Question

Sulfuryl chloride \(\left(\mathrm{SO}_{2} \mathrm{Cl}_{2}\right)\) decomposes to sulfur dioxide \(\left(\mathrm{SO}_{2}\right)\) and chlorine \(\left(\mathrm{Cl}_{2}\right)\) by reaction in the gas phase. The following pressure data were obtained when a sample containing \(5.00 \times 10^{-2}\) mol sulfury 1 chloride was heated to \(600 . \mathrm{K}\) in a \(5.00 \times 10^{-1}-\mathrm{L}\) container. Defining the rate as $$-\frac{\Delta\left[\mathrm{SO}_{2} \mathrm{Cl}_{2}\right]}{\Delta t}$$ a. determine the value of the rate constant for the decomposition of sulfuryl chloride at \(600 .\) K. b. what is the half-life of the reaction? c. what fraction of the sulfuryl chloride remains after \(20.0 \mathrm{h} ?\)

Short answer

a) The value of the rate constant for the decomposition of sulfuryl chloride at 600 K is \(0.25 \: \text{h}^{-1}\). b) The half-life of the reaction is \(2.77 \: \text{h}\). c) The fraction of sulfuryl chloride remaining after 20 hours is 0.15.

Step-by-step solution

Determine the initial concentration of sulfuryl chloride

First, we need to determine the initial concentration of \(\mathrm{SO}_{2} \mathrm{Cl}_{2}\) in the container. The concentration can be found using the formula: \[C = \frac{n}{V}\] where: \(C\) = concentration of \(\mathrm{SO}_{2} \mathrm{Cl}_{2}\) \(n\) = number of moles of \(\mathrm{SO}_{2} \mathrm{Cl}_{2}\) \(V\) = volume of the container For our problem, \(n = 5.00 \times 10^{-2} \: \text{moles}\) \(V = 5.00 \times 10^{-1} \: \mathrm{L}\) Now, plug in the values and calculate the initial concentration of \(\mathrm{SO}_{2} \mathrm{Cl}_{2}\): \[C = \frac{5.00 \times 10^{-2} \,\text{mol}}{5.00 \times 10^{-1}\, \mathrm{L}} = 0.100\, \text{M}\]

Determine the rate constant

We can now use first-order reaction kinetics, which is given by the following formula: \[k=\frac{1}{C \Delta t} \ln \frac{C_{0}}{C_{t}}\] where: \(k\) = rate constant \(C_{0}\) = initial concentration of \(\mathrm{SO}_{2} \mathrm{Cl}_{2}\) \(C_{t}\) = concentration of \(\mathrm{SO}_{2} \mathrm{Cl}_{2}\) at time \(t\) \(\Delta t\) represents the change in time from the initial time to the final time. Using the information in the exercise, we know: \(C_0 = 0.100 \: \text{M}\) \(C_t = 0.050 \: \text{M}\) (from the exercise, half of the concentration was decomposed) \(\Delta t = 12.0 \: \text{h} - 0.0 \: \text{h} = 12.0 \: \text{h}\) (when half of the concentration is decomposed) Now plug in the values and calculate the rate constant: \[k = \frac{1}{0.1 \times 12} \ln \frac{0.100}{0.050} = 0.25 \text{h}^{-1}\]

Determine the half-life of the reaction

For a first-order reaction, the half-life can be found using the formula: \[t_{1/2} = \frac{0.693}{k}\] Now, plug in the rate constant we found in step 2 and calculate the half-life: \[t_{1/2} = \frac{0.693}{0.25 \: \text{h}^{-1}} = 2.77\, h\]

Determine the fraction of sulfuryl chloride remaining after 20 hours

Finally, we can use the first-order reaction equation to find the fraction of \(\mathrm{SO}_{2} \mathrm{Cl}_{2}\) remaining after 20 hours: \[C_t = C_0e^{-kt}\] Where: \(C_t\) = concentration of \(\mathrm{SO}_{2} \mathrm{Cl}_{2}\) at time \(t\) \(C_0\) = initial concentration of \(\mathrm{SO}_{2} \mathrm{Cl}_{2}\) \(k\) = rate constant \(t\) = 20 hours Now, plug in the known values and find the final concentration of \(\mathrm{SO}_{2} \mathrm{Cl}_{2}\) after 20 hours: \[C_t = 0.100 \times e^{-0.25 \times 20} = 0.015\, \text{M}\] To find the remaining fraction, divide the concentration after 20 hours by the initial concentration: \[\text{Fraction remaining} = \frac{C_t}{C_0} = \frac{0.015}{0.100} = 0.15\] Summary: a) The value of the rate constant for the decomposition of sulfuryl chloride at 600 K is 0.25 h^-1. b) The half-life of the reaction is 2.77 hours. c) The fraction of sulfuryl chloride remaining after 20 hours is 0.15.

Key concepts

Rate Constant

In the realm of chemical kinetics, the rate constant is a crucial numerical value that provides a lot about a chemical reaction's speed. It's represented by the symbol 'k' and can be considered the speedometer for a reaction, telling you how fast the reactants are turning into products.

For a given reaction at a constant temperature, the rate constant holds steady, but this can vary widely between different reactions and conditions. It's like how two cars might have different maximum speeds; in the same way, two reactions can have different 'top speeds' or rate constants.

To determine the rate constant, we look at the initial concentration of the reactants and the change in concentration over a certain period. In mathematical terms, for a first-order reaction, it's captured by the formula:
\[k = \frac{1}{C \Delta t} \ln \frac{C_{0}}{C_{t}}\]
where \(C\) is the concentration, \(\Delta t\) is the change in time, and \(C_{0}\) and \(C_{t}\) represent the initial and at time 't' concentrations respectively. The ability to calculate 'k' gives scientists and engineers a powerful tool to predict how a reaction will progress over time.

Reaction Half-Life

Another key concept in chemical kinetics is the reaction half-life, denoted as \(t_{1/2}\). This term is a measure of the time it takes for half of the reactant to be used up in a reaction. It’s synonymous with the half-life of radioactive elements, but here it's all about chemical concentrations.

The half-life provides a snapshot of the reaction's pace; a shorter half-life means a faster reaction. For first-order reactions, the relationship between the half-life and the rate constant is beautifully simple and utterly independent of the initial concentration:
\[t_{1/2} = \frac{0.693}{k}\]
This equation is a core piece of knowledge for chemists, as it allows for easy estimation of how long a reaction will take to reach a certain point. For first-order reactions, it's especially convenient because the half-life remains constant, no matter the concentration. It’s like knowing that you’ll need the same amount of time to eat half a pizza, whether it’s a small or a family-sized one!

First-Order Reaction

Digging deeper into the kinetics universe, we encounter first-order reactions. These are reactions where the rate is directly proportional to the concentration of one reactant. In other words, if you double the amount of reactant, the reaction rate also doubles.

First-order reactions follow an exponential decay model. The concentration of reactants decreases over time in a manner that can be mapped out with a natural logarithm curve. This type of reaction is mathematically expressed as:
\[C_t = C_0e^{-kt}\]
Here, \(C_t\) denotes the concentration after time \(t\), \(C_0\) is the initial concentration, \(k\) is the rate constant, and \(e\) represents the base of the natural logarithm.

The importance of first-order reactions cannot be overstated in chemistry, as they apply to numerous processes including radioactive decay and enzyme kinetics in biology. Understanding them not only allows for control over industrial processes but also has implications for pharmacokinetics, which determines how drugs behave in the body.