on both sides of a partition, nothing seems to change. Why does entropy go up just from removing the wall?
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Good intuition. The energy really does not change for an ideal gas (Delta H_mix = 0). What changes is the number of distinguishable arrangements. The moment you remove the partition, an A molecule goes from "confined to the left" to "free over the whole volume", and same for B. Boltzmann's S = k ln W has W explode, so the entropy rises. With the defaults (n = 1 mol, x_1 = 0.50, x_2 = 0.30, x_3 = 0.20, T = 298 K) the tool reports Delta S_mix about 8.56 J/K and Delta G_mix about -2.55 kJ — mixing is spontaneous.
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If I make one component dominate by increasing x_1, what happens to the entropy?
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Move the x_1 slider. At x_1 = 0.98 (essentially pure species 1) Delta S_mix shrinks. The maximum is at x_1 = x_2 = x_3 = 1/3, where Delta S_max = n R ln 3 about 9.13 J/K. The tool also shows Delta S / Delta S_max so you can see at a glance how close you are to maximum randomness. The defaults (0.50, 0.30, 0.20) give about 93.7 percent, which is already "almost equimolar". On the ternary diagram the yellow dot moves toward the centre as the ratio climbs.
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Does changing the temperature T change Delta S_mix itself?
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Sharp question. The formula Delta S_mix = -n R Sum x_i ln x_i has no T, so the entropy of mixing itself does not depend on T (for an ideal gas). What changes is Delta G_mix = -T Delta S_mix. Raising T makes |Delta G_mix| larger and increases the driving force for mixing. Try T = 298 K -> 1000 K in the tool — Delta G_mix grows from about -2.55 kJ to -8.56 kJ, a 3.4x increase. Real gases and liquids have a T-dependent Delta H_mix, but this tool stays in the ideal-gas regime.
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What do the colour shades inside the ternary diagram represent?
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The shading shows the entropy field — namely -Sum x_i ln x_i normalised by ln 3 so it ranges from 0 at the vertices (pure species) to 1 at the centre (equimolar). Think of it as a contour map. If one component dominates (x_1 = 0.9, x_2 = 0.05, x_3 = 0.05) the yellow dot sits in a dark region near a vertex; as you approach equimolar it moves into the bright centre. Running the x_1 sweep makes the dot slide between the top vertex and the x_2 / x_3 base line, tracing the entropy landscape.
FAQ
The mixing entropy of an ideal gas is the entropy gained when separate pure species are combined into a single container, given by Delta S_mix = -n R Sum x_i ln x_i, where x_i are mole fractions (Sum x_i = 1), n is the total moles and R = 8.314 J/(mol K). Because ideal gases have no intermolecular forces Delta H_mix = 0 and Delta G_mix = -T Delta S_mix is always negative, so mixing is spontaneous. With the defaults (n = 1 mol, x_1 = 0.50, x_2 = 0.30, x_3 = 0.20, T = 298 K) the tool reports Delta S_mix about 8.56 J/K, Delta G_mix about -2.55 kJ, max Delta S_max about 9.13 J/K and a ratio of about 93.7 percent.
Maximising -Sum x_i ln x_i under Sum x_i = 1 is the same problem as maximising the Shannon information entropy. A Lagrange-multiplier calculation gives the stationary point x_1 = x_2 = x_3 = 1/3, where -Sum(1/3) ln(1/3) = ln 3, so Delta S_max / n = R ln 3 about 9.13 J/(mol K). The defaults (0.50, 0.30, 0.20) give Delta S / Delta S_max about 93.7 percent. Setting x_1 = 0.34, x_2 = 0.33 pushes the ratio close to 100 percent.
The ideal-gas assumption neglects intermolecular forces, so molecules diffusing from separate containers into a common volume do not change their potential energy. The enthalpy of mixing is therefore zero and Delta G_mix = Delta H_mix - T Delta S_mix = -T Delta S_mix. As long as T is positive Delta G_mix is negative and mixing is spontaneous. Real gases and liquids show Delta H_mix not equal to zero — ethanol-water has negative Delta H_mix (exothermic), hexane-butanol has positive Delta H_mix (endothermic). This tool only uses the ideal-gas approximation.
