The Extent of Reaction

Imagine a flask holding nitrogen, hydrogen, and ammonia, all reacting toward equilibrium. Every species changes amount at once, but not independently. Chemists needed one variable that tracks "how far" a reaction has proceeded, tying every mole change to a single stoichiometric bookkeeping rule. That variable is the extent of reaction, $\xi$ . It converts a tangle of coupled mole-number changes into one coordinate, letting us differentiate Gibbs energy with respect to a single quantity. That derivative becomes the reaction Gibbs energy, the engine of equilibrium theory. This article defines $\xi$ carefully. It derives $dn_i = \nu_i\,d\xi$ from stoichiometry. It works two full numeric examples, flags common sign traps, and sets up the leap to $\Delta_rG$ .

Key takeaways

The extent of reaction $\xi$ measures reaction progress with a single number, in moles.
For any species $i$ in a reaction, $dn_i = \nu_i\,d\xi$ , where $\nu_i$ is the stoichiometric number (negative for reactants, positive for products).
Integrating gives $n_i = n_{i,0} + \nu_i\xi$ , valid as long as the reaction proceeds by the assumed stoichiometric equation only.
$\xi$ has units of moles and is zero at the reference (usually starting) composition; it is not restricted to integers or to the range 0 to 1.
$\xi$ is the natural variable for defining the reaction Gibbs energy as $\left(\partial G/\partial \xi\right)_{T,p}$ , the slope that vanishes at equilibrium.
Key idea: one coordinate, $\xi$ , replaces $N$ separate mole-number variables because stoichiometry locks all $dn_i$ together.

The core insight

Because a balanced chemical equation fixes the ratios in which species are consumed and produced, all the mole-number changes in a reacting system collapse onto a single independent coordinate, the extent of reaction $\xi$ , through $dn_i = \nu_i\,d\xi$ .

The physical picture

Consider the synthesis of ammonia,

\text{N}_2(g) + 3\,\text{H}_2(g) \rightarrow 2\,\text{NH}_3(g).

Three mole numbers, $n_{\text{N}_2}$ , $n_{\text{H}_2}$ , and $n_{\text{NH}_3}$ , all change as the reaction runs. They are not independent. If one mole of N₂ disappears, exactly three moles of H₂ must disappear and exactly two moles of NH₃ must appear, because that is what the balanced equation demands. The system has only one true degree of freedom for the reaction itself: how far it has gone. That single degree of freedom is the extent of reaction, $\xi$ (Greek xi), introduced by Théophile de Donder in the 1920s specifically to make thermodynamic reasoning about reacting systems as clean as it already was for closed non-reacting systems. We assign each species a stoichiometric number $\nu_i$ : negative for reactants, positive for products, read directly off the balanced equation. For the ammonia reaction written as above,

\nu_{\text{N}_2} = -1, \qquad \nu_{\text{H}_2} = -3, \qquad \nu_{\text{NH}_3} = +2.

Stoichiometric numbers are pure numbers (dimensionless), fixed once you fix how the equation is written. Double every coefficient and every $\nu_i$ doubles too — so $\xi$ itself is only meaningful relative to a specific balanced equation. As the reaction proceeds by an infinitesimal amount, every species' mole number changes in exact proportion to its stoichiometric number, with the same proportionality constant $d\xi$ for all species simultaneously. That is the entire physical content of the extent of reaction: it is the common "speedometer reading" that every species shares, scaled only by its own stoichiometric weight.

Figure

Mole numbers as linear functions of the extent of reaction ξ, for N₂ + 3H₂ → 2NH₃ starting from 1 mol N₂ and 3 mol H₂

The core relations

dn_i = \nu_i\,d\xi

where: $n_i$ is the amount (mol) of species $i$ , $\nu_i$ its stoichiometric number (negative for reactants, positive for products), and $\xi$ the extent of reaction (mol).

n_i = n_{i,0} + \nu_i\,\xi

where: $n_{i,0}$ is the initial amount of species $i$ , valid when only this one reaction occurs and $\xi = 0$ at that initial state.

d\xi = \frac{dn_i}{\nu_i} \quad \text{(same value for every } i\text{)}

where: this equality across all species is the operational test that a single reaction, with these stoichiometric numbers, accounts for all the composition change.

\left(\frac{\partial G}{\partial \xi}\right)_{T,p} = \Delta_rG

where: $\Delta_rG$ is the reaction Gibbs energy, the slope of $G$ against $\xi$ at fixed temperature and pressure — the quantity that vanishes at equilibrium and that the next stage of this topic develops in full.

Derivation AFrom a balanced equation to dn_i = ν_i dξ

Goal. Show that stoichiometry, together with the definition of $\xi$ , forces every $dn_i$ to be proportional to $\nu_i$ with a universal proportionality factor $d\xi$ .

