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Reaction order refers to the relationship between the reactant's concentration and the reaction rate. In other words, it shows how the change in concentration of the reactants affects the reaction's speed. The order of a reaction is usually expressed as an integer or a sum of integers corresponding to the stoichiometric coefficients of the reactants in the rate equation. The rate constant, denoted by k, is a proportionality factor that indicates the reaction's inherent speed under a specific set of conditions. The value of the rate constant depends on factors such as temperature, pressure, and the nature of the reactants involved.

Reaction order is a result of the underlying molecularity of a reaction and is determined by the mechanism of the reaction, i.e., the sequence of elementary steps that make up the reaction. Since the reaction mechanism is generally independent of temperature, the reaction order will remain the same at any temperature. For example, consider a reaction with an order of 2. This order means that doubling the concentration of the reactant will result in a four-fold increase in the reaction rate (since 2² = 4). This relationship between the reactant's concentration and the reaction rate is solely based on the reaction's molecularity and doesn't depend on the temperature.

While the reaction order remains the same at different temperatures, the rate constant's value can change significantly. This variation is due to the temperature's effect on the reaction's kinetics, especially the activation energy (Ea), which is the minimum energy required for a reaction to occur. According to the Arrhenius equation, the rate constant k of a reaction is related to temperature T and activation energy Ea as follows: k = A * e^{(-Ea / RT)} where A is the pre-exponential factor, R is the universal gas constant, and T is the temperature in Kelvin. From the equation, we can see that the value of k increases with increasing temperature. This increase in the rate constant indicates that the reaction rate is higher at higher temperatures, which is consistent with experimental observations. In conclusion, the reaction order remains constant at different temperatures because it depends on the reaction's mechanism, which is usually independent of temperature. In contrast, the rate constant varies with temperature because it is related to the activation energy and the reactants' molecular motion, both of which are temperature-dependent.