Calculating K_c from Experimental Data

0.800 mol of sulfur dioxide and 0.500 mol of oxygen are sealed in a 2.00 dm³ vessel and heated with a catalyst. At equilibrium 0.600 mol of sulfur trioxide is present. Calculate K_c, with units.

2SO₂(g) + O₂(g) ⇌ 2SO₃(g)

Step 3. K_c = ([SO₃]²) / ([SO₂]²[O₂]) = (0.300²) / (0.100² × 0.100) = (0.0900) / (0.00100).

Step 4. Units = ((mol dm⁻³)²) / ((mol dm⁻³)³) = dm³ mol⁻¹.

Answer. K_c = 90.0 dm³ mol⁻¹.

What goes wrong

• Using moles instead of concentrations gives (0.600²) / (0.200² × 0.200) = 45, which is wrong: with three terms underneath and two on top, V does not cancel.
• Halving the oxygen change is the step most often missed. The change in O₂ is half the change in SO₃.
• Volumes in cm³ must be converted to dm³ by dividing by 1000 before the division.

The one equilibrium amount that an ICE table needs has to be measured without disturbing the equilibrium. Two techniques are used.

Quench and titrate

This suits an equilibrium that contains an acid or an alkali, such as the esterification in worked example 1.

1. Mix known amounts of the reactants, with an acid catalyst, in a stoppered flask.
2. Leave the flask at a constant temperature until equilibrium is reached, which can take days at room temperature.
3. Quench a measured sample by running it into ice-cold water. Cooling and diluting slow the reaction almost to a stop.
4. Titrate quickly with standard sodium hydroxide solution, using phenolphthalein, to find the moles of acid.
5. Subtract the moles of acid catalyst, found from a separate blank titration.
6. Scale up from the sample to the whole mixture.

In worked example 1, one tenth of the equilibrium mixture needed 15.5 cm³ of 1.00 mol dm⁻³ sodium hydroxide after the catalyst was allowed for.

Answer. Moles of ethanoic acid in the whole mixture = 0.0155 × 10 = 0.155 mol.

Colorimetry

This suits an equilibrium in which one substance is coloured. Iron(III) ions and thiocyanate ions form a deep red complex:

Fe³⁺(aq) + SCN⁻(aq) ⇌ FeSCN²⁺(aq)

1. Make solutions of known concentration of the coloured substance and measure the absorbance of each with a colorimeter.
2. Plot a calibration line of absorbance against concentration.
3. Measure the absorbance of the equilibrium mixture and read its concentration from the line.

Nothing is removed or added, so the equilibrium is not disturbed.

In one experiment the mixture started with 1.00 × 10⁻³ mol dm⁻³ of each ion, and the calibration line gave [FeSCN²⁺] = 1.10 × 10⁻⁴ mol dm⁻³ at equilibrium.

Step 1. [Fe³⁺] = [SCN⁻] = 1.00 × 10⁻³ − 1.10 × 10⁻⁴ = 8.90 × 10⁻⁴ mol dm⁻³.

Step 2. K_c = ([FeSCN²⁺]) / ([Fe³⁺][SCN⁻]) = (1.10 × 10⁻⁴) / ((8.90 × 10⁻⁴)²).

Answer. K_c = 139 dm³ mol⁻¹.

Two ways to measure an equilibrium concentration: quenching a sample and titrating the acid, and using a colorimeter with a calibration line. Either result completes the ICE table.

Say

• “Change in SO₃ = +0.600 mol, so change in O₂ = −0.300 mol (ratio 2 : 1).”
• “Equilibrium concentration = (equilibrium moles) / (volume in dm³).”
• “V cancels because there are equal numbers of moles on each side.”
• “The sample is quenched in ice-cold water to stop the position of equilibrium shifting during the titration.”

Do not say

• “K_c = (0.800²) / (…)” using initial amounts (only equilibrium values go in).
• “All the changes are the same” (they follow the balancing numbers).
• “The titration gives the acid at equilibrium” without mentioning the catalyst, when one was added.

Watch for

• Data given as a percentage: “40% of the acid reacted” means the change is 0.40 × the initial moles.
• Volumes in cm³: convert to dm³ before finding concentrations.
• A titration on a portion of the mixture: scale up to the whole before using the ICE table.
• Final answers quoted to the same number of significant figures as the data, usually three.