Question

Consider the tabulated 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, \(k\). $$ \begin{array}{cc} {[\mathrm{A}](\mathrm{M})} & \text { Initial Rate }(\mathrm{M} / \mathrm{s}) \\\ 0.12 & 0.0078 \\ \hline 0.16 & 0.0104 \\ \hline 0.20 & 0.0130 \\ \hline \end{array} $$

Short answer

The reaction is first-order with respect to [A], with a rate law of Rate = k[A] where k is approximately 0.065 M^-1s^-1.

Step-by-step solution

Analyze the Relationship Between Concentration and Initial Rate

Examine the given data to determine how the initial rate changes with the concentration of reactant A. Note that when the concentration increases, the initial rate also increases. To find the order of the reaction with respect to A, one can look for a pattern in the way the rate changes as the concentration changes.

Calculate the Rate Constant for Each Pair of Data

Use the rate equation, Rate = k[A]^n, where n is the order of the reaction. Assume n is 1 for a first-order reaction and solve for k in each case to see if it remains constant. k = Rate/[A]. Calculate the values of k for at least two different concentrations to compare.

Determine the Order of the Reaction

Compare the calculated rate constants. If they are consistent across different concentrations, the assumed order is likely correct. If the calculated values of k vary significantly, try a different reaction order and repeat step 2.

Calculate the Ratio of Rate Changes and Concentration Changes

For a reaction where the order is n, the rate changes by a factor of (concentration change)^n. Calculate the ratios of rate changes to the corresponding concentration changes. The ratio should be constant if the correct order is assumed.

Write the Rate Law Including the Value of k

Once the reaction order and the value of k are confirmed to be constant for different data points, write the rate law as Rate = k[A]^n, including the calculated value of k.

Key concepts

Chemical Kinetics

Chemical kinetics is the study of the rates at which chemical reactions proceed and the factors that affect these rates. It's crucial in understanding how reactions happen and how to control them, whether you're developing a new pharmaceutical or running a biochemical assay.

The rate of a reaction is often expressed as the change in concentration of a reactant or product per unit time. For example, if a reaction has a high rate, it means that the substances involved are being consumed or produced quickly. Factors that can influence reaction rates include reactant concentrations, temperature, the presence of a catalyst, and the physical state of the reactants. By examining how these factors affect reaction rates, chemists can deduce the mechanism of the reaction—how reactants transform into products at the molecular level.

Rate Law

A rate law is an equation that links the rate of a chemical reaction to the concentration of its reactants. It usually takes the form of \( \text{Rate} = k[\text{Reactant}]^n \), where \( k \) is the rate constant, \( [\text{Reactant}] \) is the concentration of the reactant, and \( n \) is the reaction order. The reaction order is an exponent that provides insight into the relationship between reactant concentration and reaction rate.

Determining the rate law involves identifying the reaction order and the value of \( k \) through experiments. The process typically involves measuring the initial rates of a reaction at various reactant concentrations. Through analysis, scientists can then deduce the mathematical relationship between concentration and reaction rate.

Rate Constant

The rate constant, represented by the symbol \( k \) in the rate law equation, is a proportionality constant that links the rate of the reaction to the concentrations of reactants raised to their respective powers in the rate law equation. The value of \( k \) is specific to a particular reaction at a given temperature and does not change with the concentrations of the reactants.

The units of \( k \) depend on the overall reaction order. For a first-order reaction, \( k \) has units of \( s^{-1} \); for a second-order reaction, the units are \( M^{-1}s^{-1} \), and so on. Knowing the rate constant helps chemists predict how fast a reaction will proceed under different conditions.

Initial Rate Method

The initial rate method is a practical approach to determining the reaction order by measuring the initial rate of reaction—the rate immediately after the reaction starts, before the concentrations of reactants have changed significantly. This method is useful because it eliminates the complications that arise from changes in concentrations over time.

To use this method, a series of experiments are conducted in which the concentration of one reactant is varied while holding the others constant. The initial rates of the reaction are measured for these various concentrations. By comparing how the initial rates change with the change in reactant concentrations, the reaction order with respect to each reactant can be deduced. After finding the reaction order, the rate constant \( k \) can also be calculated.