This question involves determining the half-life of reactant A in a chemical reaction using kinetic data. The reaction is: 2A + B → Products. We are given experimental data with concentrations of A and B and their initial rates.

Step 1: Determine the Order of the Reaction

The rate law for the reaction can be written as: $r=k\left[A{]}^x\right[B{]}^y$ , where x and y are the orders with respect to A and B, and k is the rate constant.

Compare Experiments I and II: [A] is constant (0.10 M), [B] changes from 0.20 to 0.25 M, but the rate remains the same (6.93 × 10⁻³). This indicates that the rate does not depend on [B], so y = 0.

Now, the rate law simplifies to: $r=k[A{]}^x$

Compare Experiments I and III to find x. Use the ratio of rates:

$\frac{r_3}{r_1}=\frac{k{[A{]}_3}^x}{k{[A{]}_1}^x}={\left(\frac{{\left[A\right]}_3}{{\left[A\right]}_1}\right)}^x$

Substitute values: r₃ = 1.386 × 10⁻², r₁ = 6.93 × 10⁻³, [A]₃ = 0.20, [A]₁ = 0.10.

$\frac{1.386\times {10}^{-2}}{6.93\times {10}^{-3}}={\left(\frac{0.20}{0.10}\right)}^x$
$2=2^x\Rightarrow x=1$

So, the reaction is first order with respect to A and zero order with respect to B. The overall order is 1.

Step 2: Calculate the Rate Constant k

Use the rate law: $r=k\left[A\right]$

Take data from Experiment I: r = 6.93 × 10⁻³ mol L⁻¹ min⁻¹, [A] = 0.10 mol L⁻¹.

$k=\frac{r}{\left[A\right]}=\frac{6.93\times {10}^{-3}}{0.10}=6.93\times {10}^{-2}\text{ min⁻¹}$

Step 3: Find the Half-Life of A

For a first-order reaction, the half-life (t₁/₂) is given by: $t_{1/2}=\frac{\ln \left(2\right)}{k}=\frac{0.693}{k}$

Substitute k = 6.93 × 10⁻² min⁻¹:

$t_{1/2}=\frac{0.693}{6.93\times {10}^{-2}}=10\text{ minutes}$

Thus, the time required to consume half of A is 10 minutes.

Related Topics and Formulae

Rate Law: Expresses the relationship between the reaction rate and the concentrations of reactants. For a reaction aA + bB → products, rate = k [A]^m [B]^n, where m and n are orders.

Order of Reaction: The power to which the concentration of a reactant is raised in the rate law. It must be determined experimentally.

Half-Life for First-Order Reactions: t₁/₂ = ln(2)/k = 0.693/k. It is constant and independent of initial concentration.

Zero-Order Reactions: Rate is independent of reactant concentration. Half-life t₁/₂ = [A]₀/(2k).