Critiquing an IA: 'How does acid concentration affect the rate of reaction with magnesium?'
A student's internal assessment investigates: 'How does the concentration of hydrochloric acid affect the rate of its reaction with magnesium?' She uses 25.0 cm³ of acid at 0.50, 1.00, 1.50 and 2.00 mol dm⁻³ with a fresh 5.0 cm length of magnesium ribbon of mass 0.098 g (0.0040 mol) for each trial, at room temperature, and records the time for the gas syringe to fill with 40.0 cm³ of hydrogen (molar volume 24.0 dm³ mol⁻¹):
0.50 mol dm⁻³: 198 s; 1.00 mol dm⁻³: 96 s; 1.50 mol dm⁻³: 61 s; 2.00 mol dm⁻³: 47 s.
She calculates 1/t for each and plots it against concentration. The four points lie close to a straight line through the origin, and she concludes: '1/t is proportional to [HCl], so the reaction is first order with respect to hydrochloric acid.' She also notes in her evaluation that the reaction mixture 'got noticeably warm, especially in the 2.00 mol dm⁻³ trial'. No repeats were performed.
1. Evaluate whether the conclusion is supported by the data. Your answer should consider what 1/t actually measures, how far the reaction has progressed by the end of each trial, and whether the acid is in excess.
Read the worked answer
Moles of H₂ collected = 40.0 × 10⁻³ / 24.0 = 1.67 × 10⁻³ mol, which consumes 3.33 × 10⁻³ mol of HCl. The available HCl is 0.0250 × C, i.e. 1.25 × 10⁻² mol at 0.50 mol dm⁻³, so the acid is in excess even in the most dilute trial (about 3.8× excess) - the concentration of H⁺ is therefore a genuine variable and is not being consumed to exhaustion. That part of the design is acceptable.
The problem is what 1/t represents. It is the mean rate over the whole time taken to collect 40.0 cm³, by which point 1.67 × 10⁻³ mol of magnesium has reacted out of 4.03 × 10⁻³ mol, i.e. about 41 % of the metal, and by the stoichiometry about 27 % of the H⁺ at 0.50 mol dm⁻³ (6.7 % at 2.00 mol dm⁻³). Because the concentration falls during each trial, 1/t is not the initial rate, and the size of the fall depends on which concentration is being used. The lowest concentration therefore suffers the largest downward bias, which flattens the low-concentration end and can easily manufacture a straight line through the origin. A mean rate over a fixed volume of gas is not the same physical quantity as a rate at a stated concentration.
Ratio check: doubling 0.50 to 1.00 increases 1/t by 2.06×; doubling 1.00 to 2.00 increases it by 2.21×. These are close to first order, but the variation between them and the bias above mean the data support 'approximately first order' rather than a demonstrated order. The conclusion overstates what has been shown.
2. The temperature rise is different in each trial. Explain how this affects the measured times, estimate its size, and state the consequence for the value of the order.
Read the worked answer
The reaction of magnesium with acid is strongly exothermic, about −462 kJ per mole of magnesium. The same amount of magnesium reacts in every trial (1.67 × 10⁻³ mol), so the same heat is released each time: roughly 0.77 kJ. If the 25 cm³ of solution has a mass of about 25 g and a specific heat capacity near 4.18 J g⁻¹ K⁻¹, the maximum temperature rise with no heat loss is about 0.77 000 / (25 × 4.18) ≈ 7 °C.
Heat is lost to the surroundings throughout the reaction, so the fraction retained depends on how long the reaction takes. The 2.00 mol dm⁻³ trial finished in 47 s and so retains far more of that 7 °C than the 0.50 mol dm⁻³ trial, which took 198 s. Rate constants increase with temperature, so the mixture speeds up during the run, more so at high concentration. The measured rate at 2.00 mol dm⁻³ is therefore inflated relative to a constant-temperature measurement.
Consequence: the temperature effect acts in the same direction as the concentration effect. The slope of 1/t against [HCl] is steeper than the true concentration dependence, so the order with respect to HCl is overestimated - the near-first-order appearance is partly coincidental. The student has identified an uncontrolled variable but drawn no conclusion from it, which is the key omission in the evaluation.
3. Redesign the measurement so that a defensible order with respect to H⁺ can be obtained, keeping the same apparatus where possible. Give the key change, how the data would be processed, and one safety precaution.
Read the worked answer
Key change: restrict each trial to a small extent of reaction so that the measured rate is an initial rate. Collect only the first 10.0 cm³ of hydrogen (about 4.2 × 10⁻⁴ mol, roughly 10 % of the magnesium) or, better, log volume against time with a gas pressure sensor or balance and draw the tangent at t = 0. Clean the ribbon with emery paper and use a constant mass and constant exposed area for every trial.
Control the temperature explicitly: stand the flask in a thermostat bath at 25.0 °C and record the actual temperature in each trial, so the small residual heating is monitored rather than inferred.
Processing: convert each initial rate to a common unit (cm³ s⁻¹) and plot log(initial rate) against log[HCl]. A gradient of 1 indicates first order, 2 second order; alternatively compare rate ratios with concentration ratios for each pair, for example r₂/r₁ against C₂/C₁. Repeat each concentration at least three times, take the mean, and put error bars on the log plot so that the gradient carries an uncertainty.
Safety: hydrogen is flammable, so no flames or spark sources near the gas outlet; 2.00 mol dm⁻³ hydrochloric acid is corrosive to skin and eyes, so goggles and gloves are required, and acid spills need to be neutralised and disposed of as directed.