Question

Consider the reaction \(\mathrm{Z} \longrightarrow\) products The data below give the rate of the reaction and the concentration of \(Z(\) as \(M)\) $$ \begin{array}{cc} \hline \text { rate (M/min) } & {[\mathrm{Z}]} \\ \hline 0.02 & 0.1 \\ 0.08 & 0.2 \\ 0.32 & 0.4 \\ 1.28 & 0.8 \\ \hline \end{array} $$ (a) What is the order of the reaction with respect to \(Z\) ? (b) What is the rate expression for the reaction? (c) Estimate the value of \(k\).

Short answer

Question: Determine the order of the reaction with respect to Z and the value of the rate constant k. Answer: (a) The order of the reaction with respect to Z is 2. (b) The value of the rate constant k is 2 M^{-1}min^{-1}.

Step-by-step solution

Determine the order of the reaction with respect to Z

To determine the order of the reaction with respect to Z, we need to find the relationship between the rate of reaction and the concentration of Z. We can do this by taking the given data and analyzing how the rate changes when the concentration of Z is changing. From the given reaction table, let's take two sets of data points and analyze them. We have: 1st set: Rate: 0.02 M/min [Z]: 0.1 M 2nd set: Rate: 0.08 M/min [Z]: 0.2 M Ratio of rates: (0.08 M/min) / (0.02 M/min) = 4 Ratio of concentrations: (0.2 M) / (0.1 M) = 2 Next, we need to determine what power the concentration ratios must be raised to, in order to equal the ratio of rates. Let's call the order of the reaction n: \((\text{concentration ratio})^n = \text{rate ratio}\) \((2)^n = 4\) The value of n is 2. Therefore, the order of the reaction with respect to Z is 2.

Write the rate expression for the reaction

Using the determined reaction order, we can now write the rate expression for the reaction: Rate = k[Z]^2

Estimate the value of the rate constant k

To estimate the value of the rate constant k, we can use the rate expression and the given data. Let's use the data from the first set: Rate = 0.02 M/min [Z]= 0.1 M 0.02 M/min = k(0.1 M)^2 Now, we can solve for k: k = (0.02 M/min) / (0.1 M)^2 k = 2 M^{-1}min^{-1} The value of k is 2 M^{-1}min^{-1}. #Summary# In conclusion: (a) The order of the reaction with respect to Z is 2. (b) The rate expression for the reaction is Rate = k[Z]^2 (c) The value of the rate constant k is estimated to be 2 M^{-1}min^{-1}.

Key concepts

Rate of Reaction

Understanding the rate of a chemical reaction is fundamental in the study of kinetics, which deals with the speed at which reactions occur. The rate of a reaction can be thought of as the change in concentration of a reactant or product per unit time. For a simple decomposition reaction such as \(\mathrm{Z} \longrightarrow\) products, the rate can be measured by how quickly \(\mathrm{Z}\) decreases over time.

When analyzing the rate, it's important to notice how changes in the concentration of \(\mathrm{Z}\) affect the speed of the reaction. In the exercise, by comparing different sets of data, we can deduce that the rate of reaction is not proportional to the concentration of \(\mathrm{Z}\) in a simple linear way; instead, it follows a pattern that indicates the reaction is of a certain order.

Concentration of Reactants

The concentration of reactants plays a crucial role in the kinetics of a chemical reaction. It's integral in determining not just the rate, but also the order of the reaction. In a given rate law equation, the exponents of the concentration terms indicate the order of the reaction with respect to each reactant.

In our exercise, we see that when the concentration of \(\mathrm{Z}\) doubles, the rate of the reaction increases by a factor of four, suggesting that the reaction order with respect to \(\mathrm{Z}\) is 2. This is indicative of a second-order reaction, where the rate is proportional to the square of the reactant's concentration, highlighting a quadratic relationship between the rate and \(\mathrm{Z}\)'s concentration. Therefore, in the rate expression \(\text{Rate} = k[\mathrm{Z}]^2\), the exponent '2' reflects this second-order behavior.

Rate Constant k

The rate constant, represented as \(k\), is a proportionality factor that relates the rate of a reaction to the concentrations of the reactants. It is unique for each chemical reaction and can be affected by factors such as temperature and the presence of a catalyst. In a rate law expression, \(k\) remains constant at a given temperature and provides the necessary link between reactant concentration and reaction rate.

To determine \(k\) for our reaction, we need to plug in the values of the rate and concentration into the rate law expression derived from the experimental data. As solved in the exercise, using the concentration and rate from the first set of data and the determined second-order rate law, we find that \(k = 2 M^{-1}min^{-1}\). This constant is crucial for predicting the behavior of the reaction under different concentrations and is the key to converting between rate and concentration in the mathematical model of a chemical reaction.