Test a constant-fraction reaction model

Repeated halving predicts a curve

The fictional reaction’s ten-second half-life means that each successive ten seconds leaves half of the A that was present at the beginning of that interval. Starting from 1 mol/L, the sequence is 1, 0.5, 0.25, 0.125 and 0.0625 mol/L at 0, 10, 20, 30 and 40 s. It does not reach zero at 20 s: the second interval removes half of the remaining amount, not another half of the original amount.

Try ten-second windows at several starting times. The ratio of ending to starting A stays at 0.5. The amount consumed and the average disappearance rate both decline. This constant fractional change is the key prediction being tested inside the model.

Connect the numerical pattern to an equation

[A]t = [A]0 × 2^(−t/10) expresses repeated halving for a half-life of 10 s. Here [A]0 is the initial concentration and t is elapsed seconds. The exponent is dimensionless because t is divided by a time. At 20 s, the exponent is −2 and the remaining fraction is 1/4. At 5 s, the fraction is 1/√2, approximately 0.707107; a half interval does not remove exactly a quarter of the original amount.

The more general first-order expression is [A]t = [A]0 exp(−kt). The exponential function exp uses the number e as its base. The rate constant k has units s⁻¹, and this model assigns k = ln(2)/10 = approximately 0.069315 s⁻¹. The natural logarithm ln is the inverse of exp. You can use the repeated-halving version for the guided investigation without deriving the exponential equation.

The instantaneous disappearance rate is k[A]. At time zero it is about 0.069315 mol/(L s). At 10 s it is about 0.034657 mol/(L s). The ten-second average between them is 0.05 mol/(L s), which lies between those endpoint rates. This relationship follows from a smoothly declining rate here; it is not a rule for every possible time course.

Distinguish assumed law from evidence

The reaction equation A → B supplies a stoichiometric relationship, not experimental proof of first-order kinetics. The model explicitly assigns the rate law. A real investigation would measure concentrations at multiple times, control relevant conditions, quantify uncertainty and compare competing predictions. A straight-looking trend over a narrow range is weak evidence for a universal law.

For example, a constant-amount-loss model that removes 0.05 mol/L each second agrees with this model at 0 and 10 s. It predicts zero A at 20 s, while the first-order model predicts 0.25 mol/L. A sufficiently precise measurement near 20 s can distinguish those predictions. Merely matching the first two observations cannot.

The ideal equation remains positive at every finite time. A plotted curve may look flat or a rounded reading may show zero while a small concentration remains. Actual detection limits, finite molecule counts, reverse reactions and side reactions can make the ideal continuous model inappropriate. This widget does not represent those effects, and its flattening is not evidence of dynamic equilibrium.

Rate and final amount are different outcomes

Changing a first-order rate constant changes how quickly conversion occurs. In the stipulated irreversible model, the limiting product concentration is still the initial A concentration when time grows without bound. That statement does not imply catalysts improve equilibrium yield in real reversible systems, nor that a slow experiment will actually reach its theoretical endpoint before other processes intervene. This widget keeps k fixed; it lets you change the observation window, not chemical conditions.

Transfer: a different sample starts at 0.8 mol/L with the same ten-second half-life. Predict A at 30 s and its average disappearance rate from 20 to 30 s. This initial concentration is a written extension, not a widget setting.

Check the half-life transfer

At 20 s, A = 0.2 mol/L. At 30 s, A = 0.1 mol/L. The average disappearance rate is (0.2 − 0.1)/10 = 0.01 mol/(L s). The fraction remaining is unchanged by initial concentration under the stipulated first-order law; the absolute amount and rate scale with it.

OpenStax integrated rate laws gives the general first-order equation and half-life relationships.