Find the binding energy per nucleon in J and eV for the three following isotopes, given their atomic masses in u: 22(A) 6(Z) of carbon 22.0572097 u
To find the binding energy per nucleon for the given isotopes, we need to follow a few steps.
Step 1: Convert the atomic mass from u (unified atomic mass unit) to kg.
We know that 1 u is equal to 1.66054 x 10^-27 kg. Therefore, we can convert the atomic mass from u to kg using the following formula:
Mass (kg) = Atomic Mass (u) x (1.66054 x 10^-27 kg/u)
For carbon-22, the atomic mass is 22.0572097 u:
Mass (kg) = 22.0572097 u x (1.66054 x 10^-27 kg/u)
Mass (kg) ≈ 3.66 x 10^-26 kg
Step 2: Calculate the total binding energy for the isotope.
The total binding energy is given by the mass defect, which is the difference between the mass of the nucleus and the sum of the masses of its individual nucleons.
Mass defect (kg) = (A x Mass of a nucleon) - Mass of the nucleus
We know that the mass of a nucleon is approximately 1.675 x 10^-27 kg.
For carbon-22:
Mass defect (kg) = (22 x 1.675 x 10^-27 kg) - (3.66 x 10^-26 kg)
Mass defect (kg) ≈ -2.75 x 10^-27 kg
Step 3: Convert the mass defect to energy using Einstein's mass-energy equivalence equation, E = mc^2.
Here, c is the speed of light, approximately 3 x 10^8 m/s.
Binding energy (J) = Mass defect (kg) x (Speed of light)^2
For carbon-22:
Binding energy (J) ≈ (-2.75 x 10^-27 kg) x (3 x 10^8 m/s)^2
Binding energy (J) ≈ -2.475 x 10^-11 J
Step 4: Convert the binding energy from joules to electron volts (eV).
1 electron volt (eV) is equal to 1.602 x 10^-19 joules.
Binding energy (eV) = Binding energy (J) / (1.602 x 10^-19 eV/J)
For carbon-22:
Binding energy (eV) ≈ (-2.475 x 10^-11 J) / (1.602 x 10^-19 eV/J)
Binding energy (eV) ≈ -154.56 MeV
The binding energy per nucleon is approximately -154.56 MeV. Remember that the negative sign indicates that energy is released when nucleons come together, as in the process of nuclear fusion.
Please note that the calculated value is negative because we calculated the binding energy released.
To find the binding energy per nucleon for an isotope, we can use the formula:
Binding energy per nucleon (in J) = (mass defect * c^2) / number of nucleons
where c is the speed of light (approximately 3 × 10^8 m/s).
First, we need to calculate the mass defect for the given isotope:
Atomic mass of carbon-22 (A) = 22.0572097 u
Atomic mass of 6 protons = 6 * 1.007276 u (mass of a proton)
Atomic mass of 16 neutrons = 16 * 1.008665 u (mass of a neutron)
Total mass of protons and neutrons = (6 * 1.007276 u) + (16 * 1.008665 u)
Total mass of the nucleus = 22.0572097 u
Mass defect = Total mass of protons and neutrons - Total mass of the nucleus
Now, let's calculate the binding energy per nucleon in joules (J):
Binding energy per nucleon (J) = (mass defect * c^2) / number of nucleons
Number of nucleons = Number of protons + Number of neutrons = 6 + 16 = 22
Binding energy per nucleon (J) = (mass defect * c^2) / 22
Finally, we can convert the binding energy per nucleon from joules (J) to electron volts (eV):
1 eV = 1.60218 × 10^-19 J
Binding energy per nucleon (eV) = (Binding energy per nucleon in J) / (1.60218 × 10^-19)
Let's calculate the binding energy per nucleon for carbon-22:
Step 1: Calculate the mass defect:
Total mass of protons and neutrons = (6 * 1.007276 u) + (16 * 1.008665 u) = 22.993036 u
Mass defect = 22.993036 u - 22.0572097 u = 0.9358263 u
Step 2: Calculate the binding energy per nucleon in joules (J):
Binding energy per nucleon (J) = (mass defect * c^2) / number of nucleons
Binding energy per nucleon (J) = (0.9358263 * (3 × 10^8 m/s)^2) / 22
Binding energy per nucleon (J) ≈ 1.224 × 10^-10 J
Step 3: Convert the binding energy per nucleon from joules (J) to electron volts (eV):
Binding energy per nucleon (eV) ≈ (1.224 × 10^-10 J) / (1.60218 × 10^-19) ≈ 763.77 eV
Therefore, the binding energy per nucleon for carbon-22 is approximately 1.224 × 10^-10 J or 763.77 eV.