Kinetics: Rate Equations and Activation Energy: Practice Questions

In the gas-phase decomposition of dinitrogen pentoxide, the following rate equation was experimentally determined:
\( \text{rate} = k[\text{N}_2\text{O}_5] \)
What are the units of the rate constant, \( k \), for this first-order reaction?

The initial rates of the reaction \( A(aq) + B(aq) \rightarrow D(aq) \) catalysed by \( C(aq) \) were measured at a constant temperature. The following results were obtained:
Experiment 1: \( [A] = 0.10, [B] = 0.10, [C] = 0.05 \), rate = \( 1.2 \times 10^{-4} \text{ mol dm}^{-3} \text{ s}^{-1} \)
Experiment 2: \( [A] = 0.20, [B] = 0.10, [C] = 0.05 \), rate = \( 4.8 \times 10^{-4} \text{ mol dm}^{-3} \text{ s}^{-1} \)
Experiment 3: \( [A] = 0.10, [B] = 0.20, [C] = 0.05 \), rate = \( 1.2 \times 10^{-4} \text{ mol dm}^{-3} \text{ s}^{-1} \)
Experiment 4: \( [A] = 0.10, [B] = 0.10, [C] = 0.10 \), rate = \( 2.4 \times 10^{-4} \text{ mol dm}^{-3} \text{ s}^{-1} \)
What is the rate equation for this reaction?

Consider the following proposed two-step mechanism for a reaction between nitrogen dioxide and carbon monoxide:
Step 1: \( \text{NO}_2 + \text{NO}_2 \rightarrow \text{NO}_3 + \text{NO} \) (slow)
Step 2: \( \text{NO}_3 + \text{CO} \rightarrow \text{NO}_2 + \text{CO}_2 \) (fast)
Which of the following correctly identifies the overall balanced equation and the role of \( \text{NO}_3 \) in the reaction?

The addition of a catalyst to a chemical reaction provides an alternative pathway with a lower activation energy. How does the addition of a catalyst affect the Maxwell-Boltzmann distribution of molecular energies and the value of the activation energy, \( E_a \), at a constant temperature?

The rate equation for the reaction between two substances, \( X \) and \( Y \), is determined to be:
\( \text{rate} = k[X][Y]^2 \)
What are the correct units for the rate constant, \( k \)?

A chemical reaction follows the rate equation \(\text{rate} = k[\text{X}][\text{Y}]^{2}\). State the overall order of the reaction and deduce the units for the rate constant \(k\), assuming concentration is in \(\text{mol dm}^{-3}\) and time is in \(\text{s}\).

The rate constant for the thermal decomposition of a hydrocarbon is \(2.10 \times 10^{-4} \text{ s}^{-1}\) at \(600\text{ K}\) and \(8.40 \times 10^{-3} \text{ s}^{-1}\) at \(650\text{ K}\).
Calculate the activation energy, \(E_a\), for this reaction in \(\text{kJ mol}^{-1}\).
(Given: \(R = 8.31 \text{ J K}^{-1} \text{ mol}^{-1}\))

A reaction of the type \(\text{P} + \text{Q} \rightarrow \text{Products}\) was studied. When the initial concentration of \(\text{P}\) is doubled with \([\text{Q}]\) kept constant, the rate quadruples. When the initial concentration of \(\text{Q}\) is tripled with \([\text{P}]\) kept constant, the initial rate remains unchanged.
Write the rate equation for this reaction and state its overall reaction order.

The reaction between peroxodisulfate(VI) ions and iodide ions is represented by the following equation:
\( S_2O_8^{2-}(aq) + 2I^-(aq) \rightarrow 2SO_4^{2-}(aq) + I_2(aq) \)
An investigation was carried out to determine the initial rate of this reaction at a constant temperature. The results are shown in the table below:

Experiment 1: \([S_2O_8^{2-}] = 0.040\text{ mol dm}^{-3}\), \([I^-] = 0.020\text{ mol dm}^{-3}\), \(\text{Initial Rate} = 1.2 \times 10^{-5}\text{ mol dm}^{-3}\text{ s}^{-1}\)
Experiment 2: \([S_2O_8^{2-}] = 0.080\text{ mol dm}^{-3}\), \([I^-] = 0.020\text{ mol dm}^{-3}\), \(\text{Initial Rate} = 2.4 \times 10^{-5}\text{ mol dm}^{-3}\text{ s}^{-1}\)
Experiment 3: \([S_2O_8^{2-}] = 0.040\text{ mol dm}^{-3}\), \([I^-] = 0.040\text{ mol dm}^{-3}\), \(\text{Initial Rate} = 2.4 \times 10^{-5}\text{ mol dm}^{-3}\text{ s}^{-1}\)

(a) Deduce the order of reaction with respect to \( S_2O_8^{2-} \) and \( I^- \). Justify your answer.
(b) Write the rate equation for the reaction.
(c) Calculate the value of the rate constant, \( k \), using the data from Experiment 1. Include units in your answer.

The decomposition of dinitrogen pentoxide, \( 2N_2O_5(g) \rightarrow 4NO_2(g) + O_2(g) \), follows first-order kinetics. A series of experiments were conducted at different temperatures to determine the activation energy, \( E_a \), for the reaction.

(a) Define the term activation energy.
(b) A plot of \( \ln k \) against \( 1/T \) (where \( T \) is the temperature in Kelvin) yielded a straight line with a gradient of \( -1.24 \times 10^4 \text{ K} \). Calculate the activation energy, \( E_a \), in \( \text{kJ mol}^{-1} \). (The gas constant \( R = 8.31 \text{ J mol}^{-1}\text{ K}^{-1} \)).
(c) Using your knowledge of the Maxwell-Boltzmann distribution, explain how an increase in temperature increases the rate of this reaction.
(d) In the presence of a catalyst, the reaction rate increases. Explain how a catalyst affects the activation energy and how this is reflected in the Arrhenius equation, \( k = Ae^{-E_a/RT} \).