Step 1. State the balanced equation in stoichiometric-number form. Write any reaction with all species on one side, reactants carrying negative coefficients and products positive ones:

0 = \sum_i \nu_i\,\text{X}_i

For $\text{N}_2 + 3\text{H}_2 \rightarrow 2\text{NH}_3$ this is $0 = -\text{N}_2 - 3\text{H}_2 + 2\text{NH}_3$ , giving $\nu_{\text{N}_2}=-1$ , $\nu_{\text{H}_2}=-3$ , $\nu_{\text{NH}_3}=+2$ . This is just stoichiometry, restated with signs so reactants and products are treated symmetrically.

Step 2. Use the meaning of "one balanced reaction event." A single "unit" of reaction, as written, consumes $|\nu_i|$ moles of each reactant and produces $\nu_i$ moles of each product, simultaneously and in fixed ratio. If the reaction advances by some infinitesimal fraction of that unit event, call that fraction $d\xi$ , then every species' change must scale by its own $\nu_i$ times that same fraction — because the reaction cannot advance "a little for N₂" and "a lot for H₂" independently; the balanced equation locks them together.

Step 3. Define ξ operationally. Define $\xi$ so that this proportionality holds exactly:

dn_i \equiv \nu_i\,d\xi

This is a definition of $\xi$ , but it is only a sensible, non-contradictory definition because Step 2 guarantees the same $d\xi$ works for every species at once. If it did not — if $dn_{\text{N}_2}/\nu_{\text{N}_2}$ differed from $dn_{\text{H}_2}/\nu_{\text{H}_2}$ — the system would not be following this single reaction, and $\xi$ would not be well defined.

Step 4. Integrate at fixed stoichiometry. Since $\nu_i$ is a constant (fixed by the balanced equation, not by composition), integrate from the reference state $\xi=0$ , $n_i = n_{i,0}$ , to any later extent $\xi$ :

\int_{n_{i,0}}^{n_i} dn_i' = \nu_i \int_0^\xi d\xi'

n_i - n_{i,0} = \nu_i\,\xi

Step 5. Box the result.

\boxed{n_i = n_{i,0} + \nu_i\,\xi}

Every species' amount is an affine (straight-line) function of the single variable $\xi$ , with slope equal to its own stoichiometric number. This is exactly the family of straight lines plotted in the figure above.

Worked examples

Example 1 — Foundational: ammonia synthesis at a specified extent. A rigid vessel starts with $n_{\text{N}_2,0} = 3.00\ \text{mol}$ , $n_{\text{H}_2,0} = 9.00\ \text{mol}$ , and no ammonia, reacting by

\text{N}_2(g) + 3\,\text{H}_2(g) \rightarrow 2\,\text{NH}_3(g).

Find all three mole numbers when $\xi = 1.20\ \text{mol}$ , and check total moles for sanity. Step 1 — stoichiometric numbers. $\nu_{\text{N}_2} = -1$ , $\nu_{\text{H}_2} = -3$ , $\nu_{\text{NH}_3} = +2$ . Step 2 — apply $n_i = n_{i,0} + \nu_i\xi$ .

n_{\text{N}_2} = 3.00\ \text{mol} + (-1)(1.20\ \text{mol}) = 1.80\ \text{mol}

n_{\text{H}_2} = 9.00\ \text{mol} + (-3)(1.20\ \text{mol}) = 9.00 - 3.60 = 5.40\ \text{mol}

n_{\text{NH}_3} = 0 + (2)(1.20\ \text{mol}) = 2.40\ \text{mol}

Step 3 — sanity check via total moles. Initial total: $3.00 + 9.00 = 12.00\ \text{mol}$ . Final total: $1.80 + 5.40 + 2.40 = 9.60\ \text{mol}$ . The change is $9.60 - 12.00 = -2.40\ \text{mol}$ , matching $\left(\sum_i \nu_i\right)\xi = (-1-3+2)(1.20) = (-2)(1.20) = -2.40\ \text{mol}$ . Consistent, as required since gas-phase moles decrease when the reaction produces fewer molecules than it consumes. Step 4 — check maximum extent. H₂ is limiting: it runs out at $\xi_{\max} = 9.00/3 = 3.00\ \text{mol}$ , while N₂ alone would allow $\xi_{\max}=3.00\ \text{mol}$ too (a coincidence of these numbers, both giving 3.00 mol here); either way $\xi = 1.20\ \text{mol}$ is comfortably below both limits, so all mole numbers found are physically valid (non-negative).

Example 2 — Applied: relating extent to a measured pressure change. Phosphorus pentachloride dissociates in the gas phase,

\text{PCl}_5(g) \rightarrow \text{PCl}_3(g) + \text{Cl}_2(g).

