Calculating Keq — Substitution & ICE Tables
In 1927, physical chemist Farrington Daniels at the University of Wisconsin used a spectrophotometer to measure [NO₂] in a sealed N₂O₄/NO₂ flask at 55°C and calculated Keq = 0.498 mol/L — the first precise experimental determination of Keq for this equilibrium. His ICE table method, setting up I-C-E rows from initial concentrations and the measured equilibrium value, became the standard technique taught today.
Practise this lesson
Four printable worksheets that build from the foundations up to exam-style questions — start at whatever level suits you.
A chemist seals 0.500 mol of N₂O₄(g) in a 1.00 L flask at 100°C. After reaching equilibrium, they measure the concentration of NO₂ as 0.300 mol/L. Before reading on: (1) How much N₂O₄ has been consumed? (Think about the mole ratio in N₂O₄ ⇌ 2NO₂.) (2) What is the equilibrium concentration of N₂O₄? Write your reasoning before reading the worked solution.
Know
- Calculate Keq from given equilibrium concentrations by direct substitution
- Set up an ICE table with correct stoichiometric ratios in the Change row
Understand
- Use an ICE table to find equilibrium concentrations when one equilibrium concentration is given
- Verify ICE table answers by substituting equilibrium concentrations back into the Keq expression
Can Do
- Solve forward ICE problems where Keq is given and initial concentrations are known
I = Initial concentration (mol/L) | C = Change (mol/L) | E = Equilibrium (mol/L)
Change row rules:
- Use stoichiometric ratios from the balanced equation — NOT ±x for every species
- For N₂ + 3H₂ ⇌ 2NH₃: Change = −x, −3x, +2x (ratios 1:3:2)
- Equilibrium row = Initial + Change (negative changes decrease the value)
Verification: always substitute final E values back into Keq — must equal the given Keq
1. Type 1 — Direct Substitution
The simplest Keq calculation requires no algebra — you are given the equilibrium concentrations directly and simply substitute them into the Keq expression.
Procedure:
- Write the correct Keq expression first
- Substitute values with correct stoichiometric powers
- Evaluate numerator and denominator separately before dividing
- Report to appropriate significant figures
Example: 2SO₂(g) + O₂(g) ⇌ 2SO₃(g) at 600°C. At equilibrium: [SO₂] = 0.200 mol/L, [O₂] = 0.100 mol/L, [SO₃] = 0.350 mol/L.
Keq expression: Keq = [SO₃]² / ([SO₂]²[O₂])
Substitution: Keq = (0.350)² / ((0.200)²(0.100)) = 0.1225 / (0.0400 × 0.100) = 0.1225 / 0.00400 = 30.6
Direct substitution: write Keq expression first (products/reactants, stoichiometric powers); substitute equilibrium concentrations with correct powers; evaluate numerator and denominator separately; for 2SO₂ + O₂ ⇌ 2SO₃ with [SO₂]=0.200, [O₂]=0.100, [SO₃]=0.350 → Keq = (0.350)²/((0.200)²(0.100)) = 30.6.
Pause — copy the highlighted direct substitution procedure into your book before moving on.
For the equilibrium 2SO₂(g) + O₂(g) ⇌ 2SO₃(g), at equilibrium [SO₂] = 0.200, [O₂] = 0.100, [SO₃] = 0.350 mol/L. What is Keq?
2. Type 2 — Introducing the ICE Table
We just saw how to calculate Keq by direct substitution when all equilibrium concentrations are given. That raises a question: what do we do when only some equilibrium concentrations are given — how do we find the rest? This card answers it → the ICE table framework using stoichiometric ratios.
When you are given initial concentrations and only some equilibrium concentrations, you need the ICE table to find the missing values. The ICE table is the single most important calculation tool in Module 5.
ICE table rows:
- Initial (I): starting concentrations before equilibrium is reached. Products start at zero if the reaction begins with pure reactants.
- Change (C): how much each concentration changes, using stoichiometric ratios. All changes are expressed in terms of one variable (x or a given equilibrium value).
- Equilibrium (E): concentration of each species at equilibrium. Equilibrium = Initial + Change.
Think First scenario — N₂O₄ ⇌ 2NO₂:
Initial: [N₂O₄] = 0.500 mol/L; [NO₂] = 0. At equilibrium, [NO₂] = 0.300 mol/L.
Change in NO₂ = +0.300 mol/L. Stoichiometric ratio: 1 mol N₂O₄ → 2 mol NO₂. So if NO₂ increased by 0.300, N₂O₄ decreased by 0.300/2 = 0.150 mol/L.
Keq = [NO₂]² / [N₂O₄] = (0.300)² / (0.350) = 0.0900 / 0.350 = 0.257
ICE table: three rows — Initial (given starting concentrations), Change (stoichiometric ratios, reactants −, products +), Equilibrium (Initial + Change); for N₂O₄ ⇌ 2NO₂ with [NO₂]eq = 0.300 → Δ[N₂O₄] = −0.150 (ratio 1:2); Keq = (0.300)²/0.350 = 0.257.
