Question

How much energy is released in the explosion of a fission bomb containing $\mathbf{3}\mathbf{.}\mathbf{0}\mathbf{}\mathit{k}\mathit{g}$ of fissionable material? Assume that $\mathbf{0}\mathbf{.}\mathbf{10}\mathbf{\%}$ of the mass is converted to released energy. (b) What mass of TNT would have to explode to provide the same energy release? Assume that each mole of TNT liberate $\mathbf{3}\mathbf{.}\mathbf{4}\mathbf{}\mathit{M}\mathit{J}$ of energy on exploding. The molecular mass of TNT is $\mathbf{0}\mathbf{.}\mathbf{227}\mathbf{}\mathit{k}\mathit{g}\mathbf{/}\mathit{m}\mathit{o}\mathit{l}$. (c) For the same mass of explosive, what is the ratio of the energy released in a nuclear explosion to that released in a TNT explosion?

Short answer

1. The fission bomb releases $2.7\times {10}^{14}J$of energy.
2. The required mass is $2.2\times {10}^{7}kg$of TNT is required to release the same amount of energy.
3. The ratio of energy released for the same mass in fission bomb to TNT bomb is $0.73\times {10}^7$.

Step-by-step solution

Relativistic Total energy:

The total energy of an object is the sum of its rest mass energy, kinetic energy if its moving, and potential energy if it is under some force.

(a) Define energy is released in the explosion of a fission bomb:

The total relativistic energy of an object moving at constant velocity will be expressed as,

$E=\gamma m{c}^2$

Here, $\gamma$is the Lorentz factor, and is the rest mass of the object. Here in the question, it is asked to determine the energy released in the explosion.

As no info about the motion of the bomb is given it is assumed that it is stationary. Therefore the value of the Lorentz factor will be unity.

Therefore the energy released in the explosion is,

role="math" localid="1663138082072" $$\begin{gathered} E=\left(0.001\right)\left(3.0kg\right){\left(3.0\times {10}^{8}m/s\right)}^2 \\ =2.7\times {10}^{14}J \end{gathered}$$

Hence, the fission bomb releases $2.7\times {10}^{14}J$of energy.

(b) Determine the mass of an equivalent TNT bomb:

One mole of a TNT bomb releases $3.4MJ$of energy and therefore the no. of moles required to release the same energy as a fission bomb is,

$\frac{2.7\times {10}^{14}J}{3.4\times {10}^{6}J}=0.79\times {10}^{8}moles$

Each mole has a mass of 0.227 kg, therefore $0.79\times {10}^{8}moles$of TNT has a mass of

$\left(0.79\times {10}^{8}mol\right)\left(0.227\frac{kg}{mol}\right)=2.2\times {10}^{7}kg$

Hence, the required mass is $2.2\times {10}^{7}kg$of TNT is required to release the same amount of energy.

(c) Ratio of a fission bomb to TNT bomb:

For the $3kg$ of fission bomb, it releases $2.7\times {10}^{14}J$ of energy. A $3kg$ of TNT bomb releases.

$\left(3.0kg\right)\left(\frac{1}{0.277}\frac{mol}{kg}\right)\left(3.4\times {10}^6\frac{J}{mol}\right)=3.68\times {10}^{7}J$

The ratio of energy released in fission bomb to TNT bomb is,

$\frac{2.7\times {10}^{14}J}{3.68\times {10}^{7}J}=0.73\times {10}^7$

Hence, the ratio of energy released for the same mass in fission bomb to TNT bomb is $0.73\times {10}^7$.