Let's solve the question step by step: --- ### (a) **Determine, with justification, the order of the reaction.** **Given:** - Rate constant \( k = 1.70 \times 10^{-2} \) **dm³ mol⁻¹ s⁻¹** at 24°C - Rate constant \( k = 2.01 \times 10^{-2} \) **dm³ mol⁻¹ s⁻¹** at 37°C #### **Justification:** The units of the rate constant tell us the order of the reaction: - **Zero order:** \( k \) has units of mol dm⁻³ s⁻¹ - **First order:** \( k \) has units of s⁻¹ - **Second order:** \( k \) has units of dm³ mol⁻¹ s⁻¹ - **Third order:** \( k \) has units of dm⁶ mol⁻² s⁻¹ **Here, the given units are dm³ mol⁻¹ s⁻¹, which matches a SECOND ORDER reaction.** **Answer:** > The reaction is **second order** because the units of the rate constant are **dm³ mol⁻¹ s⁻¹**, which are the units for a second order rate constant. --- ### (b) **Evaluate the Arrhenius parameters of the reaction.** #### **The Arrhenius equation:** \[ k = A e^{-E_a/(RT)} \] or in logarithmic form: \[ \ln k = \ln A - \frac{E_a}{R} \cdot \frac{1}{T} \] Given two temperatures and their rate constants, use: \[ \ln\left(\frac{k_2}{k_1}\right) = \frac{E_a}{R}\left(\frac{1}{T_1} - \frac{1}{T_2}\right) \] **Given:** - \( k_1 = 1.70 \times 10^{-2} \) at \( T_1 = 24^\circ C = 297 \text{ K} \) - \( k_2 = 2.01 \times 10^{-2} \) at \( T_2 = 37^\circ C = 310 \text{ K} \) - \( R = 8.314 \text{ J mol}^{-1} \text{ K}^{-1} \) #### **Step 1: Calculate \( E_a \) (activation energy):** \[ \ln\left(\frac{2.01 \times 10^{-2}}{1.70 \times 10^{-2}}\right) = \frac{E_a}{8.314}\left(\frac{1}{297} - \frac{1}{310}\right) \] \[ \ln\left(\frac{2.01}{1.70}\right) = \frac{E_a}{8.314}\left(\frac{1}{297} - \frac{1}{310}\right) \] \[ \ln(1.182) \approx 0.167 \] \[ \frac{1}{297} - \frac{1}{310} = \frac{310 - 297}{297 \times 310} = \frac{13}{92070} \approx 0.0001412 \] So, \[ 0.167 = \frac{E_a}{8.314} \times 0.0001412 \] \[ E_a = \frac{0.167}{0.0001412} \times 8.314 \] \[ E_a = (1183.3) \times 8.314 \approx 9835 \, \text{J mol}^{-1} = 9.84 \, \text{kJ mol}^{-1} \] #### **Step 2: Calculate \( A \) (pre-exponential factor):** Use the Arrhenius equation with one data point (e.g., at 24°C): \[ k = A e^{-E_a/(RT)} \] \[ 1.70 \times 10^{-2} = A e^{-9835/(8.314 \times 297)} \] Calculate the exponent: \[ \frac{9835}{8.314 \times 297} = \frac{9835}{2467.258} \approx 3.987 \] \[ e^{-3.987} \approx 0.01857 \] So, \[ 1.70 \times 10^{-2} = A \times 0.01857 \] \[ A = \frac{1.70 \times 10^{-2}}{0.01857} \approx 0.915 \, \text{dm}^3 \text{mol}^{-1} \text{s}^{-1} \] --- ## **Summary Table:** | Parameter | Value | |------------------|------------------------------------------| | **Order** | **Second Order** | | **Eₐ** | **9.84 kJ mol⁻¹** | | **A** | **0.915 dm³ mol⁻¹ s⁻¹** | --- **Full Answers:** --- ### (a) **Order of the reaction** The reaction is **second order** because the units of the rate constant are **dm³ mol⁻¹ s⁻¹**, which is characteristic of a second order reaction. --- ### (b) **Arrhenius parameters** - **Activation energy, \( E_a \) = 9.84 kJ mol⁻¹** - **Pre-exponential factor, \( A = 0.915 \) dm³ mol⁻¹ s⁻¹**