Suppose that a radioactive substance decays according to the model N = N0e–λt.(a) Show tha...

Suppose that a radioactive substance decays according to the model N = N0e–λt.

(a) Show that after a period of Tλ = 1/λ, the material has decreased to e–1 of its original value. Tλ is called the time constant and it is defined by this property.

(b) A certain radioactive substance has a half-life of 12 hours. Compute the time constant for this substance.

(c) If there are originally 1000 mg of this radioactive substance present, plot the amount of substance remaining over four time  eriods Tλ.

In the laboratory, a more useful measurement is the decay rate R, usually measured in disintegrations per second, counts per minute, etc. Thus, the decay rate is defined as R = –dN/dt. Using the equation dN/dt = –λN, it is easily seen that R = λN. Furthermore, differentiating the solution N = N0e–λt with respect to t reveals that

, (2.39)

in which R0 is the decay rate at t = 0. That is, because R and N are proportional, they both decrease with time according to the same exponential law. Use this idea to help solve Exercise.