Homework 7

Due Date: Saturday, October 29

Instructions: Feel free to work together with other students in the class, though you must turn in your own copy of the solutions, and you must acknowledge anyone that you worked with. You can turn in your homework assignment by e-mailing me your solutions.

1. When butyl chloride (\(\text{C}_4\text{H}_9\text{Cl}\)) is dissolved in water, it reacts to form butyl alcohol and hydrochloric acid: \[ \text{C}_4\text{H}_9\text{Cl} + \text{H}_2\text{O} \;\longrightarrow\; \text{C}_4\text{H}_9\text{OH} + \text{HCl}. \] Assuming water is abundant, the concentration of butyl chloride in a solution will decay exponentially with a rate constant of \(k\).

   Suppose that a tank initially holds a volume \(V\) of pure water. A solution of butyl chloride with a (constant) molar concentration of \(C_{\text{in}}\) is added to the tank at a rate of \(R\). The liquid in the tank is kept thoroughly mixed, and the mixed solution is pumped out of the tank at the same rate \(R\).

   1. Write a differential equation for the molar concentration of butyl chloride in the tank.
   2. Find the solution to your equation from part (a), assuming the tank is initially full of fresh water.
   3. Write a differential equation for the molar concentration of hydrochloric acid in the tank. You may assume the incoming solution of butyl chloride does not contain any hydrochloric acid.
   4. Find the solution to your equation from part (c).
   5. What are the equilibrium levels of butyl chloride and hydrochloric acid in the tank?
2. When heated, nitrous oxide gas (\(\text{N}_2 \text{O}\)) decomposes into nitrogen (\(\text{N}_2\)) and oxygen (\(\text{O}_2\)). This reaction occurs in two steps: \begin{align} \text{N}_2\text{O} \;&\longrightarrow\; \text{N}_2 + \text{O}\tag*{(1)} \\[6pt] \text{N}_2\text{O} + \text{O} \;&\longrightarrow\; \text{N}_2 + \text{O}_2\tag*{(2)} \end{align} Here \(\text{O}\) denotes free oxygen atoms.
   1. Write a system of differential equations for the amounts \([\text{N}_2\text{O}]\) and \([\text{O}]\). Your system should involve two rate constants \(k_1\) and \(k_2\) which govern the rates of the two steps of the reaction.
   2. Because free oxygen molecules are so reactive, the rate constant for the second step is much larger than the rate constant for the first step. Use the NDSolve command in Mathematica to simulate 10 seconds of nitrous oxide decomposition, assuming that \(k_1 = 0.2\;\mathrm{sec}^{-1}\) and \(k_2 = 100\;\mathrm{mol}^{-1}\mathrm{sec}^{-1}\). Make separate plots of \([\text{N}_2\text{O}]\) and \([\text{O}]\) over the course of ten seconds.
   3. Based on your experiment from part (b), the amount \([\text{O}]\) of free oxygen atoms is roughly constant over the course of the reaction. Find a formula for the constant value of \([\text{O}]\) in terms of \(k_1\) and \(k_2\). Does this formula agree with your results from part (b)?
   4. Assuming \([\text{O}]\) is equal to the value you found in part (c), find a general formula for \([\text{N}_2\text{O}]\) as a function of time. Does this formula agree with your results from part (b)?