Extent of Reaction

Track an entire reaction with a single variable ξ — no matter how many species are involved.

Material & Energy BalancesChemical Engineering Year 1Free preview
⏱️ About 16 min

A reactor involves five species, but a single number ξ tells you exactly how far the reaction has gone for all of them. How?

💡
The big idea: The extent of reaction ξ is one variable that tracks every species simultaneously through the stoichiometric relationship n_i = n_i0 + ν_i·ξ.
🎯 By the end, you'll be able to
  • Define the extent of reaction ξ and its sign convention
  • Apply n_i = n_i0 + ν_i·ξ to compute species amounts
  • Determine ξ from one species' change and predict all others
  • Recognize that ξ depends on the written balanced equation
📎 Helpful to know first

The Extent of Reaction Concept

The extent of reaction ξ (xi) is a single variable that describes how far a reaction has proceeded. For any species i in the reaction, its molar amount changes according to:

ni = ni,0 + νi · ξ

Here νi is the stoichiometric coefficient: negative for reactants (consumed) and positive for products (generated). If you know ξ, you can compute the amount of every species — one number tracks the whole reaction.

\[ n_i = n_{i,0} + \nu_i\, \xi \]
The extent of reaction relates every species' final amount to its initial amount through one variable ξ.
⚠️ Use the same balanced equation throughout

The value of ξ depends on the stoichiometric coefficients you wrote down. If you double all coefficients (e.g., write 2A + 2B → 2C instead of A + B → C), ξ is halved. Pick one balanced equation at the start and use it consistently for every calculation.

Computing ξ from One Species

Rearranging the extent formula: ξ = (ni − ni,0) / νi. You only need the change in one species to find ξ, then use it to predict all the others.

For example, if 6 mol of A are consumed (ΔnA = −6) and νA = −2, then ξ = (−6)/(−2) = 3. That single value of ξ = 3 now determines every other species' final amount.

📝 Worked example: Consider the reaction 2A + B → 3C. Initially, n_A0 = 10 mol, n_B0 = 5 mol, n_C0 = 0 mol. After the reaction proceeds, n_A = 4 mol. Find ξ and predict n_B and n_C.
  1. Stoichiometric coefficients: ν_A = −2, ν_B = −1, ν_C = +3
  2. Compute ξ from species A: ξ = (n_A − n_A0) / ν_A = (4 − 10) / (−2) = (−6)/(−2) = 3 mol
  3. Predict n_B: n_B = n_B0 + ν_B · ξ = 5 + (−1)(3) = 5 − 3 = 2 mol
  4. Predict n_C: n_C = n_C0 + ν_C · ξ = 0 + (3)(3) = 9 mol
  5. Verify stoichiometry: A consumed = 6, B consumed = 3, C produced = 9 → ratio 6:3:9 = 2:1:3 ✓
✓ ξ = 3 mol; n_B = 2 mol, n_C = 9 mol
✏️ Practice: For the reaction A + 2B → C, initially n_A0 = 8 mol, n_B0 = 12 mol, n_C0 = 0. After reaction, n_B = 4 mol. What is n_A (in mol)?
mol
Solution
  1. Coefficients: ν_A = −1, ν_B = −2, ν_C = +1
  2. ξ = (n_B − n_B0) / ν_B = (4 − 12) / (−2) = (−8)/(−2) = 4 mol
  3. n_A = n_A0 + ν_A · ξ = 8 + (−1)(4) = 8 − 4 = 4 mol

Check your understanding

1. In the formula n_i = n_i0 + ν_i·ξ, what is the sign of ν_i for a product?
Products have positive stoichiometric coefficients (they are generated), so ν_i > 0 for products.
2. If ξ = 0, what does this mean?
ξ = 0 means no species' amount changed, so no reaction has occurred.
🔎 Related lessons
Conversion, Selectivity & Yield · soon
✅ Key takeaways
  • Extent of reaction ξ is a single variable tracking how far a reaction has proceeded
  • n_i = n_i0 + ν_i·ξ; ν_i is negative for reactants, positive for products
  • Compute ξ from any one species, then predict all others
  • ξ depends on the specific balanced equation — use it consistently
➡️ The extent of reaction lets us predict every species' amount from a single variable. But when multiple reactions compete, we need additional metrics to characterize reactor performance — conversion, selectivity, and yield.
Want to test yourself on this? Try the Chemical Aptitude test →