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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

A sample of sodium-24 chloride contains \(0.050 \mathrm{mg}\) of \(\mathrm{Na}-24\) to study the sodium balance of an animal. After \(24.9 \mathrm{~h}, 0.016 \mathrm{mg}\) of \(\mathrm{Na}-24\) is left. What is the half- life of \(\mathrm{Na}-24 ?\)

The first-order rate constant for the decomposition of a certain drug at \(25^{\circ} \mathrm{C}\) is \(0.215 \mathrm{month}^{-1}\) (a) If \(10.0 \mathrm{~g}\) of the drug is stored at \(25^{\circ} \mathrm{C}\) for one year, how many grams of the drug will remain at the end of the year? (b) What is the half-life of the drug? (c) How long will it take to decompose \(65 \%\) of the drug?

Experimental data are listed for the hypothetical reaction $$\mathrm{A}+\mathrm{B} \longrightarrow \mathrm{C}+\mathrm{D}$$ $$\begin{array}{lcccccc}\hline \text { Time (s) } & 0 & 10 & 20 & 30 & 40 & 50 \\\\{[\mathrm{~A}]} & 0.32 & 0.24 & 0.20 & 0.16 & 0.14 & 0.12 \\ \hline\end{array}$$ (a) Plot these data as in Figure \(11.2\). (b) Draw a tangent to the curve to find the instantaneous rate at \(30 \mathrm{~s}\). (c) Find the average rate over the 10 to \(40 \mathrm{~s}\) interval. (d) Compare the instantaneous rate at \(30 \mathrm{~s}\) with the average rate over the thirty-second interval.

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

Ammonia is produced by the reaction between nitrogen and hydro- gen gases. (a) Write a balanced equation using smallest whole-number coefficients for the reaction. (b) Write an expression for the rate of reaction in terms of \(\Delta\left[\mathrm{NH}_{3}\right]\). (c) The concentration of ammonia increases from \(0.257 \mathrm{M}\) to \(0.815 \mathrm{M}\) in \(15.0 \mathrm{~min} .\) Calculate the average rate of reaction over this time interval.
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