In an experiment, a sample of \(\mathrm{NaClO}_{3}\) was \(90 \%\) decomposed in 48 min. Approximately how long would this decomposition have taken if the sample had been heated \(20^{\circ} \mathrm{C}\) higher? (Hint: Assume the rate doubles for each 10 'C rise in temperature.)

Chemical kinetics is the branch of chemistry that deals with the rates of chemical reactions. It is a crucial field of study because understanding how fast a reaction proceeds can guide us in optimizing conditions for industrial processes, preserving food, and even in developing life-saving medications.

The science of kinetics not only measures how fast a reaction occurs but also sheds light on the various factors that influence this rate. These factors include the concentration of reactants, temperature, presence of a catalyst, and the physical state of the reactants. By adjusting these variables, chemists can control the speed of chemical reactions to suit specific needs.

The temperature of a system is a critical factor affecting the rate of a chemical reaction. Generally, an increase in temperature results in a higher reaction rate. This is because molecules have greater kinetic energy at higher temperatures, which leads to more frequent and more energetic collisions between them. If these collisions have enough energy, they can overcome what is known as the activation energy barrier, leading to a successful reaction event.

In practical terms, this relationship means that a reaction can be sped up by heating the reactants, which is a principle commonly observed and applied in cooking, engine design, and many industrial chemical processes.

To quantitatively describe the temperature dependence of reaction rates, the Arrhenius equation is essential. It relates the rate of chemical reactions to the temperature and activation energy. The equation is expressed as: \[ k = A \times e^{-\frac{E_a}{RT}} \]where \(k\) is the rate constant of a reaction, \(A\) is the pre-exponential factor reflecting the frequency of collisions with proper orientation, \(E_a\) is the activation energy, \(R\) is the universal gas constant, and \(T\) is the temperature in Kelvin.

From this equation, one can understand that as the temperature increases, so does the rate constant \(k\), leading to a faster reaction rate. This relationship explains why even a modest rise in temperature can have a significant effect on how quickly a reaction happens, as illustrated in the textbook exercise.

Activation energy is the minimum amount of energy that reactant molecules must possess to undergo a chemical reaction. It acts as a barrier to the reaction; only molecules with enough kinetic energy to overcome this barrier will react.

The concept of activation energy explains why certain reactions only occur at high temperatures; the heat provides the energy needed to surpass the activation barrier. It also helps us understand why some substances are stable at room temperature but react vigorously when heated, as they require the absorption of heat to reach the threshold energy needed for the reaction to proceed. In the context of the exercise, the activation energy determines how much the rate of reaction increases with a 20°C rise in temperature.