Chapter 1.1, Problem 9E

In Exercises 6—10, we consider the phenomenon of radioactive decay which, from experimentation, we know behaves according to the law:
The rate at which a quantity of a radioactive isotope decays is proportional to the amount of the isotope present. The proportionality constant depends only on which radioactive isotope is used.
9. Engineers and scientists often measure the rate of decay of an exponentially decaying quantity using its time constant. The time constant $\tau$ is the amount of time that an exponentially decaying quantity takes to decay by a factor of l/e. Because l/e is approximately 0.368, $\tau$ is the amount of time that the quantity takes to decay to approximately 36.8% of its original amount.
(a) How are the time constant $\tau$ and the decay rate $\lambda$ related?
(b) Express the time constant in terms of the half-life.
(c) What are the time constants for Carbon 14 and Iodine 131?
(d) Given an exponentially decaying quantity r(t) with initial value r₀= r(0), show that its time constant is the time at which the tangent line to the graph of r(t)/r₀at (0, l) crosses the t-axis. [Hint: Start by sketching the graph of r(t)/r₀and the line tangent to the graph at (0, 1).]
(e) It is often said that an exponentially decaying quantity reaches its steady state in five time constants, that is, at t =5 $\tau$ . Explain why this statement is not literally true but is correct for all practical purposes.