The rate constant of a chemical reaction typically increases with temperature, a relationship that can be described by the Arrhenius equation.

The Arrhenius equation relates the rate constant $k$ to the temperature $T$ as follows:

In this equation, $A$ represents the pre-exponential factor, $E_a$ is the activation energy, $R$ is the universal gas constant, and $T$ is the temperature measured in Kelvin. This equation illustrates that the rate constant increases exponentially as temperature rises.

The derivation of the Arrhenius equation is based on the principle that molecules must overcome a certain energy barrier, referred to as the activation energy, in order to react. At elevated temperatures, a greater number of molecules possess the kinetic energy necessary to surpass this energy barrier, resulting in an increased reaction rate. This is quantitatively reflected in the rate constant, which serves as an indicator of the speed of the reaction.

The pre-exponential factor $A$ in the Arrhenius equation is indicative of the frequency of collisions between reacting molecules. While it is typically considered constant for a given reaction, it may experience slight variations with changes in temperature. Similarly, the activation energy $E_a$ is also generally regarded as a constant for a specific reaction.

The exponential component of the Arrhenius equation, $e^{-\frac{E_a}{RT}}$, is known as the Boltzmann factor. This factor represents the proportion of molecules that possess sufficient energy to overcome the activation energy barrier. As temperature increases, the Boltzmann factor also increases, indicating that a larger fraction of molecules are capable of reacting. Consequently, this leads to an increase in the rate constant.

In conclusion, the rate constant of a reaction is significantly influenced by temperature. According to the Arrhenius equation, it rises exponentially with increasing temperature due to a greater fraction of molecules having adequate energy to initiate a reaction. This relationship is a fundamental aspect of chemical kinetics, which focuses on the rates of chemical reactions.