Question

The reaction $${\mathrm{SO}}_2{\mathrm{Cl}}_2(g)⟶{\mathrm{SO}}_2(g)+{\mathrm{Cl}}_2(g)$$ is first order in ${\mathrm{SO}}_2{\mathrm{Cl}}_2$. Using the following kinetic data, determine the magnitude and units of the first order rate constant: $$\overline{)\begin{array}{rl}\text{ Time (s) }& \text{ Pressure }{\mathrm{SO}}_2{\mathrm{Cl}}_2\text{ (atm) }\\ 0& 1.000\\ 2,500& 0.947\\ 5,000& 0.895\\ 7,500& 0.848\\ 10,000& 0.803\end{array}}$$

Short answer

The magnitude of the first-order rate constant k for the given reaction is $2.2314\times {10}^{-5}$ and its units are $s^{-1}$ (per second).

Step-by-step solution

Write the first-order rate law for the reaction

The rate law for a first-order reaction is given by: $Rate=k[S{O}_{2}C{l}_2]$, where Rate is the reaction rate, k is the rate constant, and [SO_2Cl_2] is the concentration of SO2Cl2.

Express the first-order rate law in terms of pressure instead of concentration

The pressure of SO2Cl2 is proportional to its concentration and, thus, the same will hold for the rate expression: $Rate=k{P}_{S{O}_{2}C{l}_2}$, where $P_{S{O}_{2}C{l}_2}$ is the pressure of SO2Cl2.

Define the integrated rate law and its variables

To find k, we must use the integrated rate law for a first-order reaction. The integrated rate law is given by: $ln\frac{P_{S{O}_{2}C{l}_{2,initial}}}{P_{S{O}_{2}C{l}_2}}=kt$, where $P_{S{O}_{2}C{l}_{2,initial}}$ is the initial pressure at time t = 0, $P_{S{O}_{2}C{l}_2}$ is the pressure at any time t, and t is the elapsed time in seconds.

Choose data points to substitute into the integrated rate law equation

We can choose any two points from the given data to find the rate constant. For simplicity, let's choose the initial point (Time = 0 s and Pressure = 1.000 atm) and the final point (Time = 10,000 s and Pressure = 0.803 atm).

Plug the data points into the integrated rate law equation

Now we can substitute the chosen values into the equation: $ln\frac{1.000}{0.803}=k\times 10,000$

Solve the equation for the rate constant k

First, we will calculate the value of the natural logarithm and then rearrange the equation to isolate k: $ln(1.25)=0.22314=10,000k$ $k=\frac{0.22314}{10,000}$

Calculate the rate constant k

Now we can find the rate constant by dividing 0.22314 by 10,000: $k=2.2314\times {10}^{-5}$ Hence, the magnitude of the first-order rate constant k is $2.2314\times {10}^{-5}$.

Determine the units of the rate constant k

Since this is a first-order reaction, and the pressures are given in atm and time in seconds, the units of the rate constant k will be $s^{-1}$, which means the rate constant is expressed in terms of "per second". In conclusion, the magnitude of the first-order rate constant k is $2.2314\times {10}^{-5}$ and its units are $s^{-1}$.

Key concepts

Reaction Kinetics

Understanding reaction kinetics is crucial for analyzing how fast a chemical reaction proceeds. In our exercise, we're focusing on a first-order reaction, which means the rate depends linearly on the concentration of a single reactant. For the decomposition of ${\mathrm{SO}}_2{\mathrm{Cl}}_2$, the rate law is expressed as:

• $\text{Rate}=k[{\mathrm{SO}}_2{\mathrm{Cl}}_2]$

Here, $k$ is the rate constant, and $[{\mathrm{SO}}_2{\mathrm{Cl}}_2]$ is the concentration. For gases like ${\mathrm{SO}}_2{\mathrm{Cl}}_2$, concentration can be represented as pressure because of the ideal gas law. So, we adapt the law to:

• $\text{Rate}=k{P}_{{\mathrm{SO}}_2{\mathrm{Cl}}_2}$

This tells us how quickly the pressure of ${\mathrm{SO}}_2{\mathrm{Cl}}_2$ decreases over time, which is directly proportional to its concentration. Hence, understanding these basics helps us set up the equations properly for calculating the rate constant and know which variables affect reaction speed.

Integrated Rate Law

An essential aspect of first-order reactions is the integrated rate law, which connects the concentration of reactants over time using a logarithmic expression. This is very useful for calculating the rate constant $k$ when time-dependent concentration data is available. The formula is given by:

• $ln\frac{P_{{\mathrm{SO}}_{2}C{l}_{2,\text{initial}}}}{P_{{\mathrm{SO}}_{2}C{l}_2}}=kt$

Here, $P_{{\mathrm{SO}}_2{\mathrm{Cl}}_2,\text{initial}}$ denotes the initial pressure of ${\mathrm{SO}}_2{\mathrm{Cl}}_2$ at time $t=0$, and $P_{{\mathrm{SO}}_2{\mathrm{Cl}}_2}$ is the pressure at time $t$. This equation helps us determine $k$, the rate constant, by measuring how concentrations change as the reaction proceeds. Additionally, it allows us to estimate how long it will take for the reaction to reach a certain point by rearranging the equation to solve for time. These calculations are foundational for understanding chemical kinetics and predicting how reactions progress under various conditions.

Rate Constant Calculation

Calculating the rate constant $k$ for a first-order reaction involves substituting experimental data into the integrated rate law. Let's see how this is done with our given exercise. We know:

• Initial pressure, $P_{\text{initial}}=1.000\text{atm}$
• Pressure at $10,000\text{s}$, $P=0.803\text{atm}$
• Time $t=10,000\text{s}$

Insert these values into the logarithmic expression:$$ln\frac{1.000}{0.803}=k\times 10,000$$After calculating the natural logarithm, we find:$$ln(1.25)=0.22314$$Thus, solving for $k$:$$k=\frac{0.22314}{10,000}=2.2314\times {10}^{-5}{s}^{-1}$$This result gives us the magnitude and units for the rate constant. Since the reaction is first-order and measured in terms of time, the units of $k$ are $s^{-1}$, indicating how quickly the reaction proceeds in one second. Understanding these calculations enables us to accurately assess reaction kinetics in different scenarios.