# 22.5 Fusion  (Page 6/12)

 Page 6 / 12

## Conceptual questions

Why does the fusion of light nuclei into heavier nuclei release energy?

Energy input is required to fuse medium-mass nuclei, such as iron or cobalt, into more massive nuclei. Explain why.

In considering potential fusion reactions, what is the advantage of the reaction ${}^{2}\text{H}+{}^{3}\text{H}\to {}^{4}\text{He}+n$ over the reaction ${}^{2}\text{H}+{}^{2}\text{H}\to {}^{3}\text{He}+n?$

Give reasons justifying the contention made in the text that energy from the fusion reaction ${}^{2}\text{H}+{}^{2}\text{H}\to {}^{4}\text{He}+\gamma$ is relatively difficult to capture and utilize.

## Problems&Exercises

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

${}^{1}\text{H}+{}^{1}\text{H}\to {}^{2}\text{H}+{e}^{+}+{v}_{\text{e}},$
${}^{1}\text{H}+{}^{2}\text{H}\to {}^{3}\text{He}+\gamma ,$

and

${}^{3}\text{He}+{}^{3}\text{He}\to {}^{4}\text{He}+{}^{1}\text{H}+{}^{1}\text{H}.$

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

(a) $A\text{=1+1=2}$ , $Z\text{=1+1=1+1}$ , $\text{efn}=0=-1+1$

(b) $A\text{=1+2=3}$ , $Z\text{=1+1=2}$ , $\text{efn=0=0}$

(c) $A\text{=3+3=4+1+1}$ , $Z\text{=2+2=2+1+1}$ , $\text{efn=0=0}$

Calculate the energy output in each of the fusion reactions in the proton-proton cycle, and verify the values given in the above summary.

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

$\begin{array}{lll}E& =& \left({m}_{\text{i}}-{m}_{\text{f}}\right){c}^{2}\\ & =& \left[4m\left({}^{1}\text{H}\right)-m\left({}^{4}\text{He}\right)\right]{c}^{2}\\ & =& \left[4\left(1.007825\right)-4\text{.}\text{002603}\right]\left(\text{931.5 MeV}\right)\\ & =& \text{26.73 MeV}\end{array}$

Verify by listing the number of nucleons, total charge, and electron family number before and after the cycle that these quantities are conserved in the overall proton-proton cycle in $2{e}^{-}+4{}^{1}\text{H}\to {}^{4}\text{He}+{2v}_{\text{e}}+6\gamma$ .

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?

$3.12×{\text{10}}^{5}\phantom{\rule{0.25em}{0ex}}\text{kg}$ (about 200 tons)

Tritium is naturally rare, but can be produced by the reaction $n+{}^{2}\text{H}\to {}^{3}\text{H}+\gamma$ . How much energy in MeV is released in this neutron capture?

Two fusion reactions mentioned in the text are

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

and

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

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 ${}^{2}\text{H}$ .

$\begin{array}{lll}E& =& \left({m}_{\text{i}}-{m}_{\text{f}}\right){c}^{2}\\ {E}_{1}& =& \left(\text{1.008665}+\text{3.016030}-\text{4.002603}\right)\left(\text{931.5 MeV}\right)\\ & =& \text{20.58 MeV}\\ {E}_{2}& =& \left(1\text{.}\text{008665}+1\text{.}\text{007825}-2\text{.}\text{014102}\right)\left(\text{931.5 MeV}\right)\\ & =& \text{2.224 MeV}\end{array}$

${}^{4}\text{He is more tightly bound, since this reaction gives off more energy per nucleon.}$

(a) Calculate the number of grams of deuterium in an 80,000-L swimming pool, given deuterium is 0.0150% of natural hydrogen.

(b) Find the energy released in joules if this deuterium is fused via the reaction ${}^{2}\text{H}+{}^{2}\text{H}\to {}^{3}\text{He}+n$ .

(c) Could the neutrons be used to create more energy?

(d) Discuss the amount of this type of energy in a swimming pool as compared to that in, say, a gallon of gasoline, also taking into consideration that water is far more abundant.

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?

$1\text{.}\text{19}×{\text{10}}^{4}\phantom{\rule{0.25em}{0ex}}\text{kg}$

The power output of the Sun is $4×{\text{10}}^{\text{26}}\phantom{\rule{0.25em}{0ex}}\text{W}$ .