Add the highlighted ICE table structure and worked example to your notes before continuing.
True or False: In an ICE table for N₂O₄ ⇌ 2NO₂, if [NO₂] increases by 0.300 mol/L, then [N₂O₄] decreases by 0.300 mol/L.
3. ICE Table — Worked Framework with Stoichiometric Ratios
We just saw the ICE table framework with the N₂O₄/NO₂ example. That raises a question: how do we apply the full eight-step ICE procedure to a different reaction — especially getting the stoichiometric ratios right in the Change row? This card answers it → the 2SO₃ ⇌ 2SO₂ + O₂ worked example showing the 2:2:1 ratio trap.
The ICE table is most useful when you see it applied step by step. The framework is always the same, and once you have used it three or four times it becomes automatic.
Standard ICE procedure when one equilibrium concentration is given:
- Write the balanced equation and Keq expression
- Set up ICE table with all species as columns
- Fill in Initial row from problem data
- Use the given equilibrium concentration to find x directly
- Calculate all Change row values using stoichiometric ratios
- Write Equilibrium row as Initial + Change
- Substitute Equilibrium row into Keq expression
- Solve and verify
Extended example — 2SO₃(g) ⇌ 2SO₂(g) + O₂(g):
Initial: [SO₃] = 1.00 mol/L; [SO₂] = [O₂] = 0. At equilibrium, [SO₃] = 0.780 mol/L.
Change in SO₃ = 0.780 − 1.00 = −0.220 mol/L. Stoichiometric ratios — 2 SO₃ : 2 SO₂ : 1 O₂. Change in SO₂ = +0.220 mol/L; Change in O₂ = +0.110 mol/L.
Initial
Change
Equilibrium
Keq = [SO₂]²[O₂] / [SO₃]² = (0.220)²(0.110) / (0.780)² = (0.04840)(0.110) / 0.6084 = 0.005324 / 0.6084 = 8.75 × 10⁻³
Eight-step ICE procedure: balanced equation → Keq expression → set up table → fill Initial row → use given equilibrium concentration to find x → Change row using stoichiometric ratios → Equilibrium row (I + C) → substitute into Keq → verify by back-substitution; for 2SO₃ ⇌ 2SO₂ + O₂: ratios 2:2:1, so if Δ[SO₃] = −0.220, Δ[O₂] = +0.110.
Pause — copy the highlighted eight-step procedure into your book before the check below.
For 2SO₃(g) ⇌ 2SO₂(g) + O₂(g), if SO₃ decreases by 0.220 mol/L, what is the change in O₂?
4. Forward ICE Calculation — Solving for x When Keq Is Given
We just saw how to find Keq from one known equilibrium concentration. That raises a question: how do we work backwards — given Keq and initial concentrations, how do we find all the equilibrium concentrations? This card answers it → the forward ICE calculation with the H₂/I₂/HI square-root shortcut.
The harder version of the ICE table gives you Keq and initial concentrations and asks you to find the equilibrium concentrations — this requires writing the Change row in terms of x and solving algebraically.
Example: H₂(g) + I₂(g) ⇌ 2HI(g), Keq = 54.3 at 430°C.
Initial: [H₂] = [I₂] = 0.100 mol/L; [HI] = 0. Set x = decrease in [H₂] = decrease in [I₂]; increase in [HI] = 2x.
| H₂ | I₂ | HI | |
|---|---|---|---|
| Initial | 0.100 | 0.100 | 0 |
| Change | −x | −x | +2x |
| Equilibrium | 0.100−x | 0.100−x | 2x |
Keq = [HI]² / ([H₂][I₂]) = (2x)² / (0.100−x)² = 4x² / (0.100−x)² = 54.3
Take square root of both sides (since both sides are perfect squares): 2x / (0.100−x) = √54.3 = 7.369
2x = 7.369(0.100 − x) = 0.7369 − 7.369x
9.369x = 0.7369 → x = 0.0787 mol/L
Equilibrium: [H₂] = [I₂] = 0.100 − 0.0787 = 0.0213 mol/L; [HI] = 2(0.0787) = 0.1574 mol/L
Verify: Keq = (0.1574)² / (0.0213)(0.0213) = 0.02478 / 0.000454 = 54.6 ≈ 54.3 ✓
Forward ICE (given Keq): write Change row as ±nx (n = stoichiometric coefficient) → substitute Equilibrium row into Keq → solve algebraically; for H₂ + I₂ ⇌ 2HI (Keq = 54.3), Keq = 4x²/(0.100−x)² → take √ → x = 0.0787; [H₂]=[I₂]=0.0213, [HI]=0.157 mol/L.
