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Why is the half-life of a first-order reaction independent of its initial concentration?

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A graph showing a potential energy curve for a chemical reaction, with the y-axis labeled potential energy and the x-axis labeled progress of reaction. The curve rises from the initial state (reactants) to a peak labeled transition state (activated complex), indicating activation energy, then drops to the final state (products). — chemical kinetics In the Arrhenius equation, the activation energy (E) represents the minimum amount of energy required to transform reactants into products in a chemical reaction. On a potential energy curve, the value of the activation energy is equivalent to the difference in potential energy between particles in an intermediate configuration (known as the activated complex, or transition state) and particles of reactants in their initial state. The activation energy thus can be visualized as a barrier that must be overcome by reactants before products can be formed.

In a first-order chemical reaction, the reaction rate is directly proportional to the concentration of a single reactant, expressed by the rate law $v = k [A]$ .1

Because the rate of the reaction scales linearly with the concentration of the reactant, the time required for the concentration to decrease by half remains constant regardless of the starting amount. In this kinetic behavior, the half-life is mathematically determined solely by the rate constant $k$ , making it independent of the initial concentration. This is a defining characteristic of first-order kinetics, commonly observed in processes such as the decay of radioactive substances.21