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 A is tripled? If the concentration of A is halved?

Step-by-Step Solution

Answer

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

Step 1: Understand the Rate Equation

The given rate equation is \(\text{Rate} = k[A]^2\), where \(k\) is the rate constant, and \([A]\) is the concentration of \(\text{A}\). This tells us how the reaction rate depends on the concentration of \(\text{A}\).

Step 2: Rate Change When Concentration is Tripled

If the concentration of \(\text{A}\) is tripled, the new concentration, \([A']\), is \(3[A]\). Substitute \([A']\) into the rate equation: \(\text{New Rate} = k(3[A])^2 = k \, 9[A]^2 = 9 \, \text{Rate}.\) Therefore, the rate of the reaction increases by a factor of 9.

Step 3: Rate Change When Concentration is Halved

If the concentration of \(\text{A}\) is halved, the new concentration, \([A']\), is \([A]/2\). Substitute \([A'/2]\) into the rate equation: \(\text{New Rate} = k\left(\frac{[A]}{2}\right)^2 = k \, \frac{[A]^2}{4} = \frac{1}{4} \, \text{Rate}.\) Therefore, the rate of the reaction decreases by a factor of 4.

Key Concepts

Rate Equation

In chemical kinetics, the rate equation is a mathematical expression that describes how the rate of a reaction changes with the concentration of reactants. It's like a recipe card that tells us exactly how the ingredients (reactants) combine to affect the final cooking time (reaction rate). Let's break it down with an example: for the reaction given in the exercise, the rate equation is \[\text{Rate} = k [\text{A}]^2\]This equation tells us a few things:

• The reaction rate is directly proportional to the square of the concentration of A. This means if you increase or decrease the concentration of A, the rate will change in a squared manner.
• \(k\) is the rate constant, which we’ll discuss later. It remains constant as long as the temperature does not change.

By understanding this equation, we can predict how the rate will change when the concentration of A is varied.

Concentration Effects

The concentration of a reactant can greatly influence the rate of a reaction. In the given rate equation, \[\text{Rate} = k [\text{A}]^2\]the concentration of A is the main factor affecting the rate. Let's see how does this effect show in two different scenarios:

• When A is Tripled: If the concentration of A is tripled, i.e., it becomes \(3[A]\), substitute it into the equation to get:\[\text{New Rate} = k (3[A])^2 = k \cdot 9[A]^2\]This means the reaction rate becomes 9 times quicker. The squaring in the rate equation makes the effects dramatic.
• When A is Halved: If the concentration of A is halved, i.e., it becomes \([A]/2\), substitute it into the equation to find:\[\text{New Rate} = k \left(\frac{[A]}{2}\right)^2 = k \cdot \frac{[A]^2}{4}\]Therefore, the rate is a quarter of the original rate or 4 times slower.

These calculations show how even small adjustments in concentration can lead to significant changes in reaction rate.

Reaction Rate

The reaction rate refers to how quickly a reaction occurs. It's like measuring how fast you are driving a car; speed varies with different conditions. Similarly, the reaction rate varies with changes in reactant concentrations as shown in the rate equation \[\text{Rate} = k [\text{A}]^2\]The factors that influence reaction rate include:

• Concentration of Reactants: As illustrated earlier, the rate depends on the concentration of reactant A.
• Temperature: Although not part of this specific exercise, increasing temperature generally increases reaction rates.
• Catalysts: These substances can speed up a reaction without being consumed.

Thus, understanding these factors helps in controlling how fast or slow a reaction proceeds.

Rate Constant

The rate constant, denoted as \(k\), is a crucial part of the rate equation \[\text{Rate} = k [\text{A}]^2\]It is a numerical value that quantifies the speed of a reaction under certain conditions, excluding the influence of concentrations.

• Nature: \(k\) remains constant for a given reaction at a fixed temperature. However, as temperature changes, \(k\) changes, because increased kinetic energy results in a faster reaction.
• Units: The units of \(k\) depend on the overall order of the reaction. In our case, with a second-order reaction, the units would be \(\text{M}^{-1}\,\text{s}^{-1}\), highlighting its dependency to normalize the concentration effect.

Understanding the rate constant helps determine the intrinsic reactivity of the reaction happening, independent of how concentrated the reactants might be. It's a piece of the puzzle in measuring how efficient a reaction truly is.