Add the highlighted forward ICE procedure and shortcut to your notes before continuing.
For H₂ + I₂ ⇌ 2HI with Keq = 54.3, why can we take the square root of both sides of 4x²/(0.100−x)² = 54.3?
5. Measuring Keq by Colourimetry — NO₂/N₂O₄ System
We just saw the forward ICE calculation for solving equilibrium concentrations algebraically. That raises a question: how is Keq measured experimentally in the lab — what is the NESA-specified colourimetry method? This card answers it → the five-step colourimetry procedure using Beer-Lambert law and an ICE table.
Colourimetry turns an invisible number — Keq — into something you can measure by shining a light through a brown gas and reading an absorbance value.
The equilibrium N₂O₄(g) ⇌ 2NO₂(g) is ideal because only NO₂ is coloured (brown) while N₂O₄ is colourless. The concentration of NO₂ can be determined directly from absorbance using Beer-Lambert law.
Experimental procedure:
- Build a calibration curve: measure absorbance for known NO₂ concentrations
- Measure absorbance of the equilibrium mixture
- Read off [NO₂] at equilibrium from the calibration curve
- Use ICE table to find [N₂O₄] at equilibrium (from initial amount minus consumed)
- Calculate Keq = [NO₂]² / [N₂O₄] by direct substitution
Using the Think First scenario: Starting with 0.500 mol/L N₂O₄; measuring [NO₂]eq = 0.300 mol/L; ICE gives [N₂O₄]eq = 0.350 mol/L; Keq = 0.257.
In real experiments, multiple measurements at different temperatures map the temperature-dependence of Keq — confirming that only temperature changes Keq, while concentration changes shift position without changing the constant.
Colourimetry Keq method: (1) calibration curve from known [NO₂] vs absorbance; (2) measure absorbance of equilibrium mixture; (3) read [NO₂]eq from curve; (4) use ICE table to find [N₂O₄]eq; (5) calculate Keq = [NO₂]²/[N₂O₄]; works because only NO₂ is brown; N₂O₄ is colourless.
Pause — copy the highlighted colourimetry procedure into your book before the check below.
Why is the N₂O₄ ⇌ 2NO₂ equilibrium ideal for colourimetry experiments to determine Keq?
Problem: The equilibrium PCl₃(g) + Cl₂(g) ⇌ PCl₅(g) is established at 250°C. At equilibrium: [PCl₃] = 0.130 mol/L, [Cl₂] = 0.045 mol/L, [PCl₅] = 0.380 mol/L. (a) Write the Keq expression. (b) Calculate Keq. (c) What does the magnitude of Keq indicate?
Problem: 1.00 mol of SO₃(g) is placed in a 1.00 L flask at 700°C. At equilibrium, [SO₃] = 0.780 mol/L. Equilibrium reaction: 2SO₃(g) ⇌ 2SO₂(g) + O₂(g). (a) Set up and complete the ICE table. (b) Calculate Keq. (c) Verify and interpret the magnitude.
| SO₃ | SO₂ | O₂ | |
|---|---|---|---|
| Initial | 1.00 | 0 | 0 |
| Change | −0.220 | +0.220 | +0.110 |
| Equilibrium | 0.780 | 0.220 | 0.110 |
Put the ICE table method for finding Keq in the correct order.
- Fill in the Change row using stoichiometric ratios (express as ±x or known values).
- Write the balanced equilibrium equation.
- Substitute Equilibrium concentrations into the Keq expression and solve.
- Fill in the Initial row with known starting concentrations.
- Calculate the Equilibrium row (I + C for each species).
Complete the Learn phase to unlock practice questions.
A sealed 2.00 L flask initially contains 1.20 mol of H₂ and 1.20 mol of I₂. After reaching equilibrium at 430°C for H₂(g) + I₂(g) ⇌ 2HI(g), the flask is analysed and [HI] = 0.480 mol/L is measured. (a) Calculate the initial concentrations of H₂ and I₂. (b) Set up and complete the ICE table, finding x from the given [HI]. (c) Calculate Keq and compare it to the literature value of 54.3. (d) The temperature is now increased to 600°C and Keq decreases to 38.0. Explain what this tells you about whether the forward reaction is exothermic or endothermic. (e) Without re-doing the full ICE calculation, describe qualitatively how the equilibrium concentrations of H₂, I₂, and HI change when temperature increases from 430°C to 600°C.
Look back at your Think First calculation for the N₂O₄/NO₂ scenario. Recall that Farrington Daniels used this exact system in 1927 at the University of Wisconsin — measuring just one equilibrium concentration by spectrophotometry, then using the ICE method to find everything else. Did you get N₂O₄ consumed = 0.150 mol/L and [N₂O₄]eq = 0.350 mol/L, matching Daniels' systematic approach?