Zero-order: $t_{1/2} = [A]_0/2k = 0.5$ min → $[A]_0/k = 1.0$ min
75% completion: $t = 0.75[A]_0/k = 0.75 \times 1.0 = \mathbf{0.75}$ min
Answer: 0.75 min
The integrated rate laws are derived by integrating the differential rate equations and provide the relationship between reactant concentration and time, allowing calculation of concentrations at any given time or determination of how long a reaction will take to reach a certain extent of completion. Zero-order reaction: d[A]/dt = -k (rate constant with units of concentration time^-1). Integration: [A] = [A]0 - kt. A plot of [A] versus t is a straight line with slope -k and y-intercept [A]0. The reaction reaches completion at t = [A]0/k. Half-life: t1/2 = [A]0/2k. First-order reaction: d[A]/dt = -k[A] (rate constant with units of time^-1). Integration: ln[A] = ln[A]0 - kt, or [A] = [A]0 * e^(-kt). A plot of ln[A] versus t is a straight line with slope -k. Half-life: t1/2 = ln2/k = 0.693/k (constant, independent of concentration). Second-order reaction (type I, one reactant): d[A]/dt = -k[A]^2. Integration: 1/[A] = 1/[A]0 + kt. A plot of 1/[A] versus t is linear with slope k. Half-life: t1/2 = 1/k[A]0. The units of k depend on the order: zero-order: mol L^-1 s^-1. First-order: s^-1. Second-order: L mol^-1 s^-1.
A zero-order reaction is one in which the rate of reaction is completely independent of the concentration of the reactant(s): rate = k. This means that even if the concentration of the reactant doubles, the rate remains unchanged. Zero-order kinetics arise when the reaction rate is limited not by the concentration of the reactant(s) but by some other factor that remains constant regardless of reactant concentration. The most important physical situation giving rise to zero-order kinetics is a heterogeneous catalytic reaction in which the catalytic surface (or enzyme active site) is completely saturated with reactant molecules at the concentrations studied. When all active sites or surface sites are occupied, adding more reactant has no effect on the rate because there are no additional sites available for the reaction to proceed. As the reaction progresses and reactant concentration falls, zero-order kinetics continue until the concentration becomes low enough that some active sites are no longer occupied — at this point the kinetics transition to a higher-order behaviour. Classic examples: Decomposition of ammonia on a hot platinum surface: 2NH3(g) → N2(g) + 3H2(g) on Pt, which follows zero-order kinetics at moderate NH3 pressures because the Pt surface is saturated. Thermal decomposition of HI on Au or Pt surfaces. Enzyme-catalysed reactions at saturating substrate concentrations (Vmax regime in Michaelis-Menten kinetics): the enzyme is fully saturated, and the rate equals kcat*[E]total regardless of [S]. Photochemical reactions where the rate depends on light intensity (absorbed photon flux) rather than on the concentration of reactants.
The half-life (t1/2) is defined as the time required for the concentration of a reactant to decrease to exactly half of its initial value. It is one of the most important and practically useful kinetic parameters. For zero-order reactions: t1/2 = [A]0/2k. This means the half-life is proportional to the initial concentration — higher initial concentration gives longer half-life. The half-life is NOT constant for zero-order reactions: as the reaction proceeds and [A] decreases, successive half-lives become shorter and shorter until the reaction is complete. For first-order reactions: t1/2 = 0.693/k. This is constant and independent of concentration — the most important and distinctive property of first-order kinetics. Radioactive decay is the paradigmatic example of first-order kinetics with constant half-life. First-order pharmacokinetics: most drugs are eliminated from the body following first-order kinetics (constant fraction eliminated per unit time), with characteristic half-lives used to determine dosing intervals and time to steady state. For second-order reactions (one reactant): t1/2 = 1/k[A]0. The half-life is inversely proportional to initial concentration — higher initial concentration gives shorter half-life, and successive half-lives become longer as concentration decreases.
Many reactions that are actually second-order (or higher order) can appear to follow simpler (lower-order) kinetics under certain experimental conditions — these are called pseudo-order reactions. Pseudo-first-order: if a second-order reaction A + B → products has its rate as: rate = k[A][B], but [B] is kept in large excess (so [B] ≈ constant), then the rate appears to be: rate = k_obs[A] where k_obs = k[B]. This is pseudo-first-order behaviour. The pseudo-first-order rate constant k_obs depends on [B]. This approach is extensively used: to simplify kinetic analysis, to measure the rate constant k by measuring k_obs at several values of [B] and plotting k_obs versus [B] (slope = k). Pseudo-first-order reactions are very common in biological systems where enzymes and other catalysts are typically in large excess over substrate, or where one reactant is the solvent (water, which has essentially constant concentration). Environmental relevance: the hydrolysis of many pesticides, pollutants, and pharmaceuticals in water follows pseudo-first-order kinetics. This allows calculation of the environmental persistence of these compounds, half-life in water bodies, and the time required for a given percentage degradation under various pH and temperature conditions. For example, the hydrolysis of atrazine (a herbicide) follows pseudo-first-order kinetics in water, with half-life varying from days to months depending on pH and temperature.
The Arrhenius equation connects reaction kinetics to thermodynamics by relating the rate constant k to the activation energy Ea and the absolute temperature T: k = A exp(-Ea/RT). For zero-order reactions specifically: the rate itself (rate = k) is temperature-dependent through k, even though the rate does not depend on concentration. A 10°C increase in temperature can double or triple the rate constant even for zero-order reactions — the zero-order nature refers to concentration dependence, not temperature dependence. This has important practical consequences: an enzyme-catalysed reaction that is zero-order at room temperature (enzyme fully saturated) may become first-order at higher temperature if the higher temperature accelerates the enzyme's turnover rate (kcat increases) such that the enzyme becomes unsaturated at the same substrate concentration. Pharmaceutical stability: drug degradation often follows zero-order kinetics under certain conditions (solid-state decomposition), and the temperature dependence follows the Arrhenius equation, allowing prediction of shelf-life at different storage temperatures from accelerated stability studies at elevated temperatures.
The principles of chemical kinetics — rate laws, rate constants, activation energies, and reaction mechanisms — are not merely of theoretical interest but underlie the design and optimisation of virtually every chemical manufacturing process and pharmaceutical product. Industrial chemical reactors are designed based on kinetic models: continuously stirred tank reactors (CSTRs) are used when the reaction is zero-order or has relatively slow kinetics, while plug flow reactors (PFRs) are used for reactions where conversion needs to be maximised. The residence time in the reactor (analogous to the reaction time in a batch reaction) is designed based on the rate law to achieve the desired conversion. In pharmaceutical development: understanding the kinetics of drug degradation (hydrolysis, oxidation, photodegradation, isomerisation) is essential for predicting shelf-life and designing stable formulations. Most drug degradation reactions follow first-order kinetics under relevant storage conditions, and the International Conference on Harmonisation (ICH) guidelines for pharmaceutical stability testing are based on measuring degradation rate constants at multiple elevated temperatures (Arrhenius approach) to predict shelf-life at normal storage conditions. Zero-order drug delivery is highly desirable: a drug delivery device that releases drug at a constant rate (zero-order release) maintains a constant blood concentration, avoids peaks (toxicity) and troughs (insufficient effect) in drug levels, and maximises therapeutic efficacy while minimising side effects. Controlled-release formulations (osmotic pumps, erosion-controlled matrix systems, transdermal patches) are designed to achieve zero-order release kinetics.