Question

Derive the relation between atomic mass unit (u) and electron volt (eV).

Hint:

In nuclear mass defect calculations ($\Delta m$), multiplying mass defect in amu ($u$) directly by $931.5\text{ MeV}$ yields the binding energy in $\text{MeV}$ directly: $E_b = \Delta m \times 931.5\text{ MeV}$.

Answer 1

Concept:
• $1\text{ atomic mass unit (u)}$ is defined as $\frac{1}{12}\text{th}$ of the mass of an unbound neutral Carbon-12 atom: $1\text{ u} \approx 1.660539 \times 10^{-27}\text{ kg}$.

• According to Einstein's mass-energy equivalence principle, $E = m c^2$.

• $1\text{ electron-volt (eV)} = 1.60218 \times 10^{-19}\text{ J}$.

Step 1: Calculate energy equivalent in Joules
Mass $m = 1\text{ u} = 1.660539 \times 10^{-27}\text{ kg}$.
Speed of light $c = 2.99792 \times 10^8\text{ m/s}$.
Apply Einstein's equation $E = m c^2$:
\[ E = (1.660539 \times 10^{-27}\text{ kg}) \times (2.99792 \times 10^8\text{ m/s})^2 \]
\[ E = 1.660539 \times 10^{-27} \times 8.98755 \times 10^{16}\text{ J} \]
\[ E \approx 1.49242 \times 10^{-10}\text{ J} \]

Step 2: Convert energy from Joules to eV
Since $1\text{ eV} = 1.60218 \times 10^{-19}\text{ J}$:
\[ E\text{ (in eV)} = \frac{1.49242 \times 10^{-10}\text{ J}}{1.60218 \times 10^{-19}\text{ J/eV}} \]
\[ E\text{ (in eV)} \approx 931.5 \times 10^6\text{ eV} \]

Step 3: Convert to Mega electron-volts (MeV)
Since $1\text{ MeV} = 10^6\text{ eV}$:
\[ E = 931.5\text{ MeV} \]

Step 4: Conclusion
The mass-energy equivalent relation is $1\text{ u} \approx 931.5\text{ MeV} = 9.315 \times 10^8\text{ eV}$.