To find the nuclear binding energy of a [tex]\(^{56}_{26}\text{Fe}\)[/tex] (iron-56) nucleus, we need to follow these steps:
1. Determine the number of protons and neutrons:
The atomic number [tex]\(Z\)[/tex] of iron (Fe) is 26, which means there are 26 protons. The mass number [tex]\(A\)[/tex] is 56, which implies:
[tex]\[
\text{Number of neutrons} = A - Z = 56 - 26 = 30
\][/tex]
2. Calculate the total mass of the protons and neutrons:
- Mass of one proton ([tex]\(m_p\)[/tex]): [tex]\(1.673 \times 10^{-24}\)[/tex] g
- Mass of one neutron ([tex]\(m_n\)[/tex]): [tex]\(1.675 \times 10^{-24}\)[/tex] g
Total mass of protons:
[tex]\[
\text{Total mass of protons} = 26 \times 1.673 \times 10^{-24} \, \text{g} = 4.3498 \times 10^{-23} \, \text{g}
\][/tex]
Total mass of neutrons:
[tex]\[
\text{Total mass of neutrons} = 30 \times 1.675 \times 10^{-24} \, \text{g} = 5.025 \times 10^{-23} \, \text{g}
\][/tex]
Total mass of all nucleons (protons and neutrons):
[tex]\[
\text{Total mass of nucleons} = 4.3498 \times 10^{-23} \, \text{g} + 5.025 \times 10^{-23} \, \text{g} = 9.3748 \times 10^{-23} \, \text{g}
\][/tex]
3. Calculate the mass defect:
The mass defect is the difference between the total mass of the nucleons and the actual mass of the nucleus.
[tex]\[
\text{Mass defect} = \text{Total mass of nucleons} - \text{Mass of nucleus}
\][/tex]
[tex]\[
\text{Mass defect} = 9.3748 \times 10^{-23} \, \text{g} - 9.289 \times 10^{-23} \, \text{g} = 8.58 \times 10^{-25} \, \text{g}
\][/tex]
4. Convert the mass defect to kilograms:
[tex]\[
1 \, \text{g} = 1 \times 10^{-3} \, \text{kg}
\][/tex]
[tex]\[
\text{Mass defect in kg} = 8.58 \times 10^{-25} \, \text{g} \times 1 \times 10^{-3} \, \frac{\text{kg}}{\text{g}} = 8.58 \times 10^{-28} \, \text{kg}
\][/tex]
5. Calculate the nuclear binding energy using [tex]\(E = mc^2\)[/tex]:
[tex]\[
E = (\text{mass defect}) \times (c^2)
\][/tex]
where [tex]\(c = 3.00 \times 10^8 \, \text{m/s}\)[/tex].
[tex]\[
E = 8.58 \times 10^{-28} \, \text{kg} \times (3.00 \times 10^8 \, \text{m/s})^2
\][/tex]
[tex]\[
E = 8.58 \times 10^{-28} \, \text{kg} \times 9 \times 10^{16} \, \text{m}^2/\text{s}^2
\][/tex]
[tex]\[
E = 77.22 \times 10^{-12} \, \text{J}
\][/tex]
[tex]\[
E = 7.722 \times 10^{-11} \, \text{J}
\][/tex]
Based on these calculations, the correct answer is:
b. [tex]\(7.72 \times 10^{-11}\)[/tex]