A container held at constant $T$ and constant volume starts with pure PCl₅, $n_0 = 2.00\ \text{mol}$ , at total pressure $p_0 = 1.00\ \text{bar}$ . After partial dissociation the total pressure rises to $p_{\text{tot}} = 1.60\ \text{bar}$ . Find $\xi$ at this point, and the mole fraction of Cl₂. Step 1 — stoichiometric numbers. $\nu_{\text{PCl}_5} = -1$ , $\nu_{\text{PCl}_3} = +1$ , $\nu_{\text{Cl}_2} = +1$ . Step 2 — total moles as a function of ξ. Using $n_i = n_{i,0}+\nu_i\xi$ for each species and summing,

n_{\text{tot}}(\xi) = n_0 + \left(\sum_i \nu_i\right)\xi = 2.00\ \text{mol} + (-1+1+1)\xi = 2.00\ \text{mol} + \xi

Step 3 — link total moles to total pressure. At fixed $T$ and $V$ , ideal-gas pressure is proportional to total moles, so

\frac{p_{\text{tot}}}{p_0} = \frac{n_{\text{tot}}(\xi)}{n_0}

1.60 = \frac{2.00\ \text{mol} + \xi}{2.00\ \text{mol}}

Solve:

2.00\ \text{mol} + \xi = 3.20\ \text{mol} \quad \Rightarrow \quad \xi = 1.20\ \text{mol}

Step 4 — individual mole numbers.

n_{\text{PCl}_5} = 2.00 - 1.20 = 0.80\ \text{mol}, \qquad n_{\text{PCl}_3} = n_{\text{Cl}_2} = 1.20\ \text{mol}

Total: $0.80 + 1.20 + 1.20 = 3.20\ \text{mol}$ , matching $n_{\text{tot}} = 2.00+1.20=3.20\ \text{mol}$ from Step 2. ✓ Step 5 — mole fraction of Cl₂.

x_{\text{Cl}_2} = \frac{1.20\ \text{mol}}{3.20\ \text{mol}} = 0.375

Sanity check. $\xi = 1.20\ \text{mol} < n_0 = 2.00\ \text{mol}$ , so PCl₅ has not been exhausted; a pressure rise of 60% corresponds to 60% of the starting PCl₅ having dissociated, since each mole dissociated adds exactly one net mole of gas — a clean, physically reasonable result.

Limiting cases & common traps

Situation	What to watch for
$\xi = 0$	Defines the reference composition; not necessarily "reactants only," just wherever you start counting.
$\xi < 0$	Perfectly legal — means the reaction has run in reverse relative to the reference state.
Multiple reactions occurring	Each independent reaction needs its own extent, $\xi_1, \xi_2, \ldots$ ; $dn_i = \sum_j \nu_{ij}\,d\xi_j$ .
Rewriting the equation	Doubling all coefficients doubles every $\nu_i$ and halves $\xi$ for the same physical change — $\xi$ is equation-dependent.
Limiting reagent	$\xi$ cannot exceed the value that drives any reactant's mole number to zero.

The chemist behind it

The extent of reaction is the signature contribution of the Belgian thermodynamicist Théophile de Donder, who in the 1920s sought a rigorous way to apply the Gibbs formalism of equilibrium — built originally for simple closed systems — to systems undergoing chemical change. His insight was that a reacting mixture, despite having many mole numbers, has only as many true independent "reaction coordinates" as there are independent reactions. Defining $\xi$ let him write the reaction Gibbs energy as a genuine partial derivative, $(\partial G/\partial\xi)_{T,p}$ , and identify its vanishing as the equilibrium condition — a formulation still used, essentially unchanged, in every modern textbook treatment of chemical equilibrium.

Origins · 1 minThéophile de Donder and how this was discovered

Théophile de Donder

1872–1957 · Belgian mathematician and thermodynamicist who introduced the extent of reaction and founded the modern theory of chemical affinity

Ask the tutor: How does the reaction Gibbs energy, built from ξ, actually predict which way a reaction runs?

Why not just track each mole number n_i separately

You could, but it obscures the physics. A reacting system with five species would need five coupled differential relations instead of one. Stoichiometry already tells you the ratios; $\xi$ packages that information once, so you differentiate $G$ with respect to a single variable to get the reaction Gibbs energy, rather than juggling five partial derivatives that are not even independent of each other.

Does ξ depend on how you write the balanced equation

Yes. If you rewrite $\text{N}_2+3\text{H}_2\rightarrow2\text{NH}_3$ as $\tfrac12\text{N}_2+\tfrac32\text{H}_2\rightarrow\text{NH}_3$ , every $\nu_i$ is halved, so $\xi$ doubles for the same physical amount of reaction. The mole numbers $n_i$ themselves, being physically measured quantities, do not change — only the bookkeeping variable $\xi$ does. Always quote $\xi$ alongside the specific balanced equation used.

What happens with more than one reaction happening at once

Each independent reaction gets its own extent, $\xi_1,\xi_2,\ldots$ , and each species' total change is a sum over all reactions it participates in: $dn_i = \sum_j \nu_{ij}\,d\xi_j$ . This is exactly how simultaneous or coupled equilibria, such as parallel or consecutive reactions, are handled — you generalize the single-reaction result rather than replacing it.

How does ξ connect to the reaction Gibbs energy

Because composition changes at fixed $T$ and $p$ can be described entirely through $\xi$ , the Gibbs energy of the mixture becomes a function of $\xi$ alone (at fixed $T,p$ ). Its slope, $(\partial G/\partial\xi)_{T,p}$ , defines the reaction Gibbs energy $\Delta_rG$ . That slope is negative while the forward reaction is spontaneous, zero at equilibrium, and positive if the reverse reaction is favored — the subject of the next stage of this topic.