Problem — Energy of a fission bomb

Problem

In a fission bomb, the mass of the products is about $0{,}3$ g less than the initial mass. Estimate the energy released using $E = mc^2$ and compare it with that of $1$ kt of TNT ($\approx 4{,}2\cdot 10^{12}$ J).

Solution

Mass defect. $\Delta m = 0{,}3\;\text{g} = 3\cdot 10^{-4}\;\text{kg}$.

Energy released (mass-energy equivalence).

$\Delta E = \Delta m\,c^2 = (3\cdot 10^{-4}\;\text{kg})(3\cdot 10^{8}\;\text{m/s})^2 = 3\cdot 10^{-4} \times 9\cdot 10^{16}\;\text{J}$

$\ev{\Delta E \approx 2{,}7\cdot 10^{13}\;\text{J}}$

Comparison with TNT.

$\frac{\Delta E}{E_{1\,\text{kt}}} = \frac{2{,}7\cdot 10^{13}}{4{,}2\cdot 10^{12}} \approx 6{,}4$

Just $0{,}3$ g of “vanished” mass releases the energy of about 6.4 kt of TNT: this is the extraordinary efficiency of mass-energy conversion compared to any chemical explosive.

Links

Topics: Special relativity Concepts: Mass-energy equivalence Skills: Fermi estimation