Part C
Calculate (in MeV ) the total binding energy for 40 Ca .
To calculate the total binding energy for a nucleus, you need to know the mass defect of the nucleus. The mass defect is the difference between the mass of the nucleus and the combined mass of its individual protons and neutrons.
The total binding energy is then related to the mass defect through Einstein's famous equation E = mc^2, where E represents energy, m represents mass, and c represents the speed of light.
To calculate the mass defect of an isotope, you can use the atomic mass of its constituent particles. The atomic mass of a proton is approximately 1.0073 atomic mass units (u), and the atomic mass of a neutron is approximately 1.0087 u.
For ^40Ca (calcium-40), you need to know the number of protons and neutrons in its nucleus. Calcium has an atomic number of 20, which means it has 20 protons. Since the atomic mass of ^40Ca is 40.078 u, we can subtract the atomic mass of its protons to find the number of neutrons.
40.078 u - (20 protons x 1.0073 u/proton) = 40.078 u - 20.146 u = 19.932 u
So, ^40Ca has 20 protons and 20 neutrons.
Next, we can calculate the mass defect by subtracting the combined mass of the protons and neutrons from the actual mass of the ^40Ca nucleus:
Mass defect = (mass of protons x number of protons) + (mass of neutrons x number of neutrons) - actual mass of nucleus
Mass defect = (1.0073 u x 20) + (1.0087 u x 20) - 40.078 u
Mass defect = 20.146 u + 20.174 u - 40.078 u = 0.242 u
Now we can convert the mass defect into energy using Einstein's equation:
Energy = mass defect x c^2
where c is the speed of light, approximately 3 x 10^8 m/s.
Energy = 0.242 u x (1.66 x 10^-27 kg/u) x (3 x 10^8 m/s)^2
It is important to convert u to kilograms because the unit of mass in the equation is kilograms.
After calculating the multiplication, we can convert the energy from joules to MeV (Mega electron volts) by dividing it by 1.602 x 10^-13 J/MeV.
Note: 1 MeV = 1.602 x 10^-13 J
Upon performing the calculations, the total binding energy of ^40Ca is approximately 335.526 MeV (rounded to three decimal places).
Please keep in mind that these calculations are approximate and might differ slightly based on the values used for atomic masses and constants.