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.

Verified step by step guidance

1. Identify the given temperatures and times: T1 = 28°C, t1 = 4.0 h; T2 = 5°C, t2 = 48 h.
2. Convert the temperatures from Celsius to Kelvin: T1 = 28 + 273.15 K, T2 = 5 + 273.15 K.
3. 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).
4. Assume the rate constants are inversely proportional to the times: \( k_1 = \frac{1}{t_1} \) and \( k_2 = \frac{1}{t_2} \).
5. Substitute the values into the Arrhenius equation and solve for \( E_a \) in kJ/mol.

Key Concepts

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.