Problem 8 - Chemistry and Chemical Reactivity

Question

A reaction has the experimental rate equation "Rate = \(k[\mathrm{A}]^{2} .\) How will the rate change if the concentration of \(\mathrm{A}\) is tripled? If the concentration of A is halved?

Step-by-Step Solution

Answer

Tripling [A] increases rate 9 times; halving [A] decreases rate 4 times.

Step 1: Understand the Rate Equation

The provided rate equation is "Rate = \(k[\mathrm{A}]^{2} \)." This indicates that the rate of the reaction is proportional to the square of the concentration of A. This implies that the rate changes with the square of any changes made to the concentration of A.

Step 2: Analyze Tripling the Concentration of A

If the concentration of A is tripled, then the new concentration becomes \(3[\mathrm{A}]\). The new rate equation becomes:\[\text{New Rate} = k(3[\mathrm{A}])^2 = k(9[\mathrm{A}]^2).\]Thus, the rate will increase by a factor of 9 if the concentration of A is tripled.

Step 3: Analyze Halving the Concentration of A

If the concentration of A is halved, then the new concentration becomes \(\frac{1}{2}[\mathrm{A}]\). The new rate equation becomes:\[\text{New Rate} = k\left(\frac{1}{2}[\mathrm{A}]\right)^2 = k\left(\frac{1}{4}[\mathrm{A}]^2\right).\]Therefore, the rate will decrease by a factor of 4 if the concentration of A is halved.

Key Concepts

Rate Law

In the world of chemical reactions, the rate law is a fundamental concept that describes how the rate of a reaction depends on the concentration of its reactants. The rate law is an equation that expresses the rate of a reaction as a function of the concentration of the reactants raised to a specific power. These powers, also known as orders, are determined experimentally and can vary from one reaction to another.

For example, in the given reaction, the rate law is expressed as "Rate = \(k[\mathrm{A}]^{2}\)." This notation tells us that the rate of the reaction is not only directly proportional to the concentration of A but is also affected exponentially by it. The exponent, 2 in this case, indicates that the rate changes with the square of any changes to the concentration of A.

To better understand the impact of concentration changes, if the concentration of A is tripled, the rate increases by a factor of 9, as the new rate equation would be \(k(3[\mathrm{A}])^2 = k(9[\mathrm{A}]^2)\). Similarly, if the concentration is halved, the new rate would be \(k\left(\frac{1}{2}[\mathrm{A}]\right)^2 = k\left(\frac{1}{4}[\mathrm{A}]^2\right)\), reducing the rate by a factor of 4.

Concentration Effects

Concentration plays a significant role in determining the speed of a chemical reaction. As outlined in the rate law, alterations in the concentration of reactants can have a substantial impact on the reaction rate.

Using the example from the original exercise, we can see the effects of changing concentrations. If the concentration of a reactant like A is increased, such as tripling it, the rate of reaction increases dramatically because of the square dependence as shown: \([3A]^2 = 9[\mathrm{A}]^2\).

On the other hand, when the concentration of A is decreased by halving it, the rate correspondingly decreases by a factor of four, calculated as \(\left(\frac{1}{2}[\mathrm{A}]\right)^2 = \frac{1}{4}[\mathrm{A}]^2\). This behavior explains why, in experiments and industrial applications, precise control of reactant concentrations is crucial in controlling reaction rates.

Chemical Kinetics

Chemical kinetics is the study of the speed, or rate, at which chemical reactions occur and the factors that affect these rates. It provides vital insight into how reactions progress over time, helping chemists control and optimize them.

Several factors influence reaction rates in chemical kinetics, including:

• Concentration of reactants: As we see in our rate law example, increasing the concentration of reactants typically increases the rate of reaction.
• Temperature: Higher temperatures generally increase reaction rates due to increased molecular activity.
• Presence of catalysts: Catalysts can speed up reactions by lowering the activation energy required.

Simplifying these concepts, we can understand chemical kinetics as the study of the conditions affecting the speed of reactions. This includes analyzing how different concentrations and other factors speed up or slow down reaction processes, aiding in predicting and managing chemical behavior effectively.