91Ó°ÊÓ

Phosgene (COCl_) is formed by CO and Cl_ reacting in the presence of activated charcoal: $$\mathrm{CO}+\mathrm{Cl}_{2} \rightarrow \mathrm{COCl}_{2}$$ At \(T=303.8 \mathrm{K}\) the rate of formation of phosgene in the presence of 1 gram of charcoal is $$R_{\mathrm{f}}(\mathrm{mol} / \mathrm{min})=\frac{8.75 C_{\mathrm{CO}} C_{\mathrm{C}_{2}}}{\left(1+58.6 C_{\mathrm{C}_{2}}+34.3 C_{\mathrm{COC}_{2}}\right)^{2}}$$ where C denotes concentration in mollL. (a) Suppose the charge to a 3.00 -liter batch reactor is \(1.00 \mathrm{g}\) of charcoal and a gas initially containing 60.0 mole\% CO and the balance \(\mathrm{Cl}_{2}\) at \(303.8 \mathrm{K}\) and 1 atm. Calculate the initial concentrations (mol/L) of both reactants, neglecting the volume occupied by the charcoal. Then, letting \(C_{\mathrm{P}}(t)\) be the concentration of phosgene at an arbitrary time \(t,\) derive relations for \(C_{\mathrm{P}}\) \(C_{\mathrm{CO}}\) and \(C_{\mathrm{C}_{2}}\) in terms of (b) Write a differential balance on phosgene and show that it simplifies to $$\frac{d C_{\mathrm{P}}}{d t}=\frac{2.92\left(0.02407-C_{\mathrm{P}}\right)\left(0.01605-C_{\mathrm{P}}\right)}{\left(1.941-24.3 C_{\mathrm{P}}\right)^{2}}$$ Provide an initial condition for this equation. (c) A plot of \(C_{\mathrm{P}}\) versus \(t\) starts at \(C_{\mathrm{P}}=0\) and asymptotically approaches a maximum value. Explain how you could predict that behavior from the form of the equation of Part (b). Without attempting to solve the differential equation, determine the maximum value of \(C_{\mathrm{P}}\) (d) Starting with the equation of Part (b), derive an expression for the time required to achieve a \(75 \%\) conversion of the limiting reactant. Your solution should have the form \(t=a\) definite integral. (e) The integral you derived in Part (d) can be evaluated analytically; however, more complex rate laws than the one given for the phosgene formation reaction would yield an integral that must be evaluated numerically. One procedure is to evaluate the integrand at a number of points between the limits of integration and to use a quadrature formula such as the trapezoidal rule or Simpson's rule (Appendix A.3) to estimate the value of the integral. Usea spreadsheet to evaluate the integrand of the integral of Part (c) at \(n_{p}\) equally spaced points between and including the limits of integration, where \(n_{p}\) is an odd number, and then to evaluate the integral using Simpson's rule. Perform the calculation for \(n_{p}=5,21,\) and \(51 .\) What can you conclude about the number of points needed to obtain a result accurate to three significant figures?

Step by step solution

01

Calculation of initial concentrations

The initial concentration of CO, denoted as \(C_{\mathrm{COo}}\) and Cl2, denoted as \(C_{\mathrm{Cl_2o}}\), can be calculated using the ideal gas law \(PV = nRT\) where P is the pressure (1atm), V is the volume (3.00L), n is the number of moles of the gas, R is the gas constant (0.08206 L.atm/mol.K), and T is the temperature (303.8K). Given that the gas mixture contains 60 mol% of CO, the initial concentrations can be calculated as follows: \(C_{\mathrm{COo}} = \frac{(60/100)PV}{RT}\), \(C_{\mathrm{Cl_2o}} = \frac{(40/100)PV}{RT}\)

02

Derive relations for \(C_{\mathrm{P}}\), \(C_{\mathrm{CO}}\) and \(C_{\mathrm{Cl_2}}\)

Start by representing the consumption and formation rates of reactants and products. As per stoichiometry of the reaction: \(-r_{\mathrm{CO}} = -r_{\mathrm{Cl_2}} = r_{\mathrm{COCl2}}\). Then, the change in concentration with time is presented as: \(C_{\mathrm{P}}(t) = C_{\mathrm{COCl_2}} = C_{\mathrm{COo}} - C_{\mathrm{CO}} = C_{\mathrm{Cl_2o}} - C_{\mathrm{Cl_2}}\)

03

Write a differential balance on phosgene

Substitute the derived expression from step 2 into the given rate equation to get \(R_{\mathrm{f}}\). As \(R_{\mathrm{f}} = r_{\mathrm{COCl_2}}\), the rate of formation of phosgene, \(R_{\mathrm{f}}\), can be related to the rate of change of phosgene concentration, \(dC_{\mathrm{P}}/dt\) over time. The initial condition would be that at \(t = 0\), \(C_{\mathrm{P}} = 0\)

