The reaction SO2Cl2(g) → SO2(g) + Cl2(g) is first order in SO2Cl2. Using the following kinetic data, determine the magnitude and units of the first-order rate constant: Time (s) Pressure SO2Cl2 (atm) 0 1.000 2500 0.947 5000 0.895 7500 0.848 10,000 0.803

Verified step by step guidance

Step 1: Understand that for a first-order reaction, the rate law is expressed as \( \text{Rate} = k [\text{SO}_2\text{Cl}_2] \), where \( k \) is the rate constant.

Step 2: Use the integrated rate law for a first-order reaction, which is \( \ln [\text{SO}_2\text{Cl}_2]_t = -kt + \ln [\text{SO}_2\text{Cl}_2]_0 \), where \([\text{SO}_2\text{Cl}_2]_t\) is the concentration at time \( t \), and \([\text{SO}_2\text{Cl}_2]_0\) is the initial concentration.

Step 3: Rearrange the integrated rate law to solve for \( k \): \( k = \frac{\ln [\text{SO}_2\text{Cl}_2]_0 - \ln [\text{SO}_2\text{Cl}_2]_t}{t} \).

Step 4: Substitute the given pressures (which are proportional to concentrations) and times into the equation for \( k \). For example, use the data at \( t = 2500 \) s: \( k = \frac{\ln(1.000) - \ln(0.947)}{2500} \).

Step 5: Calculate \( k \) for each time interval and average the values to find the consistent rate constant. The units of \( k \) for a first-order reaction are \( \text{s}^{-1} \).

Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

First-Order Reactions

First-order reactions are chemical reactions where the rate is directly proportional to the concentration of one reactant. In this case, the rate of the reaction SO2Cl2(g) → SO2(g) + Cl2(g) depends solely on the concentration of SO2Cl2. The rate law can be expressed as rate = k[SO2Cl2], where k is the rate constant. This means that as the concentration of SO2Cl2 decreases, the rate of the reaction also decreases.

Rate Constant (k)

The rate constant (k) is a proportionality factor in the rate law that provides insight into the speed of a reaction. For first-order reactions, the units of k are typically expressed in s⁻¹, indicating the rate of change of concentration over time. The magnitude of k can be determined from experimental data, reflecting how quickly the reactant is consumed. A larger k value signifies a faster reaction.

Integrated Rate Law

The integrated rate law for a first-order reaction relates the concentration of the reactant to time. It is expressed as ln([A]₀/[A]) = kt, where [A]₀ is the initial concentration and [A] is the concentration at time t. This equation allows for the calculation of the rate constant k by plotting ln([A]) versus time, yielding a straight line with a slope of -k. This relationship is crucial for analyzing the provided kinetic data.

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Textbook Question

As described in Exercise 14.41, the decomposition of sulfuryl chloride (SO₂Cl₂) is a first-order process. The rate constant for the decomposition at 660 K is 4.5 × 10^-2 s^-1. (b) At what time will the partial pressure of SO₂Cl₂ decline to one-tenth its initial value?

Textbook Question

The first-order rate constant for the decomposition of N₂O₅, 2 N₂O₅(g) → 4 NO₂(g) + O₂(g), at 70°C is 6.82×10^-3 s^-1. Suppose we start with 0.0250 mol of N₂O₅(g) in a volume of 2.0 L. (a) How many moles of N₂O₅ will remain after 5.0 min?

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Textbook Question

From the following data for the first-order gas-phase isomerization of CH3NC to CH3CN at 215°C, calculate the first-order rate constant and half-life for the reaction:

Time (s) Pressure CH3NC (torr)

0 502

2000 335

5000 180

8000 95.5

12,000 41.7

15,000 22.4

Textbook Question

Consider the data presented in Exercise 14.19. (a) By using appropriate graphs, determine whether the reaction is first order or second order.

Textbook Question

Consider the data presented in Exercise 14.19. (c) What is the half-life for the reaction?