For the reaction $$2 \mathrm{~N}_{2} \mathrm{O}(g) \longrightarrow 2 \mathrm{~N}_{2}(g)+\mathrm{O}_{2}(g)$$ the rate constant is \(0.066 \mathrm{~L} / \mathrm{mol} \cdot \mathrm{min}\) at \(565^{\circ} \mathrm{C}\) and \(22.8 \mathrm{~L} / \mathrm{mol} \cdot \mathrm{min}\) at \(728^{\circ} \mathrm{C} .\) (a) What is the activation energy of the reaction? (b) What is \(k\) at \(485^{\circ} \mathrm{C}\) ? (c) At what temperature is \(k\), the rate constant, equal to \(11.6 \mathrm{~L} / \mathrm{mol} \cdot \mathrm{min} ?\)

Activation energy, denoted as \(E_a\), refers to the minimum amount of energy that reacting particles must have for a chemical reaction to occur. It's a crucial concept within chemical kinetics, acting as an energy barrier that must be overcome for reactants to transform into products. In the context of the Arrhenius equation, \(E_a\) can be calculated using temperature and rate constants associated with the reaction.

Understanding activation energy helps in explaining why certain reactions happen quickly while others do not. For example, when you strike a match, the activation energy required for the chemicals in the match head to react is rapidly provided by the frictional heat, initiating combustion. On the molecular level, the activation energy corresponds to the peak in potential energy that reactants must reach before descending to a lower energy, stable product state.

The rate constant, represented as \(k\), is a proportionality factor that shows up in the rate laws of chemical reactions. It's a measure of the rate at which reactants are transformed into products. In the Arrhenius equation, the rate constant directly correlates with the temperature and activation energy of a reaction.

Factors that influence the rate constant include the physical state of the reactants, the concentration of the reactants, pressure (for gases), and the presence of a catalyst. A larger rate constant indicates a faster reaction. The dependency of the rate constant on temperature is particularly significant because even a small increase in temperature can lead to a substantial increase in the rate constant, thereby speeding up the reaction.

Chemical kinetics is the branch of chemistry that deals with the rates of chemical reactions and the factors that affect these rates. It helps us understand the speed (or slowness) of reactions and how various conditions might affect this speed. Key principles within kinetics include the reactions' rate laws, the role of catalysts, and the determination of reaction mechanisms.

The study of kinetics involves not only measuring how fast a reaction occurs but also deducing the sequence of steps (mechanism) that leads to the product formation. This understanding can be applied to control reactions in industrial processes, predict the shelf life of pharmaceuticals, and even aid in environmental assessments concerning pollutant degradation.

Temperature conversion in chemistry is essential because most formulas, including the Arrhenius equation, require temperatures to be in Kelvin (K). The Kelvin scale is an absolute temperature scale, with 0 K being absolute zero, where theoretically, particles have no kinetic energy.

To convert Celsius to Kelvin, which is a common task in chemical kinetics, you add 273.15 to the Celsius temperature. This adjustment is necessary because the Arrhenius equation uses the gas constant \(R\), which is based on the Kelvin scale. Abiding by these conversion rules is vital for calculating quantities accurately, such as the activation energy or rate constant of a reaction.