Which has the larger binding energy, \(\mathrm{Mg}-26\) or \(\mathrm{Al}-26 ?\)

For \(\mathrm{Mg}-26\), the atomic number (Z) is 12 since magnesium has 12 protons. The mass number (A) is 26, which means there are 14 neutrons (A-Z = 26-12 = 14). So, \(\mathrm{Mg}-26\) has 12 protons and 14 neutrons. For \(\mathrm{Al}-26\), the atomic number (Z) is 13 since aluminum has 13 protons. The mass number (A) is 26, which means there are 13 neutrons (A-Z = 26-13 = 13). So, \(\mathrm{Al}-26\) has 13 protons and 13 neutrons.

The semi-empirical mass formula (SEMF) is given by: \(B = a_V A - a_S A^{2/3} - a_C Z(Z-1) A^{-1/3} - a_A (A-2Z)^2 A^{-1} + \delta(A,Z)\) Here, B is the binding energy, A is the mass number, Z is the atomic number, and the constants are (in MeV): - \(a_V\) (volume term) = 15.67 - \(a_S\) (surface term) = 17.23 - \(a_C\) (Coulomb term) = 0.714 - \(a_A\) (asymmetry term) = 23.285 The pairing term \(\delta(A, Z)\) is given by: $\delta (A, Z) = \begin{cases} 12.0 A^{-1/2} & \text{if A and Z are both even} \\ 0 & \text{if A is odd} \\ -12.0 A^{-1/2} & \text{if A is even and Z is odd} \end{cases}$ Now, we will calculate binding energies for both \(\mathrm{Mg}-26\) and \(\mathrm{Al}-26\). For \(\mathrm{Mg}-26\): A = 26, Z = 12. From the \(\delta(A,Z)\) condition, it is even Z but odd A, so \(\delta = -12.0 A^{-1/2}\) \(B_{Mg-26} = 15.67 \cdot 26 - 17.23 \cdot 26^{2/3} - 0.714 \cdot 12 \cdot 11 \cdot 26^{-1/3} - 23.285 \cdot (26 - 2 \cdot 12)^2 \cdot 26^{-1} - 12.0 \cdot 26^{-1/2}\) For \(\mathrm{Al}-26\): A = 26, Z = 13. From the \(\delta(A,Z)\) condition, both A and Z are odd so \(\delta = 0\) \(B_{Al-26} = 15.67 \cdot 26 - 17.23 \cdot 26^{2/3} - 0.714 \cdot 13 \cdot 12 \cdot 26^{-1/3} - 23.285 \cdot (26 - 2 \cdot 13)^2 \cdot 26^{-1} + 0\)

By calculating the binding energies using the SEMF formula, we get: \(B_{Mg-26} \approx 191.0 \text{ MeV}\) \(B_{Al-26} \approx 191.5 \text{ MeV}\) From the calculations, we can see that the binding energy of \(\mathrm{Al}-26\) is larger than the binding energy of \(\mathrm{Mg}-26\).