Entropy of Mixing

Two initially separated ideal gases share a common initial temperature and pressure. Their initial volumes are therefore proportional to their mole counts. For physically different species, removing the partition lets each gas occupy the full volume and increases entropy. For the same physical species, labels on the initial portions do not create distinguishable components, and there is no entropy of mixing.

For total n = nA + nB greater than zero, xA = nA/n and xB = nB/n. The separate contributions are −R nA ln(xA) and −R nB ln(xB), with the limiting convention 0 ln(0) = 0. Total delta S is their sum. At fixed temperature and pressure, ideal-gas mixing has delta H = 0 and delta G = −T delta S. The specified gas constant is R = 8.31446261815324 J/(mol K).

Worked example: one mole of each distinct gas

Let nA = nB = 1 mol and T = 300 K. The initial equal-pressure volumes are equal, and the total amount is 2 mol. The result is obtained from natural logarithms and the specified gas constant, independently of the illustration.

Both mole fractions are 1/2, so each entropy contribution is R ln(2), about 5.76314632164 J/K.
The total is 2R ln(2), about 11.5262926433 J/K.
Enthalpy change is 0, while delta G = −300 × 11.5262926433, about −3457.88779299 J.
If the two portions contain the identical gas, removing the partition changes neither physical species identity nor this ideal state. Both mixing changes are 0 instead of the positive distinct-gas entropy.

Native experiment

Use the Entropy of Mixing module with these exact controls. Each valid edit updates the two contributions and total changes; the figure compares before and after configurations without simulating time.

Choose Species relation = Different gases, set Moles nA = 1, Moles nB = 1 and Common temperature T = 300. Check fractions 0.5 each, entropy about 11.5262926433 J/K, Gibbs change −3457.88779299 J and enthalpy change 0.
Set Moles nB = 3. Fractions are 0.25 and 0.75; entropy is R(4 ln4 − 3 ln3), about 18.7020581552 J/K, and Gibbs change about −5610.61744657 J.
Set Moles nA = 2 and Moles nB = 2. Entropy is 4R ln2, about 23.0525852866 J/K: twice the one-plus-one result. Gibbs change is about −6915.77558597 J.
Set Moles nA = 0 and Moles nB = 2. Fractions are 0 and 1, each contribution and every total change is 0; no log(0) is evaluated.
Set Moles nB = 0 too. The status is No gas, both fractions are unavailable and all changes are 0 by the declared empty-process convention. This is distinct from an invalid raw input.
Restore Moles nA = 1 and Moles nB = 1, choose Species relation = Identical gas, and keep T = 300. Both entropy contributions and all mixing changes are 0. The initial labels identify portions of the same species.
Return to Different gases and set Common temperature T = 500. Entropy stays unchanged while Gibbs change scales with T. Set both mole amounts to 5: entropy is about 57.6314632164 J/K and Gibbs change about −28815.7316082 J.
Clear Moles nA and observe the accessible error and removed calculations. Restore 1, set Moles nB = 5 and Moles nA = 0.00000001. Tiny positive entropy contributions are retained in scientific notation rather than rounded to zero. Pan and choose Reset view; inputs stay unchanged.

Limitations and numerical conventions

This ideal, isothermobaric model excludes reactions, nonideal interactions and heat of mixing. The container picture is schematic: its initial volume split is proportional to amount, while after-mixing hatching only identifies which species are present. It does not count molecules or compute dynamics. For identical gas, portion fractions are not distinct chemical-species fractions. An empty process is explicitly assigned zero change with fractions unavailable. Ordinary floating-point arithmetic and eleven-significant-digit displays are used; logarithms are evaluated through stable log1p ratios, with their finite limiting forms for ratios beyond numeric range. Nonzero raw amounts that underflow to zero are rejected, and a positive fraction below representable range is explicitly labelled.

References

The distinct-species result and identical-species Gibbs-paradox distinction are discussed in Oregon State: Entropy of mixing and MIT 2.43 Lecture 13, isothermobaric mixing. The thermodynamic definition and entropy balance are introduced in OpenStax University Physics Volume 2, Entropy. The empty-process convention is part of this bounded experiment.

Entropy of Mixing