The microscopic definition S = k_B ln W (W is the number of microstates) and the macroscopic Delta S_mix = -n R Sum x_i ln x_i describe the same thing. The number of ways to assign N particles to three species is W = N! / (N_1! N_2! N_3!). Using Stirling's approximation ln N! about N ln N - N gives ln W = -N Sum x_i ln x_i, and multiplying by k_B with N k_B = n R recovers Delta S_mix = -n R Sum x_i ln x_i. W is maximal at equimolar mixing and equals 1 for pure species. Shannon's H = -Sum p_i log p_i has the same form.
Real-world applications
Air composition and atmospheric physics: The Earth's atmosphere is mostly a three-component mixture of N2 (about 78 percent), O2 (about 21 percent) and Ar (about 1 percent), so it carries roughly Delta S_mix about 4.32 J/(mol K) of "extra" entropy relative to pure components. Reversing this mixing — separating air into pure components — requires at minimum the corresponding work T Delta S about 298 x 4.32 about 1.29 kJ/mol. Cryogenic distillation, PSA and membrane separation plants size their theoretical minimum energy demand from this number. Setting x_1 = 0.78, x_2 = 0.21, x_3 = 0.01 in the tool reproduces this value.
Combustion and flue-gas treatment: Engine and boiler exhaust streams are multi-component (CO2, H2O, N2, O2). Taking a typical diesel exhaust with about 0.13 CO2, 0.07 H2O and 0.75 N2 in the three main components, the tool gives Delta S / Delta S_max about 64 percent. Carbon capture (CCS) must undo this mixing to recover pure CO2; the lower the CO2 fraction (below 10 percent) the harder and more expensive the separation becomes.
Alloy design and high-entropy alloys: High-entropy alloys (HEAs) such as CoCrFeMnNi mix five or more elements at near-equiatomic ratios to maximise the configurational entropy R ln n_species and stabilise a single solid solution. For three components Delta S_config = R ln 3 about 9.13 J/(mol K); for five it is R ln 5 about 13.4 J/(mol K). The -T Delta S term in Delta G = Delta H - T Delta S_config drives the free energy down at high temperature. The ternary diagram in this tool is a direct visual of the three-component case — the centre is the high-entropy composition.
Liquid-liquid extraction and Gibbs energy: In chemical engineering, liquid-liquid extraction (aromatic recovery in refineries, solvent extraction of valuable metals) is planned on a ternary phase diagram of solute, extracting solvent and original solvent, avoiding miscibility gaps. Ideal mixing always gives Delta G_mix negative, but real systems use van Laar / NRTL / UNIQUAC activity-coefficient models to correct for non-ideality. This tool is ideal-gas only, but the way to read a ternary diagram and the geometric picture of the mixing entropy carry over directly.
Common misconceptions and pitfalls
The most common pitfall is the belief that "mixing entropy depends on the kinetic energy of the particles". The mixing entropy is a configurational (positional) entropy and has nothing to do with kinetic energy directly. At constant temperature the kinetic energy distribution before and after mixing is the same; what changes is the number of ways particles can be distributed over the available volume. In S = k ln W, W counts spatial configurations, and removing a partition makes W explode. People sometimes guess "raising T should raise Delta S_mix"; the tool shows that varying T leaves Delta S_mix unchanged (only Delta G_mix = -T Delta S_mix scales).
A second pitfall is the Gibbs paradox: if you "mix" two volumes of the same gas, classical Boltzmann counting would predict an entropy gain when the partition is removed, but quantum-mechanically identical particles are indistinguishable, so "mixing" is not defined and S does not change. This is why the tool's three components must be distinguishable species (H2, O2, N2 and similar). Plugging in x_1 = 0.5, x_2 = 0.3, x_3 = 0.2 for the same gas is physically meaningless — Delta S_mix is zero in that case.
Finally, "the formula works for real gases and liquids too" is only partially true. The tool uses the ideal-gas approximation Delta H_mix = 0. Real systems show excess free energy G^E not equal to zero from intermolecular forces. Hydrogen-bonded mixtures like ethanol-water mix exothermically (Delta H_mix negative); fluorinated solvents can show partial miscibility gaps. For accurate design use activity-coefficient models (NRTL, UNIQUAC, PC-SAFT) or equations of state, and treat this tool as the "ideal baseline" against which the non-ideal corrections are measured.