Question

Sulfuryl chloride undergoes first-order decomposition at \(320 .^{\circ} \mathrm{C}\) with a half-life of 8.75 \(\mathrm{h}\) . $$ \mathrm{SO}_{2} \mathrm{Cl}_{2}(g) \longrightarrow \mathrm{SO}_{2}(g)+\mathrm{Cl}_{2}(g) $$ What is the value of the rate constant, \(k,\) in \(\mathrm{s}^{-1}\) ? If the initial pressure of \(\mathrm{SO}_{2} \mathrm{Cl}_{2}\) is 791 torr and the decomposition occurs in a \(1.25-\mathrm{L}\) container, how many molecules of \(\mathrm{SO}_{2} \mathrm{Cl}_{2}\) remain after 12.5 \(\mathrm{h}\) ?

Short answer

The rate constant, k, is approximately \(2.20 × 10^{-5} s^{-1}\). After 12.5 hours, there are approximately \(7.16 × 10^{22}\) molecules of SO2Cl2 remaining in the container.

Step-by-step solution

Convert the half-life into seconds

Since we need the rate constant in s^-1, we will first convert the given half-life from hours to seconds: \(t_{1/2}\) = 8.75 h × 3600 s/h = 31500 s

Calculate the rate constant, k

We can calculate the rate constant, k, using the half-life equation: \(t_{1/2} = \frac{ \ln(2) }{ k }\) Rearrange the equation to solve for k: \(k = \frac{ \ln(2) }{ t_{1/2} }\) Plug in the values and calculate k: \(k = \frac{ \ln(2) }{ 31500 }\) \(k \approx 2.20 × 10^{-5} s^{-1}\)

Calculate the initial concentration of SO2Cl2

We are given the initial pressure of SO2Cl2 (791 torr) and the volume of the container (1.25 L). We can calculate the initial concentration (moles per liter) of SO2Cl2 using the ideal gas law equation: PV = nRT Where P is pressure, V is volume, n is moles, R is the ideal gas constant, and T is the temperature. We can rearrange this equation to solve for concentration: \[C = \frac{n}{V} = \frac{P}{RT}\] Note that we need to convert the pressure from torr to atm (1 atm = 760 torr), and use the correct value for the gas constant, R = 0.08206 L atm/mol K. Given the temperature is 320^{\circ}C, we need to convert it to Kelvin: T = 320 + 273.15 = 593.15 K Now, calculate the initial concentration: C_0 = (791 torr / 760 torr/atm) / (0.08206 L atm/mol K × 593.15 K) = 0.2558 mol/L

Calculate the remaining concentration of SO2Cl2 after 12.5 hours

We can use the first-order integrated rate law equation to find the remaining concentration of SO2Cl2 after 12.5 hours: \( \ln(\frac{[SO_2Cl_2]_{t=12.5h} }{ [SO_2Cl_2]_0 }) = -kt\) First, convert 12.5 hours to seconds: t = 12.5 h × 3600 s/h = 45000 s Rearrange the equation to solve for [SO2Cl2]t : \([SO_2Cl_2]_{t=12.5h} = [SO_2Cl_2]_0 × e^{-kt}\) Now, plug in the values and calculate the remaining concentration: \([SO_2Cl_2]_{t=12.5h} = (0.2558 mol/L) × e^{-(2.20 × 10^{-5} s^{-1})(45000s)}\) \([SO_2Cl_2]_{t=12.5h} \approx 0.0951 mol/L\)

Calculate the number of remaining molecules of SO2Cl2

Since we have calculated the remaining concentration of SO2Cl2, we can now find the number of remaining molecules. First, we find the remaining moles of SO2Cl2 in the container: Moles remaining = [SO2Cl2]t × V = (0.0951 mol/L) × (1.25 L) = 0.1189 mol Now, we can use Avogadro's number (6.022 × 10^23 molecules/mol) to find the number of remaining molecules: Number of remaining molecules = 0.1189 mol × 6.022 × 10^23 molecules/mol ≈ 7.16 × 10^22 molecules So, after 12.5 hours, there are approximately 7.16 × 10^22 molecules of SO2Cl2 remaining in the container.

Key concepts

First-order Reactions

Chemical reactions can be categorized based on how the reaction rate depends on the concentration of reactants. In first-order reactions, the rate of the reaction is directly proportional to the concentration of one reactant. This means that as the concentration of the reactant decreases, the reaction rate also decreases, helping make these processes linear and predictable.

A good example is the decomposition of sulfuryl chloride (SO_2Cl_2g) as given in the exercise. The reaction follows the equation:\(SO_2Cl_2(g) \longrightarrow SO_2(g) + Cl_2(g)\)

For such reactions, the integrated rate equation is:\[\ln\left(\frac{[A]_t}{[A]_0}\right) = -kt\],where \([A]_{t}\) is the concentration of reactant \(A\) at time \(t\), \([A]_0\) is the initial concentration, \(k\) is the rate constant, and \(t\) is the time elapsed.

This formula indicates that the concentration decay produces a logarithmic curve when plotted against time.

Rate Constant

In kinetics, the rate constant \(k\) is a proportionality factor that connects the speed of a chemical reaction to the concentrations of the reactants. For first-order reactions like the decomposition of \(SO_2Cl_2\), the rate constant is crucial in describing how fast the reaction proceeds under specific conditions.

The equation for a first-order reaction's half-life is:\[t_{1/2} = \frac{\ln(2)}{k}\].

This relationship shows that the half-life is independent of the concentration of the reactant and solely depends on the rate constant \(k\). Given a half-life of 8.75 hours for \(SO_2Cl_2\), converting into seconds gives us the half-life in a more usable, smaller time unit for calculating the rate constant.

By using the formula, we can rearrange it to solve for \(k\), where:\[k = \frac{\ln(2)}{t_{1/2}}\],effectively showing that \(k\) is determined by dividing the natural logarithm of 2 by the half-life. This step confirms the dynamics of the reaction over time.

Ideal Gas Law

The Ideal Gas Law is a fundamental equation in chemistry that relates the pressure \(P\), volume \(V\), temperature \(T\), and number of moles \(n\) of a gas. The equation is written as:\[PV = nRT\],where \(R\) is the universal gas constant.

This law assumes that gas molecules have no volume and do not exert forces on each other, which works well for many gases at low pressure and high temperature. For the decomposition of \(SO_2Cl_2\), we use the equation to convert the initial pressure into concentration as follows:

- Pressure must first be converted from torr to atmospheres (1 atm = 760 torr)- Temperature is converted from Celsius to Kelvin by adding 273.15 to the Celsius temperature

Using the Ideal Gas Law helps us determine the initial concentration of \(SO_2Cl_2\), which is pivotal for calculating how much of the reactant remains after a specified time period using the first-order kinetics equation.

Molecular Calculations

Molecular calculations involve translating moles of substances into numbers of molecules or atoms, making chemistry tangible. This process utilizes Avogadro's number, which is \(6.022 \times 10^{23}\) molecules per mole.

Once we calculate the remaining concentration of \(SO_2Cl_2\) in the container, following the time-dependent dynamics described by first-order kinetics, the amount present is expressed in moles. These moles can be converted to the exact number of molecules by multiplying with Avogadro's number.

Consider the following steps:

• Calculate moles left after 12.5 hours using the concentration \([SO_2Cl_2]_t\) found from the first-order reaction equation.
• By multiplying the moles with Avogadro's number, convert moles to molecules.

This step is essential for grasping the tangible scope of chemical reactions at the molecular level, and it builds a concrete understanding of how a chemical reaction progresses over time.