decays to through electron capture with a decay constant of 0.257 per day.

(a) What other particle or particles are emitted in the decay?

(b) Assume that the electron is captured from the K shell. Use Mosley’s law √v = α (Z – b) with α = 4.95 × 10⁷ s^–1/2 and b = 1 to find the wavelength of the K_α X-ray emitted following the electron capture.

The count rate of nuclear radiation coming from a radioactive sample containing ¹²⁸I varies with time as follows.

(a) Plot ln(R₀/R) against t.

(b) From the slope of the best straight line through the points, find the decay constant λ.

(c) Calculate the half-life t_1/2.

The half-life of ⁴⁰K is 1.30 × 10⁹ y. A sample of 1.00 g of pure KCl gives 160 counts s^–1. Calculate the relative abundance of ⁴⁰K (fraction of ⁴⁰K present) in natural potassium.

A radioactive isotope is being produced at a constant rate dN/dt = R in an experiment. The isotope has a half-life t_1/2. Show that after a time t >> t_1/2, the number of active nuclei will become constant. Find the value of this constant.

Consider the situation of the previous problem. Suppose the production of the radioactive isotope starts at t = 0. Find the number of active nuclei at time t.