Question 30.2: Mass defect for carbon Now let’s calculate the mass defect, ...
Mass defect for carbon
Now let’s calculate the mass defect, the total binding energy, and the binding energy per nucleon for the most common isotope of carbon, \mathrm{{}_{6}^{12}C } .
SET UP The neutral carbon atom consists of six protons, six neutrons, and six electrons. The total mass of six protons and six electrons is equal to the mass of six hydrogen atoms. Therefore, the total mass of the separate parts is 6m_H + 6m_n, so the mass defect ΔM is given by ΔM = 6m_H + 6m_n – m_C. We find the masses from Table 30.2 and the preceding discussion.
SOLVE We substitute numerical values in this expression for ΔM, obtaining
ΔM = 6(1.007825 u) + 6(1.0086649 u) – 12.000000 u = 0.09894 u.
The energy equivalent of this mass is
(0.09894 u)(931.5 MeV/u) = 92.16 MeV.
Thus, the total binding energy for the 12 nucleons is 92.16 MeV, and the binding energy per nucleon is 92.16 MeV/12 = 7.68 MeV per nucleon.
Alternate Solution: We could first find the mass of the bare carbon nucleus by subtracting the mass of six electrons from the mass of the neutral atom (including six electrons), given in Table 30.2 as 12.00000 u. Then we find the total mass of six protons and six neutrons. The mass defect is that total minus the mass of the bare nucleus. This method is more complicated than the first solution because the electron masses have to be subtracted explicitly. We don’t recommend this alternate solution method.
REFLECT To pull the carbon nucleus completely apart into 12 separate nucleons would require a minimum of 92.16 MeV. The binding energy per nucleon is one-twelfth of this, or 7.68 MeV per nucleon.
Nearly all stable nuclei, from the lightest to the most massive, have binding energies in the range from 6 to 9 MeV per nucleon, for reasons we’ll discuss in the next section.
Practice Problem: Find the mass defect, the total binding energy, and the binding energy per nucleon for the common isotope of helium, 4 He. Answers: 0.03038 u, 28.30 MeV, 7.075 MeV.
| TABLE 30.2 Atomic masses of light elements | ||||
| Element | Atomic number, Z |
Mass number, N |
Atomic mass u |
Mass number, A |
| Hydrogen, H | 1 | 0 | 1.007825 | 1 |
| Deuterium, H | 1 | 1 | 2.014101 | 2 |
| Helium, He | 2 | 1 | 3.016029 | 3 |
| Helium, He | 2 | 2 | 4.002603 | 4 |
| Lithium, Li | 3 | 3 | 6.015123 | 6 |
| Lithium, Li | 3 | 4 | 7.016003 | 7 |
| Beryllium, Be | 4 | 5 | 9.012183 | 9 |
| Boron, B | 5 | 5 | 10.012937 | 10 |
| Boron, B | 5 | 6 | 11.009305 | 11 |
| Carbon, C | 6 | 6 | 12.000000 | 12 |
| Carbon, C | 6 | 7 | 13.003355 | 13 |
| Nitrogen, N | 7 | 7 | 14.003074 | 14 |
| Nitrogen, N | 7 | 8 | 15.000109 | 15 |
| Oxygen, O | 8 | 8 | 15.994915 | 16 |
| Oxygen, O | 8 | 9 | 16.999132 | 17 |
| Oxygen, O | 8 | 10 | 17.999160 | 18 |
| Source: Atomic Mass Evaluation 2013. M. Wang et al 2012 Chinese Phys. C 36 1603 doi:10.1088/1674-1137/36/12/003 | ||||