At 28 C, raw milk sours in 4.0 h but takes 48 h to sour in a refrigerator at 5 C. Estimate the activation energy in kJ>mol for the reaction that leads to the souring of milk.

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Identify the given temperatures and times: T1 = 28°C, t1 = 4.0 h; T2 = 5°C, t2 = 48 h.

Convert the temperatures from Celsius to Kelvin: T1 = 28 + 273.15 K, T2 = 5 + 273.15 K.

Use the Arrhenius equation in the form of \( \ln \left( \frac{k_2}{k_1} \right) = \frac{E_a}{R} \left( \frac{1}{T_1} - \frac{1}{T_2} \right) \), where \( k \) is the rate constant, \( E_a \) is the activation energy, and \( R \) is the gas constant (8.314 J/mol·K).

Assume the rate constants are inversely proportional to the times: \( k_1 = \frac{1}{t_1} \) and \( k_2 = \frac{1}{t_2} \).

Substitute the values into the Arrhenius equation and solve for \( E_a \) in kJ/mol.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Activation Energy

Activation energy is the minimum energy required for a chemical reaction to occur. It represents the energy barrier that reactants must overcome to transform into products. In the context of the souring of milk, a higher activation energy indicates that the reaction proceeds more slowly at lower temperatures, which is why souring takes longer in the refrigerator.

Arrhenius Equation

The Arrhenius equation relates the rate of a chemical reaction to temperature and activation energy. It is expressed as k = A * e^(-Ea/RT), where k is the rate constant, A is the pre-exponential factor, Ea is the activation energy, R is the gas constant, and T is the temperature in Kelvin. This equation helps in estimating activation energy by comparing reaction rates at different temperatures.

Temperature Dependence of Reaction Rates

The rate of chemical reactions typically increases with temperature due to the increased kinetic energy of molecules, which leads to more frequent and effective collisions. This principle is crucial for understanding why raw milk sours faster at higher temperatures, as the increased thermal energy facilitates the reaction that causes souring.

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Textbook Question

The following mechanism has been proposed for the reaction of NO with H₂ to form N₂O and H₂O:

NO(g) + NO(g) → N₂O₂(g)

N₂O₂(g) + H₂(g) → N₂O(g) + H₂O(g)

(a) Show that the elementary reactions of the proposed mechanism add to provide a balanced equation for the reaction.

Textbook Question

The following mechanism has been proposed for the reaction of NO with H₂ to form N₂O and H₂O:

NO(g) + NO(g) → N₂O₂(g)

N₂O₂(g) + H₂(g) → N₂O(g) + H₂O(g)

(d) The observed rate law is rate = k[NO]²[H₂]. If the proposed mechanism is correct, what can we conclude about the relative speeds of the first and second reactions?