(a) If 90% of this is supplied by the proton-proton cycle, how many protons are consumed per second?

(b) How many neutrinos per second should there be per square meter at the Earth from this process? This huge number is indicative of how rarely a neutrino interacts, since large detectors observe very few per day.

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

$\begin{array}{lll}{}^{\text{12}}\text{C}+{}^{1}\text{H}& \to & {}^{\text{13}}\text{N}+\gamma ,\\ {}^{\text{13}}\text{N}& \to & {}^{\text{13}}\text{C}+{e}^{+}+{v}_{e},\\ {}^{\text{13}}\text{C}+{}^{1}\text{H}& \to & {}^{\text{14}}\text{N}+\gamma ,\\ {}^{\text{14}}\text{N}+{}^{1}\text{H}& \to & {}^{\text{15}}\text{O}+\gamma ,\\ \text{}{}^{\text{15}}\text{O}& \to & {}^{\text{15}}\text{N}+{e}^{+}+{v}_{e},\\ {}^{\text{15}}\text{N}+{}^{1}\text{H}& \to & {}^{\text{12}}\text{C}+{}^{4}\text{He.}\end{array}$

Write down the overall effect of the carbon cycle (as was done for the proton-proton cycle in $2{e}^{-}+4{}^{1}\text{H}\to {}^{4}\text{He}+{2v}_{e}+6\gamma$ ). Note the number of protons ( ${}^{1}\text{H}$ ) required and assume that the positrons ( ${e}^{+}$ ) annihilate electrons to form more $\gamma$ rays.

$2{e}^{-}+4{}^{1}\text{H}\to {}^{4}\text{He}+7\gamma +{2v}_{e}$

(a) Find the total energy released in MeV in each carbon cycle (elaborated in the above problem) including the annihilation energy.

(b) How does this compare with the proton-proton cycle output?

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.)

(a) $A\text{=12+1=13}$ , $Z\text{=6+1=7}$ , $\text{efn}=0=0$

(b) $A\text{=13=13}$ , $Z\text{=7=6+1}$ , $\text{efn}=0=-1+1$

(c) $A\text{=13}+\text{1=14}$ , $Z\text{=6+1=7}$ , $\text{efn}=0=0$

(d) $A\text{=14}+\text{1=15}$ , $Z\text{=7+1=8}$ , $\text{efn}=0=0$

(e) $A\text{=1}5\text{=15}$ , $Z\text{=8=7+1}$ , $\text{efn}=0=-1+1$

(f) $A\text{=15}+\text{1=12}+4$ , $Z\text{=7+1=6}+2$ , $\text{efn}=0=0$

Integrated Concepts

The laser system tested for inertial confinement can produce a 100-kJ pulse only 1.00 ns in duration. (a) What is the power output of the laser system during the brief pulse?

(b) How many photons are in the pulse, given their wavelength is $1.06 µm$ ?

(c) What is the total momentum of all these photons?

(d) How does the total photon momentum compare with that of a single 1.00 MeV deuterium nucleus?

Integrated Concepts

Find the amount of energy given to the ${}^{4}\text{He}$ nucleus and to the $\gamma$ ray in the reaction $n{+}^{3}\text{He}{\to }^{4}\text{He}+\gamma$ , 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.

${E}_{\gamma }=\text{20.6 MeV}$

${E}_{{}^{4}\text{He}}=5.68×{10}^{-2}\mathrm{MeV}$

Integrated Concepts

(a) What temperature gas would have atoms moving fast enough to bring two ${}^{3}\text{He}$ nuclei into contact? Note that, because both are moving, the average kinetic energy only needs to be half the electric potential energy of these doubly charged nuclei when just in contact with one another.

(b) Does this high temperature imply practical difficulties for doing this in controlled fusion?

Integrated Concepts

(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 $8×{\text{10}}^{\text{19}}$ 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 natural hydrogen to find the mass of deuterium available. Note that approximate energy yield of deuterium is $3\text{.}\text{37}×{\text{10}}^{\text{14}}$ J/kg.

(b) Comment on how much time this is by any human measure. (It is not an unreasonable result, only an impressive one.)

(a) $3×{\text{10}}^{9}\phantom{\rule{0.25em}{0ex}}\text{y}$

(b) This is approximately half the lifetime of the Earth.

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