Reactive Systems Supplementary on Rates of Reactions, Differential and Integrated Rate Laws First-order reactions For first-order reactions: AB The time rate of change of the concentration of A, (given by [A]) is: d[𝐴] d[𝐵] = −𝑘 𝐴 = d𝑡 d𝑡 Differential rate law, describing rates ∞ Rearrangement of this equation yields: 𝐴 ln 𝐴 𝑡 = −𝑘𝑡 d[𝐴] =− 𝐴 𝐴 𝑡 = 𝐴 0𝑒 −𝑘𝑡 Integral rate law, describing concentrations ∞ 𝑘𝑑𝑡 First-order reactions Plots the [A(t)]/[A(0)] in the upper panel and the natural log of [A(t)]/[A(0)] in the lower panel as a function of time for a first-order reaction Note that the slope of the line in the lower panel is -k and that the concentration falls to 1/e of its initial value after a time =1/k, often called the lifetime of the reactant A related quantity is the time it takes for the concentration to fall to half of its value, obtained from: 𝐴 (𝑡 = 𝜏1/2 ) = = exp⁡(−𝜏1/2 ) [A]0 𝜏1/2 ln⁡(2) = 𝑘 The quantity 1/2 is known as the half-life of the reactant Pseudo first-order reactions For pseudo first-order reactions: A + B  products The time rate of change of the concentration of A, (given by [A]) is: d[𝐴] = −𝑘 𝐴 [𝐵] d𝑡 If [B] remains constant (by either setting [B]0 >> [A]0 or recycling B): ∞ Rearrangement yields: 𝐴 ln 𝐴 𝑡 = −𝑘 𝐵 0𝑡 𝐴 𝑡 = 𝐴 0𝑒 −𝑘 𝐵 0𝑡 d[𝐴] =− 𝐴 ∞ 𝑘 𝐵 0𝑑𝑡 Opposing reactions, equilibrium For pseudo first-order reactions: A⇋B The time rate of change of the concentration of A, (given by [A]) is: d[𝐴] d[𝐵] =− − 𝑘1 𝐴 + 𝑘−1 𝐵 d𝑡 d𝑡 Defining the equilibria as: and [𝐴]𝑒 = [𝐴]0 −𝑥𝑒 [𝐵]𝑒 = [𝐵]0 +𝑥𝑒 𝐴 𝑡 = 𝐴 + 𝑥𝑒 𝑒 −(𝑘1+𝑘−1)⁡𝑡 This defines the characteristic time for a system to reach equilibrium It is dominated by the largest rate constant Parallel reactions For parallel reactions: A  C (k1) and B  D (k2) Consecutive reactions and steadystate approximation For consecutive reactions: A  B (k1) and B  C (k2) Consecutive reactions and steadystate approximation For parallel reactions: A  C (k1) For k1 = 10k2 and B  D (k2) For k2 = 10k1