Problem 89 - Chemistry: The Central Science in SI Units, Global Edition

Question

Consider the reaction \(2 \mathrm{~A} \longrightarrow \mathrm{B}\). Is each of the following statements true or false? (a) The rate law for the reaction must be, Rate \(=k[\mathrm{~A}]^{2} .(\mathbf{b})\) If the reaction is an elementary reaction, the rate law is second order. \((\mathbf{c})\) If the reaction is an elementary reaction, the rate law of the reverse reaction is first order. (d) The activation energy for the reverse reaction must be smaller than that for the forward reaction.

Step-by-Step Solution

Answer

(a) The rate law for the reaction could potentially be Rate = \(k[A]^{2}\) if it is second order in A. (b) True. If the reaction is elementary, the rate law must be second order (Rate = \(k[A]^2\)). (c) True. The rate law for the reverse reaction must be first order (Rate = \(k'[B]\)). (d) False. The activation energy of the reverse reaction doesn't necessarily need to be smaller than that for the forward reaction.

Step 1: This statement is true. The rate law for a reaction is given by the rate constant (k) multiplied by the concentration of the reactants raised to their respective orders. Since the stoichiometric coefficient for A in the balanced reaction is 2, the reaction can be second order in A. However, we must also consider that the order of the reaction is determined experimentally, and the stoichiometric coefficients do not always correspond to the order of the reaction. Therefore, this statement should be clarified as "the rate law for the reaction could potentially be Rate = k[A]^2 if it is second order in A." #Statement B: If the reaction is an elementary reaction, the rate law is second order#

This statement is true. An elementary reaction is a single step reaction and the reaction order can be determined based on the stoichiometric coefficients of the reactants. In this case, the stoichiometric coefficient for A is 2, so if the reaction is elementary, the rate law must be second order (Rate = k[A]^2). #Statement C: If the reaction is an elementary reaction, the rate law of the reverse reaction is first order#

Step 2: This statement is true. The reverse reaction has the form: B -> 2A. For elementary reactions, the reaction order corresponds to the stoichiometric coefficients of the reactants. Since the stoichiometric coefficient for B in the reverse reaction is 1, the rate law for the reverse reaction must be first order (Rate = k'[B]). #Statement D: The activation energy for the reverse reaction must be smaller than that for the forward reaction#

This statement is false. There is no rule that states the activation energy of the reverse reaction must be smaller than that for the forward reaction. The activation energy of a reaction is determined by the energy barrier that needs to be surpassed for the reaction to occur. The activation energies for the forward and reverse reactions can be different, but there is no requirement that one must be smaller than the other.

Key Concepts

Elementary Reaction

Elementary reactions are fundamental building blocks of chemical reactions. They occur in a single step and their rate laws can be written directly from their balanced chemical equations. This is because, for elementary reactions, the reaction order is the same as the stoichiometric coefficient of each reactant. For example, in a reaction like \(2\, \text{A} \rightarrow \text{B}\), if it is an elementary reaction, the rate law will be Rate \(= k[\text{A}]^2\), indicating a second-order reaction. Elementary reactions provide a direct insight into the molecularity of the reaction, which refers to the number of reactant molecules involved in the process.

A unique feature of elementary reactions is that everything happens in a single collision event between molecules. This is in contrast to complex reactions, which involve multiple steps and intermediate species. Hence, it's crucial to experimentally determine whether a reaction is elementary or not, as this impacts the reaction order reflected in the rate law.

Reaction Order

The reaction order indicates how the rate of a reaction depends on the concentration of each reactant. For the reaction \(2\, \text{A} \rightarrow \text{B}\), the order can be directly obtained from the stoichiometric coefficients if it is known to be an elementary reaction. So, if this reaction is elementary, it is a second-order reaction with respect to A, and its rate law can be represented as Rate \(= k[\text{A}]^2\).

Reaction order can greatly affect the kinetics and behavior of chemical reactions:

• A zero-order reaction means the rate is independent of the concentration of reactants.
• A first-order reaction depends linearly on the concentration of a single reactant.
• A second-order reaction might depend on the concentration of two separate reactants or, as in the example, the square of one reactant's concentration.

Note that while stoichiometry often suggests the reaction order, it doesn't always dictate it, especially in complex reactions that aren't elementary. Experimental data are crucial to confirm the actual reaction order.

Activation Energy

Activation energy is the minimum energy that reacting species must absorb to start a chemical reaction. It represents the energy barrier to be overcome for reactants to transform into products. In the scenario of \(2\, \text{A} \rightarrow \text{B}\), the activation energy would influence how quickly \(\text{A}\) becomes \(\text{B}\).

For reversible reactions, it's essential to understand that the activation energy for the forward reaction can be quite different from that of the reverse reaction. While some might assume that the reverse reaction (\(\text{B} \rightarrow 2\, \text{A}\)) should have a smaller activation energy, there's no rule mandating this. Instead, the relative activation energies depend on the specific energy landscape of that chemical process.

Understanding activation energy helps in the design of catalysts, which act by lowering the activation energy barrier of a reaction without being consumed, thus speeding up the reaction. Moreover, it also aids in predicting how temperature changes can affect reaction rates, based on the Arrhenius equation \(k = Ae^{-E_{a}/RT}\), where \(E_{a}\) stands for activation energy, \(R\) is the gas constant, and \(T\) is the temperature.