The isotope bismuth-212 undergoes α-decay to an isotope of thallium. In this decay a gamma-ray photon is also produced. The isotope potassium-40 undergoes β⁺ decay to an isotope of argon.

Outline how the

(i) α particle spectrum and the gamma spectrum of the decay of bismuth-212 give evidence for the existence of discrete nuclear energy levels.

(ii) β⁺ spectrum of the decay of potassium-40 led to the existence of the neutrino being postulated.

The isotope potassium-40 occurs naturally in many rock formations. In a particular sample of rock it is found that, out of the total number of argon plus potassium-40 atoms, 23% are potassium-40 atoms.

Determine the age of the rock sample. The decay constant for potassium-40 is 5.3×10^–10yr^–1.

(i) the α particles produced have discrete energies;
the gamma rays produced have discrete energies;
since the energies of the α particles and of the photons are determined by the difference in nuclear energy levels this implies that nuclear energy levels are also discrete;

(ii) the β⁺ spectrum is continuous;
the neutrino was postulated to account for those β⁺ with less energy than the maximum;

recognize that \(\frac{N}{{{N_0}}}\)=0.23;
\(0.23 = {e^{ - 5.3 \times {{10}^{ - 10}}t}}\);
t=2.8×10⁹ yr;