Question

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

Short answer

Answer: No, the spontaneous α-decay of \(^{19} \mathrm{O}\) is not possible because the energy difference is negative, indicating that the decay is energetically unfavorable.

Step-by-step solution

Identify the parent and daughter nuclei and the α-particle

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.

Calculate the mass difference between the parent nucleus and the products

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.

Convert mass difference to energy

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.