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Is the rate constant a constant?

A basic principle of chemical kinetics is that rate constants are constant. Of course, that is true only if the conditions are constant because rate constants clearly depend on experimental factors such as temperature, and (for ion channels) voltage, binding of ligands, etc.

What is meant by the rate constant k of a reaction?

The rate constant, k, is a proportionality constant that indicates the relationship between the molar concentration of reactants and the rate of a chemical reaction.

What is the importance of the rate constant?

This is the value that tells us how fast or slow a reaction is.