Marble Chips and Hydrochloric Acid: Judging Four Rate Experiments
A group is investigating the rate of reaction between marble chips (calcium carbonate) and dilute hydrochloric acid by collecting carbon dioxide in a 500 cm3 gas syringe. Each run uses exactly 2.0 g of marble and 50.0 cm3 of acid.
Run A: large chips, 1.0 mol dm−3 acid, 30 °C - 480 cm3 of gas collected after 240 s.
Run B: powdered marble, 1.0 mol dm−3 acid, 30 °C - 480 cm3 after 52 s.
Run C: large chips, 1.0 mol dm−3 acid, 40 °C - 480 cm3 after 105 s.
Run D: large chips, 2.0 mol dm−3 acid, 30 °C - 480 cm3 after 150 s.
One student says Run C proves that 'raising the temperature by 10 °C always doubles the rate'. Another suggests that repeating Run D with 4.0 mol dm−3 acid will give twice as much gas in half the time. The teacher asks the group to test both statements against the data before writing their report. [Molar volume of gas at room conditions = 24 dm3 mol−1]
1. Explain, using collision theory, why Run B finishes far sooner than Run A, and why both still produce exactly 480 cm3 of gas.
Read the worked answer
Why the final volume is the same
n(CaCO3) = 2.0 ÷ 100 = 0.020 mol
CaCO3 + 2HCl → CaCl2 + H2O + CO2 gives 1 mol CO2 per mol CaCO3, so n(CO2) = 0.020 mol.
V = 0.020 × 24 = 0.48 dm3 = 480 cm3
There is 0.050 mol HCl available in Run A, enough to react with 0.025 mol CaCO3, so the marble is the limiting reactant. The total gas depends only on how much marble is present, not on how fast it reacts.
Why B is faster
Both runs use the same mass and the same acid, so the only difference is the size of the pieces. Powder has a much larger total surface area exposed to the acid. More acid particles are in contact with the solid at any instant, so the frequency of collisions between H+ ions and carbonate ions increases. A greater fraction of those collisions exceeds the activation energy per unit time, so the rate is higher and the same 480 cm3 is delivered in 52 s instead of 240 s.
2. Use Runs A and C to evaluate the statement that a 10 °C rise always doubles the rate.
Read the worked answer
Comparison
With the same mass, particle size and concentration, time to completion is inversely proportional to the average rate:
rate C ÷ rate A = t(A) ÷ t(C) = 240 ÷ 105 = 2.3
So the rate increased by a factor of about 2.3, not exactly 2.
Evaluation
The 'rate doubles for every 10 °C' statement is a useful rule-of-thumb, not a law. The actual factor depends on the activation energy of the particular reaction: a reaction with a high activation energy is more sensitive to temperature because a small shift in the energy distribution markedly increases the fraction of particles with energy ≥ Ea.
Two further points
- The explanation in terms of collision theory still holds: at 40 °C more particles have kinetic energy greater than the activation energy and collisions are also slightly more frequent.
- Comparing times to completion measures the average rate over the whole reaction. The doubling rule is usually applied to the initial rate, taken from the steepest tangent at t = 0. A fair test of the rule would compare initial gradients, so the statement should not be accepted or rejected too confidently from one pair of numbers.
3. Evaluate the suggestion that 4.0 mol dm−3 acid will give twice the gas in half the time, and state what the group should change if they genuinely want twice the volume of gas.
Read the worked answer
The volume claim is wrong
Gas volume is fixed by the limiting reactant. With 2.0 g of marble (0.020 mol), only 0.020 mol of CO2 can ever form - 480 cm3 - no matter how concentrated the acid is. Doubling the acid concentration doubles the amount of acid present (50.0 cm3 of 4.0 mol dm−3 = 0.20 mol HCl, far more than the 0.040 mol needed) but adds no extra carbonate, so the final volume stays 480 cm3. Only more marble raises the volume.
The time claim is only partly right
Increasing concentration does increase the rate, because more H+ ions per unit volume collide with the solid surface more frequently, so the syringe would fill sooner than 150 s. But there is no reason for it to be exactly half the time; the relationship between concentration and rate is not necessarily proportional, and the surface area becomes the limiting factor once the acid is in large excess.
Practical caution
At very high rates, gas is lost before the syringe is connected and readings become unreliable, and the acid is more corrosive.
To double the gas
Use 4.0 g of marble (0.040 mol), which needs at least 0.080 mol HCl - for example 50.0 cm3 of 2.0 mol dm−3 acid, which supplies 0.10 mol.