Question

(a) Define atomic mass unit (u).
(b) Calculate the energy required to separate a deuteron into its constituent parts (a proton and a neutron).
\( m_D = 2.014102 \, u} \)
\( m_H = 1.007825 \, u} \)
\( m_n = 1.008665 \, u} \)

Hint:

Understanding the concept of mass defect and binding energy is crucial for explaining why nuclei are stable and the energy processes involved in nuclear reactions.

Answer 1

The atomic mass unit (u), also known as the unified atomic mass unit, is defined as one twelfth of the mass of an unbound neutral atom of carbon-12. In other words: \[ 1 \, u} = \frac{1}{12} \times mass of one carbon-12 atom} \] The atomic mass unit is approximately equal to: \[ 1 \, u} = 1.66053906660 \times 10^{-27} \, kg} \] It is a standard unit of mass used to express atomic and molecular weights. % Part (b) Calculation of Separation Energy (b) Calculation of Separation Energy:
First, calculate the mass defect (\(\Delta m\)): \[ \Delta m = \left( m_{H}} + m_{n}} \right) - m(D) \] Substitute the given values: \[ \Delta m = (1.007825 \, u} + 1.008665 \, u}) - 2.014102 \, u} \] \[ \Delta m = 2.016490 \, u} - 2.014102 \, u} = 0.002388 \, u} \] Next, convert the mass defect from atomic mass units to kilograms. 1 u is approximately \(1.660539 \times 10^{-27}\) kg: \[ \Delta m = 0.002388 \, u} \times 1.660539 \times 10^{-27} \, kg/u} = 3.965 \times 10^{-30} \, kg} \] Now, calculate the energy using \(E = \Delta m \cdot c^2\), where \(c = 3 \times 10^8 \, m/s}\): \[ E = 3.965 \times 10^{-30} \, kg} \times (3 \times 10^8 \, m/s})^2 \] \[ E = 3.965 \times 10^{-30} \, kg} \times 9 \times 10^{16} \, m}^2/s}^2 = 3.5685 \times 10^{-13} \, J} \] To express the energy in MeV (1 MeV = \(1.60218 \times 10^{-13}\) J): \[ E = \frac{3.5685 \times 10^{-13} \, J}}{1.60218 \times 10^{-13} \, J/MeV}} \approx 2.23 \, MeV} \] Therefore, the energy required to separate a deuteron into a proton and a neutron is approximately \boxed{2.23 \, MeV}}.