Question

The rate of a reaction is calculated as a change in concentration per time. What are the units of the rate constant, $k$, in a reaction that is second order overall with respect to one species? (Note: A second-order reaction of this type has a rate law with the form rate $=k[\mathrm{A}{]}^2,$ where $[\mathrm{A}]$ is the concentration of the species. A. $\frac{d}{2}$ B. $\frac{M}{s}$ C. 236 92 ${\mathrm{U}}^{\ast }$ D. $\frac{{\mathrm{L}}^2}{{\mathrm{mol}}^2\cdot \mathrm{s}}$

Short answer

The units of the rate constant for a second-order reaction are $\frac{\text{L}}{\text{mol}\cdot \text{s}}$.

Step-by-step solution

Understand the Rate Law

The rate law for a second-order reaction with respect to one species is given by: $$\text{rate}=k[\text{A}{]}^2$$ where $[\text{A}]$ is the concentration of the species.

Identify the Units of Rate

The rate of a reaction is expressed as change in concentration over time. Therefore, its units are: $$\frac{\text{Molarity}}{\text{seconds}}=\frac{\text{M}}{\text{s}}$$ where $\text{M}$ stands for molarity (mol/L).

Identify the Units for Concentration

The concentration of species $[\text{A}]$ is measured in molarity (M), with units of: $$\text{M}\text{ or }\frac{\text{mol}}{\text{L}}$$

Set Up the Units of the Rate Law Equation

Substitute the units for rate and concentration into the rate law: $$\frac{\text{M}}{\text{s}}=k[\text{M}{]}^2$$

Solve for the Units of the Rate Constant (k)

Isolate $k$ and solve for its units: $$k=\frac{\text{rate}}{[\text{M}{]}^2}=\frac{\frac{\text{M}}{\text{s}}}{{\text{M}}^2}=\frac{1}{\text{M}\times \text{s}}=\frac{1}{\text{M}\times \text{s}}$$ Therefore, the units of $k$ are: $$\frac{1}{\text{Molarity}\times \text{seconds}}=\frac{1}{\text{M}\times \text{s}}=\frac{\text{L}}{\text{mol}\times \text{s}}$$

Compare With Given Options

Examine the given options to find the correct one: - A. $\frac{d}{2}$ - B. $\frac{\text{M}}{\text{s}}$ - C. 236 - D. $\frac{{\text{L}}^2}{{\text{mol}}^2\times \text{s}}$ The units found are $\frac{\text{L}}{\text{mol}\times \text{s}}$, which matches none of the given options. So, there might be a typographical issue as none matches exactly.

Key concepts

second-order reaction

A second-order reaction is one where the rate of reaction is proportional to the square of the concentration of one reactant. In mathematical terms, this rate law is written as:
$$\text{rate}=k[\text{A}{]}^2$$
Here, $k$ is the rate constant, and $[\text{A}]$ is the concentration of the reactant.

Second-order reactions are intriguing because the reaction rate increases dramatically with an increase in the concentration of the reactant. This is unlike first-order reactions, where the rate simply doubles when the concentration is doubled.

Commonly, a second-order reaction might involve two molecules of the same substance reacting together. Understanding the nature of this reaction is key, as it greatly influences how you predict reaction rates and calculate various parameters rationally.

rate law

The rate law of a chemical reaction expresses the relationship between the rate of the reaction and the concentrations of the reactants. For a second-order reaction, the rate law can be written as:
$$\text{rate}=k[\text{A}{]}^2$$

Understanding the rate law helps in determining how fast a reaction proceeds and how it depends on the concentration of the reactants involved. It is essential to note that:

• $k$ is the rate constant, which is specific to a particular reaction at a particular temperature.
• The square brackets $[\text{A}]$ denote the concentration of reactant $A$.

The rate law provides insight into the reaction mechanism, helping to infer the steps involved in the reaction and predict how changes in conditions may affect the reaction rate.

reaction kinetics

Reaction kinetics studies the rates of chemical reactions and the factors that affect these rates. It provides a thorough understanding of how different variables such as temperature, pressure, concentration, and catalysts affect the speed at which reactions occur.

For second-order reactions, reaction kinetics helps explain why these reactions speed up more noticeably than first-order reactions as concentration increases. By mastering reaction kinetics concepts, you can predict the behavior of chemical systems under different conditions and understand the processes at a molecular level.

Techniques like measuring the rate constant $k$, determining rate laws, and using integrated rate equations are crucial for analyzing reaction kinetics effectively.

concentration units

In chemistry, the concentration of a substance in a solution denotes how much of that substance is present in a given volume. The commonly used unit for concentration is molarity $\text{M}$, defined as moles of solute per liters of solution:
$$\text{M}=\frac{\text{mol}}{\text{L}}$$

When dealing with rate laws in second-order reactions, concentration units are essential for calculating and understanding the reaction rate. For instance:

• In the rate law $\text{rate}=k[\text{A}{]}^2$, $[\text{A}{]}^2$ indicates the concentration squared.
• The unit for the rate is usually $\frac{\text{M}}{\text{s}}$, where M stands for molarity and s stands for seconds.

To find the units of the rate constant $k$ in a second-order reaction, you need to set up and solve the equation properly, ensuring that concentration units are consistently applied.