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Extent of Reaction

General Chemistry • Chemical Reactions

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Reaction progress

Extent of Reaction

Extent of reaction measures how far a balanced reaction has proceeded. It connects all reactant decreases and product increases using the stoichiometric coefficients.

2H2 + O2 → 2H2O

Learning target: use reaction extent, \( \xi \), to calculate remaining reactants and formed products.

Progress

How far the reaction goes

A larger \( \xi \) means more reactants have been consumed.

Ratios

Coefficients control changes

Reactants decrease and products increase in fixed mole ratios.

Limit

Reaction stops at a limiter

The maximum \( \xi \) is set by the limiting reactant.

Why it matters

Extent organizes every mole change in one variable

Instead of calculating each reactant and product separately, extent of reaction uses one progress variable to track the entire balanced equation.

N2 + 3H2 → 2NH3

If \( \xi = 0.50\ \mathrm{mol} \), then \(0.50\ \mathrm{mol}\) N2 is consumed, \(1.50\ \mathrm{mol}\) H2 is consumed, and \(1.00\ \mathrm{mol}\) NH3 is formed.

Extent of reaction helps with

  • initial-change-final tables,
  • limiting reactant analysis,
  • reaction progress diagrams,
  • equilibrium setup in later chemistry,
  • tracking all species consistently.

Core concept

Every species changes by coefficient × extent

For a balanced equation, each substance has a stoichiometric coefficient. Reactants have negative changes because they are consumed. Products have positive changes because they form.

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

Here, \(n_i\) is final moles, \(n_{i,0}\) is initial moles, \(\nu_i\) is the signed stoichiometric coefficient, and \(\xi\) is the extent of reaction.

Extent of reaction coefficient model A diagram showing extent of reaction multiplying the coefficients for hydrogen, oxygen, and water. 2H2 change: −2ξ O2 change: −ξ 2H2O change: +2ξ initial reaction progress final One value of ξ determines all mole changes.

Vocabulary

Variables and signs in extent calculations

Symbol or term Meaning How it is used Unit
\(\xi\) Extent of reaction Measures how far the reaction has proceeded mol
\(n_{i,0}\) Initial amount of species \(i\) Starting moles before reaction progress mol
\(n_i\) Final amount of species \(i\) Amount remaining or formed after reaction progress mol
\(\nu_i\) Signed stoichiometric coefficient Negative for reactants, positive for products unitless
Maximum extent Largest possible \( \xi \) Set by the limiting reactant mol
Sign rule: reactants decrease with negative changes, while products increase with positive changes.

Main relationship

Use an initial-change-final table

For \(2H_2 + O_2 \rightarrow 2H_2O\), each change is tied to the same reaction extent \( \xi \).

Species Initial moles Change Final moles
H2 \(n_{\mathrm{H_2},0}\) \(-2\xi\) \(n_{\mathrm{H_2},0} - 2\xi\)
O2 \(n_{\mathrm{O_2},0}\) \(-\xi\) \(n_{\mathrm{O_2},0} - \xi\)
H2O \(0\) \(+2\xi\) \(2\xi\)
\[ \xi_{\max} = \min\left(\frac{n_{\mathrm{H_2},0}}{2},\ n_{\mathrm{O_2},0}\right) \]

Interactive simulation

Move the reaction forward and track all mole changes

Reaction setup

2H2 + O2 → 2H2O

Reaction table result

Maximum extent 2.00 mol
Final H2 2.00 mol
Final O2 1.00 mol
H2O formed 2.00 mol

The reaction has progressed by 1.00 mol of extent, consuming 2.00 mol H2 and 1.00 mol O2.

Static fallback: if \( \xi = 1.00\ \mathrm{mol} \), then H2 decreases by \(2.00\ \mathrm{mol}\), O2 decreases by \(1.00\ \mathrm{mol}\), and H2O increases by \(2.00\ \mathrm{mol}\).

Dynamic relationship

Reactants decrease while products increase

The bars show final mole amounts for the current reaction extent. The progress line marks the fraction of the maximum possible extent.

Extent of reaction mole bar graph A graph showing remaining hydrogen, remaining oxygen, and formed water as the extent of reaction changes. 8 mol 6 4 2 0 H2 left O2 left H2O made 2.00 mol 1.00 mol 2.00 mol 50% of maximum extent

Interpretation: all bars change together because one value of \( \xi \) controls every species in the balanced equation.

