Show that the spontaneous \(\alpha\) decay of \(^{19} \mathrm{O}\) is not possible.

In this spontaneous α-decay of \(^{19} \mathrm{O}\), the parent nucleus is \(^{19} \mathrm{O}\) and the α-particle is \(^{4} \mathrm{He}\). The daughter nucleus can be determined by subtracting the atomic number and mass number of the α-particle from the parent nucleus. The daughter nucleus will have atomic number 8-2 = 6 (carbon) and mass number 19-4 = 15.

To determine if this decay is energetically possible, we need to calculate the mass difference between the parent nucleus and the products. The mass difference is given by: Δm = \(m(^{19}\mathrm{O}) - (m(^{15}\mathrm{C}) + m(^{4}\mathrm{He}))\) Where \(m(^{19}\mathrm{O})\), \(m(^{15}\mathrm{C})\), and \(m(^{4}\mathrm{He})\) are the masses of the parent nucleus, daughter nucleus, and α-particle, respectively. We will obtain these masses from the atomic mass table: \(m(^{19}\mathrm{O}) = 19.003577 \mathrm{u}\) \(m(^{15}\mathrm{C}) = 15.010599 \mathrm{u}\) \(m(^{4}\mathrm{He}) = 4.002603 \mathrm{u}\) Plugging these values into the mass difference equation, we get: Δm = \(19.003577\mathrm{u} - (15.010599\mathrm{u} + 4.002603\mathrm{u}) = -0.009625\mathrm{u}\) The negative mass difference indicates that the products have more mass than the parent nucleus, and therefore, the decay is energetically forbidden.

To calculate the energy difference, we can use Einstein's mass-energy equivalence formula: \(E = mc^2\) Where \(m\) is the mass difference, \(c\) is the speed of light, and \(E\) is the energy difference. First, we need to convert the mass difference from atomic mass units (u) to kilograms (kg). The conversion factor is: 1 u = \(1.660539040 \times 10^{-27}\) kg So, Δm in kg = \(-0.009625\mathrm{u} \times \frac{1.660539040 \times 10^{-27}\mathrm{kg}}{1\mathrm{u}} = -1.599 \times 10^{-29}\mathrm{kg}\) Now, we can calculate the energy difference: \(E = (-1.599 \times 10^{-29}\mathrm{kg}) \times (3 \times 10^8\mathrm{m/s})^2 = -4.79 \times 10^{-12}\mathrm{J}\) Since the energy difference is negative, it means that the decay is energetically unfavorable, and the spontaneous α-decay of \(^{19} \mathrm{O}\) is not possible.