In a first-order decomposition reaction, A →products, the amount of substance A at time t is x(t

step 1 In this question, we're going to be focusing on a first order decomposition reaction where we've

In a first-order decomposition reaction, A →products, the...

In a first-order decomposition reaction, $\mathrm{A} \rightarrow$ products, the amount of substance $\mathrm{A}$ at time $t$ is $$ x(t)=x(0) e^{-k t} $$ where $x(0)$ is the initial amount of $\mathrm{A}$, and $k$ is the rate constant. The time taken for the amount of A to fall to half of its initial value is called the half-life, $\tau_{1 / 2^{2}}$ of the reaction. Find the half-life for rate constants: (i) $k=3 \mathrm{~s}^{-1}$, (ii) $k=10^{-5} \mathrm{~s}^{-1}$.

In a first-order decomposition reaction, $\mathrm{A} \rightarrow$ products, the amount of substance $\mathrm{A}$ at time $t$ is
$$
x(t)=x(0) e^{-k t}
$$
where $x(0)$ is the initial amount of $\mathrm{A}$, and $k$ is the rate constant. The time taken for the amount of A to fall to half of its initial value is called the half-life, $\tau_{1 / 2^{2}}$ of the reaction. Find the half-life for rate constants: (i) $k=3 \mathrm{~s}^{-1}$, (ii) $k=10^{-5} \mathrm{~s}^{-1}$.

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Key Concepts

First-Order Reaction Kinetics

This concept refers to chemical reactions where the rate of reaction is directly proportional to the concentration of one reactant. In such reactions, the speed at which the reactant is consumed follows a predictable exponential pattern over time, and the governing differential equation produces an exponential decay function for the concentration.

Exponential Decay

Exponential decay describes the process by which a quantity decreases at a rate proportional to its current value. In the context of reaction kinetics, it is represented by the equation x(t) = x(0) e^(-kt), where the concentration of the reactant declines continuously over time, and the rate constant k determines the speed of the decay.

Half-Life

The half-life is the time required for a reactant's concentration to reduce to half of its initial value. In first-order reactions, it is mathematically derived from the exponential decay formula and is given by ?1/2 = ln(2)/k. This concept is essential because it provides a characteristic timescale for the reaction, independent of the starting concentration.

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