Worked example

Use extent to calculate remaining and formed amounts

Problem: For \(2H_2 + O_2 \rightarrow 2H_2O\), suppose the initial amounts are \(4.00\ \mathrm{mol}\) H2, \(2.00\ \mathrm{mol}\) O2, and \(0\ \mathrm{mol}\) H2O. If \( \xi = 1.25\ \mathrm{mol} \), find the final amounts.

  1. 1. Write the signed changes from the balanced equation. \[ \Delta n_{\mathrm{H_2}} = -2\xi,\qquad \Delta n_{\mathrm{O_2}} = -\xi,\qquad \Delta n_{\mathrm{H_2O}} = +2\xi \]
  2. 2. Substitute \( \xi = 1.25\ \mathrm{mol} \). \[ \Delta n_{\mathrm{H_2}} = -2(1.25) = -2.50\ \mathrm{mol} \] \[ \Delta n_{\mathrm{O_2}} = -1.25\ \mathrm{mol} \qquad \Delta n_{\mathrm{H_2O}} = +2.50\ \mathrm{mol} \]
  3. 3. Add initial amount plus change. \[ n_{\mathrm{H_2}} = 4.00 - 2.50 = 1.50\ \mathrm{mol} \] \[ n_{\mathrm{O_2}} = 2.00 - 1.25 = 0.750\ \mathrm{mol} \] \[ n_{\mathrm{H_2O}} = 0 + 2.50 = 2.50\ \mathrm{mol} \]
  4. Final answer: \(1.50\ \mathrm{mol}\) H2, \(0.750\ \mathrm{mol}\) O2, and \(2.50\ \mathrm{mol}\) H2O.

Common mistake

Do not subtract the same amount from every reactant

Incorrect reasoning

A student says that when \( \xi = 1.00\ \mathrm{mol} \), both H2 and O2 decrease by \(1.00\ \mathrm{mol}\).

This ignores the coefficient 2 in front of H2.

Correct reasoning

Use the balanced coefficients:

\[ \Delta n_{\mathrm{H_2}} = -2\xi \qquad \Delta n_{\mathrm{O_2}} = -\xi \qquad \Delta n_{\mathrm{H_2O}} = +2\xi \]

Key idea: extent is not the amount each species changes; it is the progress value multiplied by each coefficient.

Practice check

Use extent in an ammonia reaction

For the reaction below, suppose \( \xi = 0.400\ \mathrm{mol} \).

N2 + 3H2 → 2NH3

If the initial amounts are \(1.00\ \mathrm{mol}\) N2, \(2.00\ \mathrm{mol}\) H2, and \(0\ \mathrm{mol}\) NH3, what are the final amounts?

Show answer and reasoning

Write the changes from the coefficients:

\[ \Delta n_{\mathrm{N_2}} = -\xi,\qquad \Delta n_{\mathrm{H_2}} = -3\xi,\qquad \Delta n_{\mathrm{NH_3}} = +2\xi \]

Substitute \( \xi = 0.400\ \mathrm{mol} \):

\[ n_{\mathrm{N_2}} = 1.00 - 0.400 = 0.600\ \mathrm{mol} \]

\[ n_{\mathrm{H_2}} = 2.00 - 3(0.400) = 0.800\ \mathrm{mol} \]

\[ n_{\mathrm{NH_3}} = 0 + 2(0.400) = 0.800\ \mathrm{mol} \]

Answer: \(0.600\ \mathrm{mol}\) N2, \(0.800\ \mathrm{mol}\) H2, and \(0.800\ \mathrm{mol}\) NH3.

Apply the topic

A reliable strategy for extent-of-reaction tables

Step 1

Balance the equation

The coefficients are required before writing changes.

Step 2

Assign signs

Reactants are negative; products are positive.

Step 3

Multiply by \( \xi \)

Each mole change equals coefficient times extent.

Step 4

Add initial plus change

Final moles must not become negative.

In future stoichiometry and equilibrium problems, extent of reaction becomes a compact way to write all changes from a balanced equation.

Summary

What to remember

Extent measures reaction progress

The variable \( \xi \) tells how far the reaction has proceeded.

Coefficients control changes

Each mole change is the signed coefficient multiplied by \( \xi \).

Reactants decrease, products increase

Use negative signs for consumed reactants and positive signs for formed products.

Maximum extent connects to limiting reactant

The reaction cannot proceed beyond the point where a reactant would become negative.

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