Einstein’s mass-energy relation emerging out of his famous theory of relativity relates mass ($m$) to energy ($E$) as $ E = mc^2 $, where $c$ is the speed of light in vacuum. At the nuclear level, the magnitudes of energy are very small. The energy at the nuclear level is usually measured in MeV, where $1$ MeV $= 1.6 \times 10^{-13}$ J; the masses are measured in unified atomic mass unit ($u$) where, $1u = 1.67 \times 10^{-27}$ kg.
(a) Show that the energy equivalent of $1u$ is $931.5$ MeV.
(b) A student writes the relation as $1u = 931.5$ MeV. The teacher points out that the relation is dimensionally incorrect. Write the correct relation.
In this problem, we have to apply Einstein’s mass-energy relation. $E = mc^2$, to calculate the energy equivalent of the given mass.
(a) We know that
$$ 1 \text{ amu} = 1u = 1.67 \times 10^{-27} \text{kg} $$
Applying $ E = mc^2 $
Energy $E$ = $( 1.67 \times 10^{-27})( 3 \times 10^8)^2$ J
= $ 1.67 \times 9 \times 10^{-11} $ J
$$ E = \dfrac{ 1.67 \times 9 \times 10^{-11}}{ 1.6 \times 10^{-13}} \text{MeV} \approx 939.4 \text{MeV} \approx 931.5 \text{MeV} $$
(b) The dimensionally correct relation is
$$ 1 \text{ amu} \times c^2 = 1u \times c^2 = 931.5 \text{MeV} $$