32.5 Fusion

When deuterium and tritium nuclei fuse to produce helium, what else is produced?

1. positron
2. proton
3. α-particle
4. neutron

Verify that the total number of nucleons, total charge, and electron family number are conserved for each of the fusion reactions in the proton-proton cycle in

and

(List the value of each of the conserved quantities before and after each of the reactions.)

Show that the total energy released in the proton-proton cycle is 26.7 MeV, considering the overall effect in $^{1} \text{H } + ^{1} \text{H } \rightarrow ^{2} \text{H} + e^{+ } + v_{\text{e}}$, $^{1} \text{H} + ^{2} \text{H} \rightarrow ^{3} \text{He } + \gamma$, and $^{3} \text{He } + ^{3} \text{He} \rightarrow ^{4} \text{He} + ^{1} \text{H} + ^{1} \text{H}$ and being certain to include the annihilation energy.

The energy produced by the fusion of a 1.00-kg mixture of deuterium and tritium was found in Example Calculating Energy and Power from Fusion. Approximately how many kilograms would be required to supply the annual energy use in the United States?

Two fusion reactions mentioned in the text are

$n + ^{3} \text{He} \rightarrow ^{4} \text{He} + \gamma$

and

$n + ^{1} \text{H} \rightarrow ^{2} \text{H} + \gamma$.

Both reactions release energy, but the second also creates more fuel. Confirm that the energies produced in the reactions are 20.58 and 2.22 MeV, respectively. Comment on which product nuclide is most tightly bound, $^{4} \text{He}$ or $^{2} \text{H}$.

How many kilograms of water are needed to obtain the 198.8 mol of deuterium, assuming that deuterium is 0.01500% (by number) of natural hydrogen?

Another set of reactions that result in the fusing of hydrogen into helium in the Sun and especially in hotter stars is called the carbon cycle. It is
$\begin{matrix}^{\text{12}} \text{C} + ^{1} \text{H } & \rightarrow & ^{\text{13}} \text{N} + \gamma , \\ ^{\text{13}} \text{N } & \rightarrow & ^{\text{13}} \text{C} + e^{+ } + v_{e} , \\ ^{\text{13}} \text{C } + ^{1} \text{H } & \rightarrow & ^{\text{14}} \text{N} + \gamma , \\ ^{\text{14}} \text{N} + ^{1} \text{H } & \rightarrow & ^{\text{15}} \text{O} + \gamma , \\ ^{\text{15}} \text{O } & \rightarrow & ^{\text{15}} \text{N} + e^{+ } + v_{e} , \\ ^{\text{15}} \text{N} + ^{1} \text{H } & \rightarrow & ^{\text{12}} \text{C} + ^{4} \text{He}.\end{matrix}$
Write down the overall effect of the carbon cycle (as was done for the proton-proton cycle in $2 e^{- } + 4 ^{1} \text{H} \rightarrow ^{4} \text{He} + \left(2 v\right)_{e} + 6γ$). Note the number of protons ( $^{1} \text{H}$) required and assume that the positrons ( $e^{+ }$) annihilate electrons to form more $\gamma$ rays.

Verify that the total number of nucleons, total charge, and electron family number are conserved for each of the fusion reactions in the carbon cycle given in the above problem. (List the value of each of the conserved quantities before and after each of the reactions.)

Find the amount of energy given to the $^{4} \text{He}$ nucleus and to the $\gamma$ ray in the reaction $n +^{3} \text{He} \rightarrow^{4} \text{He} + \gamma$, using the conservation of momentum principle and taking the reactants to be initially at rest. This should confirm the contention that most of the energy goes to the $\gamma$ ray.

(a) Estimate the years that the deuterium fuel in the oceans could supply the energy needs of the world. Assume world energy consumption to be ten times that of the United States which is $8 \times \text{10}^{\text{19}}$ J/y and that the deuterium in the oceans could be converted to energy with an efficiency of 32%. You must estimate or look up the amount of water in the oceans and take the deuterium content to be 0.015% of na…