The rate law just given can be translated as: "The rate of disappearance of carbon-14 atoms is proportional to the number of carbon-14 atoms that are present per liter and available for decay."
Since each atom has the same inherent probability of decaying during a specified time interval, and since the probability of one atom's decaying is independent of the presence or absence of other atoms, this is the rate law that would be expected intuitively. [C-14] represents the concentration of carbon-14, and k is the rate constant.
Since the carbon-14 concentration decreases with time, d[C-14]/dt is a negative number;thus a minus sign is required on the right side of the equation. It is common to bring the minus sign to the left side, beside the rate term, and to write the rate equation as