Question

Consider the data showing the initial rate of a reaction (A \longrightarrow products) at several different concentrations of A. What is the order of the reaction? Write a rate law for the reac- tion, including the value of the rate constant, \(k\). $$ \begin{array}{cc} {[\mathrm{A}](\mathrm{M})} & \text { Initial Rate }(\mathrm{M} / \mathrm{s}) \\\ \hline 0.100 & 0.053 \\ \hline 0.200 & 0.210 \\ \hline 0.300 & 0.473 \\ \hline \end{array} $$

Short answer

The reaction is second order. The rate law is: rate = k[A]^2, and the rate constant k is approximately 5.3 M^-1s^-1.

Step-by-step solution

Examine Initial Rates vs. Concentration

Analyze the relationship between the initial concentrations of A and the corresponding initial rates of reaction. Create a table separating concentration values and initial rates to identify a pattern.

Compare Rates and Concentrations

Compare the initial rates when the concentration of A is doubled (from 0.100 M to 0.200 M) and tripled (from 0.100 M to 0.300 M). Calculate the fold increase in rate for each of these changes in concentration.

Determine Reaction Order

Use the fold increases in rates from step 2 to infer the order of the reaction. If the rate increases by a factor that is the square of the concentration change, the reaction is second-order. If the increase is the same as the concentration change, it is first-order.

Calculate the Rate Constant

Use the rate law for the determined order of the reaction and one set of concentration and rate data to solve for the rate constant, k.

Write the Rate Law

Using the order of the reaction and the calculated rate constant, write the complete rate law for the reaction.

Key concepts

Understanding Reaction Order

The concept of reaction order is fundamental to understanding the rate at which a chemical reaction proceeds. It's an exponent that indicates how the rate of a reaction is affected by the concentration of its reactants. To determine the reaction order from experimental data, you look at how changes in the concentration of a reactant affect the rate of the reaction.

For instance, if doubling the concentration of a reactant doubles the rate of the reaction, the reaction is said to be first-order with respect to that reactant. If the rate quadruples, the reaction is second-order with respect to that reactant. In more complex reactions, you may encounter mixed-order or fractional orders, indicating more nuanced relationships between concentration and rate.

In the given exercise, identifying the reaction order involves examining how the initial rate changes as the concentration of reactant A changes. This insight provides a crucial foundation for formulating the rate law, which quantifies the relationship between reactant concentration and reaction rate.

Formulating the Rate Law

The rate law is an essential equation in chemical kinetics that relates the rate of a chemical reaction to the concentration of its reactants using rate constants and reaction orders. It takes the general form \( rate = k [A]^{n} \) where \(k\) is the rate constant, \(n\) is the reaction order, and \([A]\) represents the concentration of reactant A.

Creating a rate law from experimental data involves combining your understanding of the reaction order with the rate constant to construct a mathematical representation of how fast the reaction occurs under different concentrations. In practice, this requires you to analyze how different initial rates correspond with varying initial concentrations of reactants, as shown in the exercise solution. This step-by-step analysis helps to visualize and substantiate the relationship expressed in the rate law.

By determining the reaction order and calculating the rate constant, as exhibited in the exercise, you arrive at a specific rate law for the reaction in question. This tailored rate law can predict the reaction rate at any given concentration of reactants, making it an indispensable tool in the field of chemistry.

Determining the Rate Constant

The rate constant, symbolized by \(k\), is a crucial component of the rate law equation and offers insightful information about the reaction. It is specific to each chemical reaction and varies with temperature and the presence of catalysts. The rate constant helps quantify the speed of the reaction and remains unchanged for a given reaction at a constant temperature.

In this exercise, after establishing the order of the reaction, the next step is the calculation of the rate constant. This involves using the mathematical form of the rate law and substituting in the values from one of the experiments to solve for \(k\). Once obtained, the rate constant becomes part of the finalized rate law, providing a complete description of the reaction's kinetics.

Understanding the role of the rate constant is vital for predicting the outcomes of chemical reactions under various conditions. Moreover, it is a reflection of the inherent properties of the reaction system being studied. When you've computed the rate constant accurately, as guided by the exercise, it serves as a robust predictive measure for the reaction rate at any given concentration of reactants.