The most common isotope of uranium, ${}_{92}^{238}U$, has atomic mass $238.050788$ u. Calculate (a) the mass defect; (b) the binding energy (in MeV); (c) the binding energy per nucleon.

Verified step by step guidance

1. Step 1: Understand the problem. The mass defect is the difference between the mass of the nucleus and the sum of the masses of its individual protons and neutrons. The binding energy is the energy equivalent of the mass defect, and the binding energy per nucleon is the binding energy divided by the total number of nucleons.
2. Step 2: Calculate the total mass of the individual nucleons. The nucleus of 238 92U contains 92 protons and 238 - 92 = 146 neutrons. Use the mass of a proton (1.007276 u) and the mass of a neutron (1.008665 u) to compute the total mass of the nucleons: \( \text{Total mass of nucleons} = 92 \times 1.007276 + 146 \times 1.008665 \).
3. Step 3: Compute the mass defect. The mass defect is the difference between the total mass of the nucleons and the actual mass of the nucleus: \( \text{Mass defect} = \text{Total mass of nucleons} - \text{Atomic mass of uranium nucleus} \).
4. Step 4: Convert the mass defect to energy using Einstein's equation \( E = \Delta m c^2 \). Since 1 u corresponds to 931.5 MeV/c^2, the binding energy can be calculated as \( \text{Binding energy} = \text{Mass defect} \times 931.5 \).
5. Step 5: Calculate the binding energy per nucleon. Divide the total binding energy by the number of nucleons (238) to find \( \text{Binding energy per nucleon} = \frac{\text{Binding energy}}{238} \).

Key Concepts

Mass Defect

The mass defect is the difference between the mass of an atomic nucleus and the sum of the masses of its individual protons and neutrons. This discrepancy arises because some mass is converted into energy when nucleons bind together, according to Einstein's equation E=mc². Understanding mass defect is crucial for calculating the binding energy of a nucleus.

Binding Energy

Binding energy is the energy required to disassemble a nucleus into its constituent protons and neutrons. It is a measure of the stability of a nucleus; a higher binding energy indicates a more stable nucleus. The binding energy can be calculated using the mass defect and is often expressed in mega-electronvolts (MeV).

Binding Energy per Nucleon

Binding energy per nucleon is the binding energy divided by the total number of nucleons (protons and neutrons) in the nucleus. This value provides insight into the stability of the nucleus relative to its size, allowing for comparisons between different isotopes. A higher binding energy per nucleon generally indicates a more stable nucleus.