Chapter 5, Problem 5.2Q

If it takes 11 minutes to cook spaghetti in Ann Arbor, Michigan, and 14 minutes in Boulder, Colorado, how long would it take in Cuzco, Peru? Discuss ways to make the spaghetti more tasty. If you prefer to make a creative spaghetti dinner for family or friends rather than answering this question, that’s OK, too; you’ll get full credit—but only if you turn in your recipe and bring your instructor a taste.

Interpretation Introduction

Interpretation:

The time takes to cook spaghetti in Cuzco, Peru should be calculated. 

Concept Introduction:

The reaction in which its rate is constant and independent of reactant concentration is zero order reaction.

For zero order reaction the combined rate law and mole balance is as follows,

    $-k=\frac{d{C}_A}{dt}$

Answer to Problem 5.2Q

The time takes to cook spaghetti in Cuzco, Peru is $\text{21}$ minutes.

Explanation of Solution

Given:

In Ann Arbor, Michigan it takes 11 minutes to cook spaghetti and in Boulder, Colorado it takes 14 minutes.

The boiling point present in these regions are represented as follows,

    $\begin{array}{ccc}\text{Location}& \text{Boiling point}\left({}^{\text{o}}\text{C}\right)& \text{Time}\left(\text{min}\right)\\ \text{Ann Arbor}& 99.2& 15\\ Boulder& 94.6& 17\\ Cuzco& 88.3& ?\end{array}$

Let’s assume that given reaction is zero order conversion, and then rate law is depicted as follows,

(For zero order reaction the rate constant $k$ is equal to the reaction rate.)

    $\begin{array}{c}-r_A=k\\ =A{e}^{\frac{-E}{RT}}\\ =-\frac{d{C}_A}{dt}\\ Then,\\ {\left[-C\right]}_{C_{A0}}^{C_A}=\left(A{e}^{\frac{-E}{RT}}\right)t\end{array}$

For complete conversion means the spaghetti is well cooked the $C_A$ becomes 0 at time $t$

  $\begin{array}{c}{C}_{A0}=tA{e}^{\frac{-E}{RT}}\\ \frac{C_{A0}}{A}=t{e}^{\frac{-E}{RT}}\\ \ln \left(\frac{C_{A0}}{A}\right)=\ln k\\ =\ln t\text{-}\frac{E}{R}\frac{1}{T_b}\end{array}$

Hence plotting $\ln t$ versus $\frac{1}{T_b}$ result to give cooking time for spaghetti in Cuzco, Peru.

  $\begin{array}{l}\Rightarrow {99.2}^{\circ }C=273+99.2=372.2K\\ \Rightarrow {94.6}^{\circ }C=273+94.6=367.6K\\ \Rightarrow {88.3}^{\circ }C=273+88.3=361.3K\\ \begin{array}{cccc}\text{Location}& t-Time\left(\text{min}\right)& T_b-\text{Boiling point}\left(K\right)& \frac{\text{1}}{T_b}\\ \text{Ann Arbor}& 15& 372.2K& 2.69\times {10}^{-3}\\ Boulder& 17& 367.6K& 2.72\times {10}^{-3}\\ Cuzco& ?& 361.3K& 2.767\times {10}^{-3}\end{array}\end{array}$

    

Therefore, the time takes to cook spaghetti in Cuzco, Peru is $\text{21}$ minutes.

Conclusion

The time takes to cook spaghetti in Cuzco, Peru was calculated.