Question

Consider the reaction: $$ \mathrm{A} \longrightarrow \mathrm{B} $$ The rate of the reaction is \(1.6 \times 10^{-2} \mathrm{M} / \mathrm{s}\) when the concentration of A is \(0.15 M\). Calculate the rate constant if the reaction is (a) first order in \(\mathrm{A}\) and \((\mathrm{b})\) second order in \(\mathrm{A}\).

Short answer

First order: \(k = 0.107 \, \text{s}^{-1}\). Second order: \(k = 0.711 \, \text{M}^{-1} \text{s}^{-1}\).

Step-by-step solution

Review Given Data

The rate of reaction is provided as \(1.6 \times 10^{-2} \, \text{M/s}\) and the concentration of \(\mathrm{A}\) is \(0.15 \, \text{M}\). We need to calculate the rate constant for both first-order and second-order reactions.

Identify Rate Equation for First Order

For a first-order reaction, the rate equation is given by \( \text{rate} = k[A] \), where \( k \) is the rate constant and \([A]\) is the concentration of reactant A.

Calculate First Order Rate Constant

Substitute the given values into the rate equation for a first-order reaction: \(1.6 \times 10^{-2} = k \times 0.15\). Solve for \( k \): \[ k = \frac{1.6 \times 10^{-2}}{0.15} = 0.107 \, \text{s}^{-1} \].

Identify Rate Equation for Second Order

For a second-order reaction, the rate equation is \( \text{rate} = k[A]^2 \), where \( k \) is the rate constant and \([A]\) is the concentration of reactant A.

Calculate Second Order Rate Constant

Substitute the known values into the rate equation for a second-order reaction: \(1.6 \times 10^{-2} = k \times (0.15)^2\). Solve for \( k \): \[ k = \frac{1.6 \times 10^{-2}}{(0.15)^2} = 0.711 \, \text{M}^{-1} \text{s}^{-1} \].

Key concepts

First-Order Reaction

In a first-order reaction, the rate at which the reaction occurs depends linearly on the concentration of one reactant. This means that if the concentration of reactant A is doubled, the rate of reaction also doubles. The mathematics behind a first-order reaction is relatively straightforward.

**Rate Equation:** The rate equation for a first-order reaction is expressed as \( \text{rate} = k[A] \). Here, \( k \) is the rate constant, and \([A]\) is the concentration of the reactant.

**Calculation:** Using the provided data where the rate of the reaction is \(1.6 \times 10^{-2} \text{ M/s}\) and \([A] = 0.15 \text{ M}\), we find the rate constant \( k \) by rearranging the rate equation:

• \( k = \frac{1.6 \times 10^{-2}}{0.15} = 0.107 \text{ s}^{-1} \)

It's important to note that the unit for the rate constant in a first-order reaction is \( \text{s}^{-1} \).

This simple relationship helps us to predict how quickly a reaction will proceed under changing conditions.

Second-Order Reaction

Second-order reactions have a reaction rate that depends on the square of the concentration of a single reactant or the product of two reactant concentrations. Essentially, if the concentration of A is doubled, the reaction rate increases fourfold. This provides a more dramatic effect on the reaction speed.

**Rate Equation:** In this type of reaction, the rate equation is \( \text{rate} = k[A]^2 \).

**Calculation:** Given that \(1.6 \times 10^{-2} \text{ M/s}\) is the reaction rate and the concentration is still \(0.15 \text{ M}\):

• Calculate the rate constant \( k \) using the equation: \( k = \frac{1.6 \times 10^{-2}}{(0.15)^2} = 0.711 \text{ M}^{-1} \text{s}^{-1} \).

The unit for the rate constant in second-order reactions is \( \text{M}^{-1} \text{s}^{-1} \), indicating the dependency of the rate on the concentration squared.

This deeper concentration dependency enables us to better predict outcomes in chemical reactions involving two molecules of the same or different types.

Reaction Rate

The reaction rate is a measure of how quickly reactants are converted to products in a chemical reaction. Understanding the reaction rate is crucial for predicting how fast a reaction will complete under various conditions.

**Factors Influencing Reaction Rate:**

• The concentration of reactants: Higher concentrations typically increase reaction rates.
• Temperature: Higher temperatures usually speed up reactions.
• Presence of a catalyst: Catalysts can significantly reduce the time needed for a reaction by lowering the activation energy.

In our exercise, the reaction rate is given as \(1.6 \times 10^{-2} \text{ M/s}\), which provides a snapshot of how fast the reaction progresses at a given instant. This value integrates crucial information needed to calculate the rate constant.

Rate Equation

The rate equation, also known as the rate law, is an expression that links the reaction rate with the concentrations of reactants. It helps us understand the relationship between reactant concentration and the speed of the reaction.

**General Form:** The general form of a rate equation is \( \text{rate} = k[A]^m[B]^n \) for a reaction involving reactants A and B, where \( m \) and \( n \) denote the order of the reaction with respect to each reactant.

**Order of Reaction:**

• First-order reactions depend on a single reactant raised to the first power.
• Second-order reactions depend on the square of the concentration of one reactant or the product of two reactants' concentrations.

The rate equation allows chemists to analyze and synthesize reactions by predicting how changes in concentrations impact the rate. In practical terms, it helps create conditions for desired reaction speeds.