Question

Question: Calculate the amount of energy released in kilojoules per gram of deuterium (${}^{\mathbf{2}}\mathbf{H}$), for the fusion reaction

${}_{\mathbf{1}}{}^{\mathbf{2}}\mathbf{H}\mathbf{+}{}_{\mathbf{1}}{}^{\mathbf{2}}\mathbf{H}\mathbf{\to }{}_{\mathbf{2}}{}^{\mathbf{4}}\mathbf{H}\mathbf{e}$

Use the atomic masses in Table 19.1. Compare your answer with that of the preceding problem.

Short answer

The energy released is $5.6\times {10}^8\mathrm{kJ}/\mathrm{g}$.

Step-by-step solution

Calculating the mass difference

In a nuclear fusion reaction, two lighter nuclides combine to form a heavier nuclide. The equation given below represents a nuclear fusion reaction.

$2_1^2\mathrm{H}\to {}_2{}^4\mathrm{He}$

We have to calculate the mass difference for the fusion reaction.

$\begin{array}{c}\mathrm{m}=\mathrm{m}\left({}_2{}^4\mathrm{He}\right)-2\mathrm{m}\left({}_1{}^2\mathrm{H}\right)\\ =\left(4.0026-2\times 2.0141\right)\mathrm{u}\\ =0.0256\mathrm{u}\end{array}$

We have to calculate the mass difference for the nuclear reaction resulting from 1 gram deuterium.

role="math" localid="1661235526363" $\begin{array}{rcl}∆\mathrm{m}& =& \frac{1\mathrm{g}{}^2\mathrm{H}}{2.041\mathrm{g}/\mathrm{mol}}\times \frac{6.023\times {10}^{23}\mathrm{atoms}}{1\mathrm{mol}}\times \left(\frac{-0.0256\mathrm{u}}{2\mathrm{atoms}}\right)\times \frac{1\times {10}^{-3}\mathrm{kg}}{6.023\times {10}^{23}\mathrm{u}}\\ & =& -6.27\times {10}^{-27}\mathrm{kg}\end{array}$

Calculating the energy released in the fusion reaction

From Einstein’s mass-energy relation, we get:

$\begin{array}{rcl}∆\mathrm{E}& =& {\mathrm{mc}}^2\\ & =& -6.27\times {10}^{-6}\mathrm{kg}\times {\left(2.99\times {10}^8\mathrm{m}/\mathrm{s}\right)}^2\\ & =& -5.6\times {10}^{11}\mathrm{J}/\mathrm{g}\\ & =& -5.6\times {10}^8\mathrm{kJ}/\mathrm{g}\end{array}$

Therefore, the energy released in the fusion reaction is$5.6\times {10}^8\mathrm{kJ}/\mathrm{g}$.

Comparison with energy released from the fission reaction

The energy released in the fission reaction is $7.5\times {10}^8\mathrm{kJ}/\mathrm{g}$, while the energy released in the fusion reaction is$5.6\times {10}^8\mathrm{kJ}/\mathrm{g}$.