The naturally occurring thorium series begins with 90232Th and terminates at the stable nuclide 82208Pb. Assume that only α-decays and β⁻-decays occur between these two nuclides, and that no daughter nuclide escapes from the series.
a. (i) If n α-particles and m β⁻-particles are emitted in the series, derive expressions for the net change in (a) mass number and (b) atomic number in terms of n and m.
b. (ii) Using the data for 90232Th and 82208Pb, determine the numerical values of n and m.
c. (iii) Write a condensed nuclear equation summarising the overall transformation of thorium-232 to lead-208, clearly indicating the numbers of α-particles and β⁻-particles emitted.
d. (iv) Briefly explain why the total number of nucleons is conserved, even though energy is released during the decay chain.