A sample of milk splits after 60 min. at 300 K and after 40 min. at 400 K when the populatio

step 1 Hello students, in this question I'll use the formula which is LN K2 upon K1 this is equal to EA

A sample of milk splits after 60 min. at 300 K and after...

A sample of milk splits after $60 \mathrm{~min}$. at $300 \mathrm{~K}$ and after $40 \mathrm{~min}$. at $400 \mathrm{~K}$ when the population of lactobacillus acidophilus in it doubles. The activation energy (in $\mathrm{kJ} / \mathrm{mol}$ ) for this process is closest to (Given, $\left.\mathrm{R}=8.3 \mathrm{~J} \mathrm{~mol}^{-1} \mathrm{~K}^{-1}, \ln \left(\frac{2}{3}\right)=0.4, \mathrm{e}^{-3}=4.0\right)$

Key Concepts

Temperature Dependence of Reaction Rates

Temperature affects reaction rates by influencing the kinetic energy of molecules. As temperature increases, a greater fraction of molecules have energies equal to or greater than the activation energy, thereby accelerating the reaction. This concept is fundamental in kinetics, as it explains why reactions occur faster at higher temperatures, a behavior that is captured by the Arrhenius equation.

Rate Constant and Reaction Time

The rate constant is a measure of the speed of a reaction under given conditions. It is inversely related to the characteristic time of the process; a faster reaction (shorter time) corresponds to a larger rate constant. This relationship is particularly useful when experimental observations, such as the time taken for a noticeable change, are used to compare reaction rates at different temperatures.

Arrhenius Equation

The Arrhenius equation quantitatively describes the temperature dependence of reaction rate constants. It is expressed as k = A exp(-Ea/RT), where k is the rate constant, A is the pre-exponential factor, Ea is the activation energy, R is the universal gas constant, and T is the absolute temperature. This expression is essential for understanding how a change in temperature affects the reaction speed by altering the proportion of molecules that have sufficient energy to overcome the activation barrier.

Activation Energy

Activation energy is the minimum energy required for a reaction to occur. It represents the energy barrier that reactants must surpass to transform into products. In the context of reaction kinetics, determining the activation energy is crucial because it provides insight into the sensitivity of the reaction rate with respect to temperature changes.

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