Question

For a reaction involving the decomposition of \(Z\) at a certain temperature, the following data are obtained: $$ \begin{array}{llccc} \hline \text { Rate } & 6.27 \times 10^{-3} & 5.33 \times 10^{-3} & 4.58 \times 10^{-3} & 3.54 \times 10^{-3} \\ (\mathrm{~mol} / \mathrm{L} \cdot \mathrm{min}) & & & & \\ {[\mathrm{Z}]} & 0.700 & 0.645 & 0.598 & 0.526 \\ \hline \end{array} $$ (a) What is the order of the reaction? (b) Write the rate expression for the decomposition of \(Z\) (c) Calculate \(k\) for the decomposition at that temperature.

Short answer

a) The reaction order is approximately 1, making it a first-order reaction. b) The rate expression for this first-order reaction is Rate = k[Z]. c) The rate constant 'k' for the decomposition of Z at the given temperature is approximately \(8.96 \times 10^{-3} \frac{\textrm{mol}}{\textrm{L} \cdot \textrm{min}}\).

Step-by-step solution

Calculate the reaction order

To determine the reaction order, we'll first look into the concentration and the rate changes. We can calculate the ratio of the reaction rates during two different concentration intervals. Ratio of rates: $$\frac{Rate_2}{Rate_1} = \frac{5.33 \times 10^{-3}}{6.27 \times 10^{-3}} \approx 0.85$$ Ratio of concentrations: $$\frac{[\textrm{Z}]_2}{[\textrm{Z}]_1} = \frac{0.645}{0.700} \approx 0.92$$ We want to find the reaction order, n, such that: $$(\frac{[\textrm{Z}]_2}{[\textrm{Z}]_1})^n = \frac{Rate_2}{Rate_1}$$ $$n = \log_{(\frac{[\textrm{Z}]_2}{[\textrm{Z}]_1})}(\frac{Rate_2}{Rate_1}) = \log_{0.92}(0.85) $$ By calculating the value of n, we get approximately n = 1. So, the reaction is a first-order reaction.

Write the rate expression

For a first-order reaction, the rate expression can be written as: $$Rate = k[\textrm{Z}]^n$$ Since n = 1, $$Rate = k[\textrm{Z}]$$

Calculate the rate constant 'k'

Using the rate equation and data from any row in the given table, we can solve for 'k'. Let's choose the first row (Rate = 6.27 x 10^{-3} mol/L.min and [Z] = 0.700 mol/L): $$Rate = k[\textrm{Z}]$$ $$6.27 \times 10^{-3} = k(0.700)$$ Now, solving for 'k': $$k = \frac{6.27 \times 10^{-3}}{0.700} \approx 8.96 \times 10^{-3}$$ So, the rate constant 'k' for the decomposition of Z at the given temperature is approximately \(8.96 \times 10^{-3} \frac{\textrm{mol}}{\textrm{L} \cdot \textrm{min}}\).

Key concepts

Reaction Rate

Understanding the speed at which chemical reactions occur is crucial in the field of chemistry. This speed is known as the reaction rate. It refers to the rate at which the reactants are consumed or the products are formed over time. Typically, we express this rate in terms of concentration change per unit time, such as moles per liter per minute (mol/L.min).

In our exercise, for example, the decomposition of the substance Z is analyzed by looking at how the concentration of Z decreases over time. By examining how much the concentration changes in regular intervals, scientists can get a better understanding of the reaction kinetics, which is essential for controlling industrial reactions or predicting how a reaction proceeds in nature.

Rate Expression

The relationship between the reaction rate and the concentrations of reactants is given by a rate expression, also known as a rate law. This expression mathematically defines the dependencies of the reaction rate on the concentration of each reactant.

For a simple reaction where the decomposing substance is Z, we derived a rate expression based on the order of the reaction. The order of a reaction indicates the power to which the concentration of a reactant is raised in the rate equation. In this case, we determined the reaction to be first-order with respect to Z. Hence, the rate expression becomes Rate = k[Z], where k is the rate constant, and [Z] is the concentration of the substance Z.

This rate expression allows us to predict the rate of a reaction at any given concentration of Z, provided we know the value of the rate constant.

Rate Constant (k)

The rate constant, symbolized by k, is a proportionality factor that relates the rate of a reaction to the concentrations of the reactants as per the rate expression. It is a measure of the intrinsic speed of the reaction and remains constant at a given temperature. The value of k is determined experimentally, as seen in our exercise where k was calculated using the given data points.

In first-order reactions, such as the decomposition of Z, the rate constant has units of time^-1, which signifies that the rate of the reaction is directly proportional to the concentration of Z alone. Knowing the rate constant allows chemists to calculate the time required for a certain amount of reactant to be consumed or for a given amount of product to form, which is invaluable in both research and industry.