Question

A kinetic study of the reaction: \(\mathrm{A} \rightarrow\) products provides the data: \(t=0 \mathrm{~s},[\mathrm{~A}]=2.00 \mathrm{M} ;\) \(\begin{array}{llll}500 \mathrm{~s}, & 1.00 \mathrm{M} ; 1500 \mathrm{~s}, 0.50 \mathrm{M} ; 3500 \mathrm{~s}\end{array}\) \(0.25 \mathrm{M}\). In the simplest possible way determine, whether this reaction is of (a) zero order (b) first order (c) second order (d) third order

Short answer

The reaction is first order because the concentration of reactant A halves at regular time intervals, which is characteristic of a first-order reaction.

Step-by-step solution

Understanding Rate Laws

The rate of a reaction can be described by its rate law, which shows how the rate depends on the concentration of the reactants. A zero-order reaction has a constant rate regardless of the concentration of reactants. A first-order reaction's rate is directly proportional to the concentration of one reactant. A second-order reaction's rate is proportional to either the square of the concentration of one reactant or to the product of the concentrations of two reactants. A third-order reaction's rate is proportional to the cube of the concentration of one reactant or some other combination of reactants' concentrations.

Identify Reaction Order

To determine the order of the reaction, look for a pattern in the concentration of A versus time. For a zero-order reaction, the concentration decreases linearly over time. For a first-order reaction, a plot of the natural logarithm of concentration versus time yields a straight line. For a second-order reaction, a plot of the inverse of the concentration versus time should be linear. For a third-order reaction, the relationship would be more complex and less common.

Analyze the Given Data

Assess the data provided for the reaction A -> products: at t=0s, [A]=2.00M; at t=500s, [A]=1.00M; at t=1500s, [A]=0.50M; at t=3500s, [A]=0.25M. Look for linear relationships between [A] and time (zero-order), ln[A] and time (first-order), or 1/[A] and time (second-order).

Checking for Zero-order Reaction

Check if the concentration of A decreases linearly with time. For zero-order kinetics, the rate of reaction equals k[A]⁰, which simplifies to rate=k. This implies that the concentration of A should decrease by a constant amount for equal time intervals. The given data does not show equal decrements in concentration for equal time intervals (the concentration does not halve at consistent time intervals), so the reaction is not zero-order.

Checking for First-order Reaction

First-order kinetics follow the relationship ln[A] = -kt + ln[A]₀ , where [A]₀ is the initial concentration, k is the rate constant, and t is time. The relationship between ln[A] and time (t) should be linear if the reaction is first-order. For the given concentration values, the natural log values decrease and when plotted against time will likely yield a straight line, indicating the reaction could be first-order.

Checking for Second-order Reaction

A second-order reaction can be represented as 1/[A] = kt + 1/[A]₀. If the reaction is second-order, plotting 1/[A] versus time should yield a straight line. But given that the values double for each consistent increment in time (e.g., the time interval to drop from 2.00 M to 1.00 M is the same as from 1.00 M to 0.50 M, and 0.50 M to 0.25 M), this suggests that the reaction does not follow second-order kinetics because we would expect non-uniform changes in 1/[A] over these intervals.

Checking for Third-order Reaction

Due to the low frequency of third-order reactions and the complexity of the relationships involved, in addition to the lack of a typical graph representation, the lack of fit for zero, first, and second-order models suggests the reaction is also not third-order, especially considering the consistency of halving concentrations at consistent time intervals uniquely fitting first-order kinetics best.

Conclusion Based on Data

The data suggests that the reaction is first-order since the concentration of A halves for every consistent time period, which is characteristic of first-order reactions. To confirm, one should calculate the rate constant k for the periods and verify if it remains constant.

Key concepts

Rate Laws

Rate laws are fundamental to understanding how chemical reactions occur and at what pace. A rate law is an equation that relates the rate of a chemical reaction to the concentration of its reactants. It's represented by the general formula:
rate = k[A]^x[B]^y...
where k is the rate constant, A and B are the concentrations of the reactants, and x and y are the reaction orders with respect to each reactant.
The reaction order is the sum of these exponents and indicates the reaction's sensitivity to changes in the concentrations of its reactants. Understanding the rate law of a reaction is crucial because it allows chemists to predict how a reaction will proceed under different conditions, which is essential for everything from industrial synthesis to pharmaceutical production.

Zero-Order Reaction

In a zero-order reaction, the rate is independent of the concentration of the reactants. The rate law for a zero-order reaction is given by the formula:
rate = k
This suggests that the rate constant k is the rate of the reaction itself. As such, if we were to graph the concentration of the reactant versus time, we'd expect to see a straight line, indicating a constant rate of decrease in concentration irrespective of how much reactant is left. Common examples of zero-order reactions are seen in certain biochemical processes where a catalyst (like an enzyme) is saturated with the substrate.

First-Order Reaction

A first-order reaction has a rate directly proportional to the concentration of one reactant. The rate law can be written as:
rate = k[A]
Moreover, if you take the natural logarithm of the concentrations of the reactant and plot it against time, a hallmark straight line will emerge for a first-order reaction. This straight-line graph represents an exponential decrease in concentration over time. The reactions that follow first-order kinetics are common, including radioactive decay processes and many common chemical reactions. The key indicator is that the half-life of the reactant is constant - the time it takes for the concentration to reduce by half does not change as the reaction proceeds.

Second-Order Reaction

The rate of a second-order reaction depends on the concentration of one reactant squared or the product of the concentrations of two reactants. Here, the rate law looks like:
rate = k[A]^2 or rate = k[A][B]
For a pure second-order reaction, where the rate is proportional to the square of one reactant's concentration, the plot of the inverse of the concentration ([A]^-1) versus time will yield a straight line. Unlike first-order reactions, the half-life of a second-order reaction depends on the initial concentration of the reactant - as the reaction moves forward, the half-life becomes longer.

Third-Order Reaction

Third-order reactions are relatively rare and involve either one reactant with a rate law of k[A]^3 or a mixture of reactants each contributing to the overall reaction order of three. The mechanics behind third-order reactions can be complex, and there's no straightforward graphical method for their kinetics, as seen with zero, first, or second-order reactions. Given this complexity, it's often more challenging to conclusively determine whether a reaction is third-order without extensive analysis and often more experimental data beyond simple concentration versus time plots.