04

Understand nature of phosgene concentration with time

As per the differential rate equation in step 3, as time increases, \(C_{\mathrm{P}}(t)\) increases, and the denominator of the rate of change of phosgene decreases. Therefore, \(dC_{\mathrm{P}}/dt\) decreases, which means the rate of formation of phosgene decreases with time. So even though \(C_{\mathrm{P}}\) keeps increasing as time goes on, it does so at an increasingly slower rate, leading to the asymptotic behavior. The maximum value of \(C_{\mathrm{P}}\) can be obtained by solving the equation obtained in the denominator with \(dC_{\mathrm{P}}/dt = 0\)

05

Derive expression for conversion time

75% conversion of limiting reactant implies that 75% of the initial concentration of the limiting reactant has been consumed and is present in the product. Setting up an equation with initial and final concentrations and integrating, an equation in the form of \(t = a \int{}{}\) can be attained.

06

Evaluate integral numerically

To calculate the conversion time evaluated previously, one must evaluate the integral. However, this may not be feasible analytically where certain complex rate equations are concerned. In such cases, numerical methods such as Simpson's rule or Trapezoidal rule can be applied. Using Simpson's rule, the integral is evaluated at a number of equally spaced intervals. Through comparison of results obtained with different number of points, the necessary number of points needed to achieve a specific level of accuracy can be determined.

Unlock Step-by-Step Solutions & Ace Your Exams!

• Full Textbook Solutions

  Get detailed explanations and key concepts

• Unlimited Al creation

  Al flashcards, explanations, exams and more...

• Ads-free access

  To over 500 millions flashcards

• Money-back guarantee

  We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91Ó°ÊÓ!

Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Phosgene Formation

Phosgene is a toxic and corrosive gas, often formed industrially through the reaction of carbon monoxide (CO) and chlorine gas (\(Cl_2\)). The reaction typically occurs in the presence of a catalyst such as activated charcoal, which can enhance the rate of reaction by providing a large surface area for the gases to react. The chemical equation for this reaction is:\[\mathrm{CO} + \mathrm{Cl}_2 \rightarrow \mathrm{COCl}_2\]This simple mole-to-mole reaction means one mole of carbon monoxide reacts with one mole of chlorine to produce one mole of phosgene.Understanding the conditions needed for the reaction is important:

• Temperature: The reaction is often conducted at moderately elevated temperatures to increase reaction rates.
• Pressure: Pressure influences concentration and thus affects reaction kinetics, with higher pressures usually increasing the rate.
• Catalysts: Substances like charcoal help speed up the reaction without being consumed.

Awareness of these parameters is crucial since phosgene is hazardous, requiring careful control and monitoring during synthesis.

Differential Rate Equations

Differential rate equations describe how the concentration of a reactant or product changes over time during a chemical reaction. In the context of phosgene formation, such an equation helps us understand the kinetics of the reaction. Considering the rate of formation of phosgene, the differential rate equation is given by:\[\frac{dC_{\mathrm{P}}}{dt} = \frac{8.75 C_{\mathrm{CO}} C_{\mathrm{Cl_2}}}{(1 + 58.6 C_{\mathrm{Cl_2}} + 34.3 C_{\mathrm{CO}})^2}\]This equation can be simplified and rearranged to provide insights into various kinetic parameters:

• The numerator reflects the reaction's dependency on the initial concentrations of CO and \(Cl_2\).
• The denominator shows how the reaction rate decreases as concentrations of reactants increase, due to saturation effects in the catalyst surface.

Integrating this differential equation can predict the concentration profiles of reactants and products over time, but this often requires numerical methods due to the complexity of real-world situations.

Ideal Gas Law Calculations

The Ideal Gas Law is a fundamental principle in chemistry that relates the pressure, volume, temperature, and the number of moles of a gas. The formula is:\[PV = nRT\]where

• \(P\) is the pressure of the gas, typically in atmospheres (atm),
• \(V\) represents the volume of the gas in liters (L),
• \(n\) is the number of moles of the gas,
• \(R\) is the gas constant (0.08206 \(L \, atm / mol \, K\)), and
• \(T\) is the temperature in Kelvin (K).

In the context of the exercise, the initial concentrations of CO and \(Cl_2\) are calculated by assuming the gases behave ideally. By knowing the percentage composition and the initial conditions, we can set up formulas for initial concentration:\[C_{\mathrm{COo}} = \frac{(60/100)PV}{RT}, \quad C_{\mathrm{Cl_2o}} = \frac{(40/100)PV}{RT}\]These equations allow us to determine the starting point of the reaction in terms of concentration, which is essential for analyzing the kinetics and the progression of the reaction over time.