The binding energy per nucleon for magnesium- 27 is \(1.326\) \(\times 10^{-12} \mathrm{~J} /\) nucleon. Calculate the atomic mass of \({ }^{27} \mathrm{Mg} .\)

To find the total binding energy, multiply the binding energy per nucleon by the total number of nucleons. For Mg-27, there are 27 nucleons (12 protons and 15 neutrons). Total Binding Energy = Binding energy per nucleon x Total number of nucleons Total Binding Energy = \(1.326 \times 10^{-12}\) J/nucleon × 27 nucleons #Step 2: Convert the total binding energy to mass difference#

Use the mass-energy equivalence equation \(E = mc^2\), where E is the total binding energy, m is the mass difference, and c is the speed of light. First, rearrange the equation to solve for the mass difference: m = \(\frac{E}{c^2}\) Then, plug in the values for total binding energy (E) and the speed of light (c = \(3 \times 10^8\) m/s): m = \(\frac{(1.326 \times 10^{-12}\) J/nucleon × 27 nucleons}{(3 \times 10^8 m/s)^2}\) #Step 3: Calculate the mass of Magnesium-27#

Now that we have the mass difference, we can find the mass of Magnesium-27 by adding the mass difference to the mass of the 27 nucleons (the sum of the individual proton and neutron masses). The mass of a proton is approximately \(1.673 \times 10^{-27}\) kg, and the mass of a neutron is approximately \(1.675 \times 10^{-27}\) kg. Since Magnesium-27 has 12 protons and 15 neutrons, we can find the total mass of 27 nucleons: Total mass of nucleons = (12 × mass of a proton) + (15 × mass of a neutron) Total mass of nucleons = (12 × \(1.673 \times 10^{-27}\) kg) + (15 × \(1.675 \times 10^{-27}\) kg) Then, add the mass difference to the total mass of nucleons to find the atomic mass of Magnesium-27: Atomic mass of Mg-27 = Total mass of nucleons + Mass difference This will give us the atomic mass of Magnesium-27 in kg.