The decomposition of XY is second order in XY and has a rate constant of 7.02⨉10^-3 M^-1• s^-1 at a certain temperature. a. What is the half-life for this reaction at an initial concentration of 0.100 M?

Verified step by step guidance

1. Identify the order of the reaction. Since the decomposition of XY is second order, we will use the second-order half-life formula.
2. The formula for the half-life \( t_{1/2} \) of a second-order reaction is given by: \[ t_{1/2} = \frac{1}{k[A]_0} \] where \( k \) is the rate constant and \( [A]_0 \) is the initial concentration.
3. Substitute the given values into the formula: \( k = 7.02 \times 10^{-3} \text{ M}^{-1}\cdot\text{s}^{-1} \) and \( [A]_0 = 0.100 \text{ M} \).
4. Calculate the half-life using the substituted values. This will involve performing the division to find \( t_{1/2} \).
5. Interpret the result to understand the time it takes for the concentration of XY to decrease to half of its initial value at the given conditions.

Key Concepts

Second-Order Reactions

Second-order reactions are characterized by a rate that depends on the concentration of one reactant raised to the second power or on the concentrations of two reactants, each raised to the first power. The rate law for a second-order reaction can be expressed as rate = k[A]^2 or rate = k[A][B], where k is the rate constant. Understanding this concept is crucial for determining how the concentration of reactants affects the reaction rate.

Rate Constant (k)

The rate constant (k) is a proportionality factor in the rate law that is specific to a given reaction at a particular temperature. It provides insight into the speed of the reaction; a larger k indicates a faster reaction. For second-order reactions, the units of k are M^-1•s^-1, which reflects the relationship between concentration and time in the rate equation.

Half-Life of a Reaction

The half-life of a reaction is the time required for the concentration of a reactant to decrease to half of its initial value. For second-order reactions, the half-life is inversely proportional to the initial concentration and can be calculated using the formula t_1/2 = 1/(k[A]₀), where [A]₀ is the initial concentration. This relationship is essential for predicting how long it will take for a reactant to diminish in concentration.