Problem 8 - Chemistry Structure and Properties

Question

Consider a simple reaction in which a reactant A forms products: $$\mathrm{A} \longrightarrow products$$ What is the rate law if the reaction is zero order with respect to \(A\) ? First order? Second order? For each case, explain how a doubling of the concentration of A would affect the rate of reaction.

Step-by-Step Solution

Answer

Zero order: Rate = k, rate unchanged if [A] is doubled. First order: Rate = k[A], rate doubles if [A] is doubled. Second order: Rate = k[A]^2, rate quadruples if [A] is doubled.

Step 1: Zero Order Reaction

For a zero order reaction, the rate of the reaction is independent of the concentration of reactant A. The rate law can be expressed as: \[Rate = k\] where k is the rate constant. If the concentration of A is doubled, the rate of reaction remains the same because it does not depend on the concentration of A.

Step 2: First Order Reaction

For a first order reaction, the rate of reaction is directly proportional to the concentration of reactant A. The rate law is given by: \[Rate = k[A]\] where [A] is the concentration of A. If the concentration of A is doubled, the rate of reaction would also double, as the rate is directly proportional to [A].

Step 3: Second Order Reaction

For a second order reaction, the rate of reaction is proportional to the square of the concentration of reactant A. The rate law is: \[Rate = k[A]^2\] where [A] is the concentration of A. If the concentration of A is doubled, the rate of reaction would increase by four times (since 2 squared equals 4).

Key Concepts

Zero Order Reaction

When studying chemical kinetics, a zero order reaction is one where the rate at which the reactant is consumed to form the product is constant. This means the rate does not depend on the concentration of the reactant. The rate law for a zero order reaction is simply:
\[\begin{equation}Rate = k\text{ [constant]}\end{equation}\]
where 'k' is the rate constant with units of concentration/time. This is quite unique because, unlike reactions of other orders, changing the concentration of the reactant A doesn't change the rate. If the concentration of A is doubled, tripled, or halved, the rate of the zero order reaction remains unchanged.
This can be less intuitive because we often expect that adding more reactant will speed up a reaction. However, zero order kinetics can occur in situations where a catalyst is saturated, or all the active sites are occupied.
A practical example of a zero order reaction is the decomposition of nitrogen oxide on a hot platinum surface. In such a scenario, no matter how much more reactant you add, the rate of the formation of the products doesn't increase because the surface is already fully utilized.

First Order Reaction

A first order reaction demonstrates a direct relationship between the concentration of the reactant and the rate of reaction. In this kind of reaction, the rate law is given by:
\[\begin{equation}Rate = k[A]\text{ [proportional to ]}[A]\end{equation}\]
Here, 'k' is the rate constant with units of 1/time, and represents concentration. For first order reactions, if you double the concentration of reactant A, the rate of the reaction will double as well. This one-to-one correspondence makes predictions straightforward.
A classic example of a first order reaction is radioactive decay, where the rate of decay is directly proportional to the amount of the radioactive substance.
In a broader context, many biological systems follow first order kinetics as they often have mechanisms to maintain a consistent reaction rate, adjusting to changes in reactant concentration.

Second Order Reaction

Second order reactions are characterized by the rate being proportional to the square of the concentration of a single reactant, or the product of the concentrations of two reactants. The rate law for a second order reaction with respect to reactant A is expressed as:
\[\begin{equation}Rate = k[A]^2\end{equation}\]
where 'k' has units of 1/(concentration*time), ensuring that the rate has the correct units of concentration/time. If the concentration of A is doubled, the rate of the reaction increases by four times due to the squaring in the rate law. Intuitively, this means that second order reactions are very sensitive to changes in concentration.
Reactions involving bimolecular collisions, such as the reaction between nitric oxide and hydrogen:\[\begin{equation}2 NO + 2 H_2 \rightarrow N_2 + 2 H_2O\end{equation}\]can exhibit second order kinetics. These types of reactions often pertain to scenarios where the probability of two reactant molecules colliding becomes a significant factor in determining the rate.