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 \(\mathrm{h}\) ?

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

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: \(k = \frac{ \ln(2) }{ 31500 }\) \(k \approx 2.20 × 10^{-5} s^{-1}\)

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

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: \([SO_2Cl_2]_{t=12.5h} = (0.2558 mol/L) × e^{-(2.20 × 10^{-5} s^{-1})(45000s)}\) \([SO_2Cl_2]_{t=12.5h} \approx 0.0951 mol/L\)

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.