Question

The time required for a classifier to lose half its strength under the action of taxation alone is called its half-life. Derive a general expression for the half- life as a function of the classifier's tax rate Crax. Plot a graph of half-life as a function of Crax

Answer 1

The half-life (t1/2) of a substance, particularly for first-order reactions such as radioactive decay, is determined by the decay constant (λ) using the formula t1/2 = ln(2) / λ, showing an inverse relationship between half-life and decay constant.

Step-by-step explanation:

The half-life (t1/2) is the period required for the concentration of a reactant in a first-order reaction to reduce to half of its initial value. For a first-order reaction, which includes radioactive decay, the half-life is related to the decay constant (λ) through the formula t1/2 = ln(2) / λ. By understanding the half-life, we can predict how quickly a radioactive substance will decay over time.

The decay rate is defined as decay rate = AN, where A is the activity and N is the number of undecayed nuclei. Since the activity is directly proportional to the number of nuclei, it also decreases exponentially over time, halving after each half-life. The decay constant λ provides a crucial link between the measurable decay rate and the half-life of the substance, with the equation derived as ln(2) / t1/2.

By plotting the half-life against the decay constant λ, one would see a hyperbolic relationship, as the half-life is inversely proportional to λ.