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Chapter 11: Problem 65

The activation energy for the reaction involved in the souring of raw milk is \(75 \mathrm{~kJ}\). Milk will sour in about eight hours at \(21^{\circ} \mathrm{C}\left(70^{\circ} \mathrm{F}=\right.\) room temperature). How long will raw milk last in a refrigerator maintained at \(5^{\circ} \mathrm{C}\) ? Assume the rate constant to be inversely related to souring time.

Short Answer

Expert verified
Answer: Approximately 81.56 hours, or about 3.4 days.

Step by step solution

01

Convert temperatures to Kelvin

To work with absolute temperatures, we need to convert the given temperatures in Celsius to Kelvin. To do so, simply add 273.15 to the Celsius temperatures. $$ T_1 = 21^{\circ}\mathrm{C} + 273.15 = 294.15\mathrm{K} $$ $$ T_2 = 5^{\circ}\mathrm{C} + 273.15 = 278.15\mathrm{K} $$
02

Use the Arrhenius equation to find the relationship between rate constants and temperature

We will use the following form of the Arrhenius equation: $$ \frac{k_2}{k_1} = \exp \left(\frac{-E_\mathrm{a}}{R} \left(\frac{1}{T_2} - \frac{1}{T_1}\right) \right) $$ Where \(k_1\) is the rate constant at temperature \(T_1\), \(k_2\) is the rate constant at temperature \(T_2\), \(E_\mathrm{a}\) is the activation energy, and \(R\) is the ideal gas constant (8.314 J/mol K). We are given that \(E_\mathrm{a} = 75\mathrm{kJ/mol}\) which is equal to 75,000 J/mol, and since the rate constant \(k\) is inversely related to souring time: $$ \frac{t_1}{t_2} = \exp \left(\frac{-75,000}{8.314} \left(\frac{1}{278.15} - \frac{1}{294.15}\right) \right) $$
03

Calculate the souring time at the refrigerator temperature

We are given that \(t_1\), the souring time at \(21^{\circ}\mathrm{C}\), is 8 hours. We'll substitute this value and solve for \(t_2\), the souring time at \(5^{\circ}\mathrm{C}\): $$ \frac{8}{t_2} = \exp \left(\frac{-75,000}{8.314} \left(\frac{1}{278.15} - \frac{1}{294.15}\right) \right) $$ Solve for \(t_2\): $$ t_2 = \frac{8}{\exp \left(\frac{-75,000}{8.314} \left(\frac{1}{278.15} - \frac{1}{294.15}\right) \right)} $$ When we plug the numbers into the equation, we get: $$ t_2 \approx 81.56 \mathrm{~hours} $$ So, raw milk will last about 81.56 hours, or approximately 3.4 days, in a refrigerator maintained at \(5^{\circ}\mathrm{C}\).

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Most popular questions from this chapter

For a first-order reaction \(a \mathrm{~A} \longrightarrow\) products, where \(a \neq 1\), the rate is \(-\Delta[\mathrm{A}] / a \Delta t\), or in derivative notation, \(-\frac{1}{a} \frac{d[\mathrm{~A}]}{d t} .\) Derive the integrated rate law for the first-order decomposition of \(a\) moles of reactant.

The decomposition of sulfuryl chloride, \(\mathrm{SO}_{2} \mathrm{Cl}_{2}\), to sulfur dioxide and chlorine gases is a first-order reaction. $$\mathrm{SO}_{2} \mathrm{Cl}_{2}(g) \longrightarrow \mathrm{SO}_{2}(g)+\mathrm{Cl}_{2}(g)$$ At a certain temperature, the half-life of \(\mathrm{SO}_{2} \mathrm{Cl}_{2}\) is \(7.5 \times 10^{2} \mathrm{~min}\). Consider a sealed flask with \(122.0 \mathrm{~g}\) of \(\mathrm{SO}_{2} \mathrm{Cl}_{2}\) (a) How long will it take to reduce the amount of \(\mathrm{SO}_{2} \mathrm{Cl}_{2}\) in the sealed flask to \(45.0 \mathrm{~g}\) ? (b) If the decomposition is stopped after \(29.0 \mathrm{~h}\), what volume of \(\mathrm{Cl}_{2}\) at \(27^{\circ} \mathrm{C}\) and \(1.00\) atm is produced?

Write the rate expression for each of the following elementary steps: (a) \(\mathrm{NO}_{3}+\mathrm{CO} \longrightarrow \mathrm{NO}_{2}+\mathrm{CO}_{2}\) (b) \(\mathrm{I}_{2} \longrightarrow 2 \mathrm{I}\) (c) \(\mathrm{NO}+\mathrm{O}_{2} \longrightarrow \mathrm{NO}_{3}\)

Consider the following hypothetical reaction: $$\mathrm{X}(g) \longrightarrow \mathrm{Y}(g)$$ A 200.0-mL flask is filled with \(0.120\) moles of \(\mathrm{X}\). The disappearance of \(\mathrm{X}\) is monitored at timed intervals. Assume that temperature and volume are kept constant. The data obtained are shown in the table below. $$\begin{array}{lccccc}\hline \text { Time }(\min ) & 0 & 20 & 40 & 60 & 80 \\\ \text { moles of } \mathrm{X} & 0.120 & 0.103 & 0.085 & 0.071 & 0.066 \\ \hline\end{array}$$ (a) Make a similar table for the appearance of Y. (b) Calculate the average disappearance of \(\mathrm{X}\) in \(\mathrm{M} / \mathrm{s}\) in the first two twenty-minute intervals. (c) What is the average rate of appearance of \(\mathrm{Y}\) between the 20 - and 60-minute intervals?

The decomposition of ethane, \(\mathrm{C}_{2} \mathrm{H}_{6}\), is a first-order reaction. It is found that it takes 212 s to decompose \(0.00839 \mathrm{M} \mathrm{C}_{2} \mathrm{H}_{6}\) to \(0.00768 \mathrm{M}\). (a) What is the rate constant for the reaction? (b) What is the rate of decomposition (in \(\mathrm{mol} / \mathrm{L} \cdot \mathrm{h}\) ) when \(\left[\mathrm{C}_{2} \mathrm{H}_{6}\right]=\) \(0.00422 \mathrm{M} ?\) (c) How long (in minutes) will it take to decompose \(\mathrm{C}_{2} \mathrm{H}_{6}\) so that \(27 \%\) remains? (d) What percentage of \(\mathrm{C}_{2} \mathrm{H}_{6}\) is decomposed after \(22 \mathrm{~min}\) ?
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