(a) The reaction \(\mathrm{H}_{2} \mathrm{O}_{2}(a q) \longrightarrow \mathrm{H}_{2} \mathrm{O}(l)+\frac{1}{2} \mathrm{O}_{2}(g)\) is first order. At 300 \(\mathrm{K}\) the rate constant equals \(7.0 \times 10^{-4} \mathrm{s}^{-1}\) . Calculate the half-life at this temperature. (b) If the activation energy for this reaction is \(75 \mathrm{kJ} / \mathrm{mol},\) at what temperature would the reaction rate be doubled?

When studying chemical kinetics, a pivotal concept is the rate constant, symbolized by k. This value is intrinsic to a reaction and provides a measure of its speed; essentially, it tells us how fast a reaction will proceed under certain conditions. For a first-order reaction, like the decomposition of hydrogen peroxide (\textsf{\textbf{H}}\(_2\)\textsf{\textbf{O}}\(_2\)) in water, the rate constant is particularly important since it directly relates to the reaction's half-life (t_{1/2}).

The half-life is the time taken for the concentration of a reactant to decrease to half its initial concentration and for a first-order reaction, it is calculated using the equation:
\[ t_{1/2} = \frac{0.693}{k} \]
With a higher rate constant, the half-life decreases, indicating a faster reaction. It is essential for students to grasp that the rate constant, although a 'constant', is actually dependent on factors like temperature, which leads us into the Arrhenius equation and how it describes the temperature dependence of k.

Delving deeper into chemical kinetics, the Arrhenius equation establishes a quantitative link between the rate constant (\textsf{\textbf{k}}) and temperature. It is expressed as:
\[ k = Ae^{-\frac{E_a}{RT}} \]
where A is the pre-exponential factor, E_a represents the activation energy, R is the universal gas constant, and T is the absolute temperature in Kelvin.

The equation essentially says that the rate constant will increase with a rise in temperature, leading to a faster reaction rate. For instance, if we know the activation energy and the rate constant at a particular temperature, we can predict the change in the rate constant with a change in temperature.

This aspect is vital when considering real-world applications where temperature control is used to optimize reaction rates, such as in the synthesis of chemicals, pharmaceuticals, or even in the biological processes occuring within our own bodies.

Integral to the Arrhenius equation is the concept of activation energy (E_a), the minimum energy that must be supplied to reactants for a chemical reaction to take place. It's akin to a barrier that reactants must overcome to transform into products. The higher the activation energy, the slower the reaction rate, as fewer molecules possess the necessary kinetic energy to react at a given temperature.

Given that the activation energy for the hydrogen peroxide decomposition reaction is 75 kJ/mol, we can infer it's relatively high, meaning that under typical conditions, the reaction is slow. However, by increasing temperature, more molecules gain the required energy to overcome this barrier, thereby increasing the reaction rate.

This relationship between activation energy and temperature is crucial in industries and research where altering reaction rates is necessary, and it is beautifully illustrated when calculations show that doubling the reaction rate of our given exercise only requires a slight increase in temperature to 332.34 K, proving how sensitive reaction rates are to temperature alterations.