10.6 Nuclear fusion  (Page 3/8)

Solution

1. The decrease in mass for the fusion reaction is
   
   $\begin{array}{cc}\hfill \text{Δ}m& =4m\left({}_1^1\text{H}\right)-m\left({}_2^4\text{H}\text{e}\right)-2m\left({}_{\text{+}1}^0\text{e}\right)\hfill \\ & =4\left(1.007825\text{u}\right)-4.002603\text{u}-2\left(0.000549\text{u}\right)\hfill \\ & =0.0276\text{u}.\hfill \end{array}$
   
   The energy released per fusion reaction is
   
   $Q=\left(0.0276\text{u}\right)\left(931.49\text{MeV/u}\right)=25.7\text{MeV}.$
   
   Thus, to supply $3.8\times {10}^{26}\text{J/s}=2.38\times {10}^{39}\text{MeV/s},$ there must be
   
   $\frac{2.38\times {10}^{39}\text{MeV/s}}{25.7\text{MeV/reaction}}=9.26\times {10}^{37}\text{reaction/s}.$
2. The Sun’s mass decreases by $0.0276\text{u}=4.58\times {10}^{\text{−}29}\text{kg}$ per fusion reaction, so the rate at which its mass decreases is
   
   $\left(9.26\times {10}^{37}\text{reaction/s}\right)\left(4.58\times {10}^{\mathrm{-29}}\text{kg/reaction}\right)=4.24\times {10}^9\text{kg/s}.$
3. In $5\times {10}^9\text{y}=1.6\times {10}^{17}\text{s},$ the Sun’s mass will therefore decrease by
   
   $\text{Δ}M=\left(4.24\times {10}^9\text{kg/s}\right)\left(1.6\times {10}^{17}\text{s}\right)=6.8\times {10}^{26}\text{kg}.$
   
   The current mass of the Sun is about $2.0\times {10}^{30}\text{kg},$ so the percentage decrease in its mass when its hydrogen fuel is depleted will be
   
   $\left(\frac{6.8\times {10}^{26}\text{kg}}{2.0\times {10}^{30}\text{kg}}\right)\times 100\text{\%}=0.034\text{\%}.$

Significance

The hydrogen bomb

In 1942, Robert Oppenheimer suggested that the extremely high temperature of an atomic bomb could be used to trigger a fusion reaction between deuterium and tritium, thus producing a fusion (or hydrogen) bomb. The reaction between deuterium and tritium, both isotopes of hydrogen, is given by

Deuterium is relatively abundant in ocean water but tritium is scarce. However, tritium can be generated in a nuclear reactor through a reaction involving lithium. The neutrons from the reactor cause the reaction

to produce the desired tritium. The first hydrogen bomb was detonated in 1952 on the remote island of Eniwetok in the Marshall Islands. A hydrogen bomb has never been used in war. Modern hydrogen bombs are approximately 1000 times more powerful than the fission bombs dropped on Hiroshima and Nagasaki in World War II.

The fusion reactor

The fusion chain believed to be the most practical for use in a nuclear fusion reactor    is the following two-step process:

This chain, like the proton-proton chain, produces energy without any radioactive by-product. However, there is a very difficult problem that must be overcome before fusion can be used to produce significant amounts of energy: Extremely high temperatures $\left(~{10}^7\text{K}\right)$ are needed to drive the fusion process. To meet this challenge, test fusion reactors are being developed to withstand temperatures 20 times greater than the Sun’s core temperature. An example is the Joint European Torus (JET) shown in [link] . A great deal of work still has to be done on fusion reactor technology, but many scientists predict that fusion energy will power the world’s cities by the end of the twentieth century.

Summary

• Nuclear fusion is a reaction in which two nuclei are combined to form a larger nucleus; energy is released when light nuclei are fused to form medium-mass nuclei.
• The amount of energy released by a fusion reaction is known as the Q value.
• Nuclear fusion explains the reaction between deuterium and tritium that produces a fusion (or hydrogen) bomb; fusion also explains the production of energy in the Sun, the process of nucleosynthesis, and the creation of the heavy elements.