Question

Rearrange the integrated rate equations for (a) a first-order reaction, (b) a second-order reaction, and (c) a zero-order reaction to calculate \([A]_{t}\). Use the symbol \([A]_{0}\) to represent the initial concentration if needed.

Short answer

\([A]_t = [A]_0 e^{-kt}\) for first-order, \([A]_t = \frac{1}{\frac{1}{[A]_0} + kt}\) for second-order, and \([A]_t = [A]_0 - kt\) for zero-order reactions.

Step-by-step solution

First-Order Reaction

The integrated rate equation for a first-order reaction is given by \(\ln\left([A]_t\right) = \ln\left([A]_0\right) - kt\). To calculate \([A]_t\), exponential both sides to undo the natural logarithm, resulting in \([A]_t = [A]_0 e^{-kt}\).

Second-Order Reaction

For a second-order reaction, the integrated rate law is \(\frac{1}{[A]_t} = \frac{1}{[A]_0} + kt\). To isolate \([A]_t\), take the reciprocal of both sides, yielding \([A]_t = \frac{1}{\frac{1}{[A]_0} + kt}\).

Zero-Order Reaction

The integrated rate equation for a zero-order reaction is \([A]_t = [A]_0 - kt\). Since this is already solved for \([A]_t\), no further steps are required.

Key concepts

Understanding First-Order Reactions

In the realm of chemical kinetics, first-order reactions are characterized by their property that the rate at which they proceed is directly proportional to the concentration of a single reactant. This is mathematically represented by the rate law: Rate = k[A], where 'k' is the rate constant and '[A]' represents the concentration of the reactant.

With first-order reactions, the time taken for the concentration of the reactant to decrease by half is constant, regardless of the starting concentration. This is referred to as the half-life of the reaction. The integrated rate equation, \[ \ln\left([A]_t\right) = \ln\left([A]_0\right) - kt \], succinctly captures the relationship between the reactant concentration at any time (\([A]_t\)) and the initial concentration (\([A]_0\)).

By taking the exponential of both sides, we find a straightforward formula for calculating the concentration at any given time, which is \([A]_t = [A]_0 e^{-kt}\). This equation implies that as time progresses, the concentration of the reactant decreases exponentially, and it aids in predicting the extent of reaction over time.

Diving into Second-Order Reactions

Second-order reactions differ from first-order reactions in that the rate of the reaction is proportional to the square of the concentration of one reactant or to the product of the concentrations of two reactants. The general rate law for a single reactant is given by: Rate = k[A]^2.

The integrated rate equation for second-order reactions looks like this: \[ \frac{1}{[A]_t} = \frac{1}{[A]_0} + kt \]. This relationship reflects how the inverse of the reactant concentration changes linearly with time, unlike the exponential change seen in first-order reactions.

To find the concentration of the reactant at a particular time (\([A]_t\)), you take the reciprocal of the equation, resulting in: \([A]_t = \frac{1}{\frac{1}{[A]_0} + kt}\). As time advances, the concentration diminishes, but unlike in first-order reactions, the rate at which the concentration reduces increases over time. This dynamic is crucial for understanding how certain reactions, like those that form complex molecules, progress.

Zero-Order Reactions Simplified

Zero-order reactions are conceptually simpler in comparison to first and second-order reactions, as the rate at which they proceed is independent of the concentration of the reactant. For these reactions, the rate law is: Rate = k, with 'k' being the rate constant, signifying a constant rate of reaction as long as the reactant is present.

The integrated rate equation for zero-order reactions is a direct expression of this: \([A]_t = [A]_0 - kt\), where \([A]_t\) represents the concentration of the reactant at time 't' and \([A]_0\) is the initial concentration. This linear decay in concentration makes zero-order reactions predictable over time; the concentration decreases at a constant rate until the reactant is depleted.

These types of reactions often occur under conditions where the reaction rate is limited by a factor other than the reactant's concentration, for example, surface catalyzed reactions where the reaction rate is limited by the surface area of the catalyst.