The decay constant of \({ }_{6} \mathrm{C}^{14}\) is \(2.31 \times 10^{-4}\) year \(^{-1}\). Its half life is: (a) \(2 \times 10^{3}\) yrs (b) \(2.5 \times 10^{3}\) yrs (c) \(3 \times 10^{3}\) yrs (d) \(3.5 \times 10^{3} \mathrm{yrs}\)

The concept of the decay constant, symbolized by \(\lambda\), is crucial in understanding radioactive decay. It represents the probability per unit time of a single atom decaying. One might wonder how it's possible to assign a time value to the decay of individual atoms, which seem to act unpredictably. Indeed, it is unpredictable for one atom, but when we scale up to a sample containing a large number of atoms, the average behavior becomes remarkably consistent and can be described mathematically.

The decay constant is unique to each radioactive isotope and essentially sets the pace at which the isotope will decay. A larger decay constant indicates that the isotope will decay more quickly, whereas a smaller value suggests that the atoms will remain unchanged for longer periods. In mathematical terms, if one has the decay constant value, one can employ it to calculate not only the half-life of the isotope but also to predict the remaining quantity of a substance after a certain period using the exponential decay formula.

Radioactive isotope decay is the process by which an unstable atomic nucleus loses energy by emitting radiation in the form of particles or electromagnetic waves. This transformation often results in the nucleus changing to a different element or a different isotope of the same element.

Understanding the decay process is important for a variety of applications, including radioactive dating techniques, medical treatments like cancer therapy, and ensuring safety around nuclear materials. What essentially happens during decay is that the unstable isotope seeks a more stable configuration.

Examples in Nature and Technology

In nature, carbon-14 (\( _{6}\mathrm{C}^{14} \)) decays to nitrogen-14, while in technology, uranium-235 is used in nuclear reactors and atomic bombs due to its propensity to undergo fission when its nucleus decays. These decays happen at rates defined by each isotope's decay constant.

Exponential decay is a fundamental concept in chemistry that describes the decrease in the amount of a substance as time progresses. It is exponential because the quantity of the substance does not decrease by a constant amount each period; rather, it decreases by a constant proportion of its current value. This pattern of decay is different from linear decay, where the amount would decrease by the same absolute amount over each time interval.

The formula describing exponential decay is \( N(t) = N_0 e^{-\lambda t} \) where \( N(t) \) represents the remaining quantity of the substance at time \( t \), \( N_0 \) is the initial quantity, \( \lambda \) is the decay constant, and \( e \) is the base of the natural logarithm. When applied, this formula can predict how much of a radioactive substance remains after a given period, which is vital in fields like pharmacokinetics, where it is used to model drug elimination from the body.