# 31.4 Nuclear decay and conservation laws  (Page 7/25)

 Page 7 / 25

What do the three types of beta decay have in common that is distinctly different from alpha decay?

## Problems&Exercises

In the following eight problems, write the complete decay equation for the given nuclide in the complete ${}_{Z}^{A}{X}_{N}$ notation. Refer to the periodic table for values of $Z$ .

${\beta }^{-}$ decay of ${}^{3}\text{H}$ (tritium), a manufactured isotope of hydrogen used in some digital watch displays, and manufactured primarily for use in hydrogen bombs.

${}_{1}^{3}{\text{H}}_{2}\to {}_{2}^{3}{\text{He}}_{1}+{\beta }^{-}+{\overline{\nu }}_{e}$

${\beta }^{-}$ decay of ${}^{\text{40}}K$ , a naturally occurring rare isotope of potassium responsible for some of our exposure to background radiation.

${\beta }^{+}$ decay of ${}^{\text{50}}\text{Mn}$ .

${}_{\text{25}}^{\text{50}}{M}_{\text{25}}\to {}_{\text{24}}^{\text{50}}{\text{Cr}}_{\text{26}}+{\beta }^{+}+{\nu }_{e}$

${\beta }^{+}$ decay of ${}^{\text{52}}\text{Fe}$ .

Electron capture by ${}^{7}\text{Be}$ .

${}_{4}^{7}{\text{Be}}_{3}+{e}^{-}\to {}_{3}^{7}{\text{Li}}_{4}+{\nu }_{e}$

Electron capture by ${}^{\text{106}}\text{In}$ .

$\alpha$ decay of ${}^{\text{210}}\text{Po}$ , the isotope of polonium in the decay series of ${}^{\text{238}}\text{U}$ that was discovered by the Curies. A favorite isotope in physics labs, since it has a short half-life and decays to a stable nuclide.

${}_{\text{84}}^{\text{210}}{\text{Po}}_{\text{126}}\to {}_{\text{82}}^{\text{206}}{\text{Pb}}_{\text{124}}+{}_{2}^{4}{\text{He}}_{2}$

$\alpha$ decay of ${}^{\text{226}}\text{Ra}$ , another isotope in the decay series of ${}^{\text{238}}\text{U}$ , first recognized as a new element by the Curies. Poses special problems because its daughter is a radioactive noble gas.

In the following four problems, identify the parent nuclide and write the complete decay equation in the ${}_{Z}^{A}{X}_{N}$ notation. Refer to the periodic table for values of $Z$ .

${\beta }^{-}$ decay producing ${}^{\text{137}}\text{Ba}$ . The parent nuclide is a major waste product of reactors and has chemistry similar to potassium and sodium, resulting in its concentration in your cells if ingested.

${}_{\text{55}}^{\text{137}}{\text{Cs}}_{\text{82}}\to {}_{\text{56}}^{\text{137}}{\text{Ba}}_{\text{81}}+{\beta }^{-}+{\overline{\nu }}_{e}$

${\beta }^{-}$ decay producing ${}^{\text{90}}\text{Y}$ . The parent nuclide is a major waste product of reactors and has chemistry similar to calcium, so that it is concentrated in bones if ingested ( ${}^{\text{90}}\text{Y}$ is also radioactive.)

$\alpha$ decay producing ${}^{\text{228}}\text{Ra}$ . The parent nuclide is nearly 100% of the natural element and is found in gas lantern mantles and in metal alloys used in jets ( ${}^{\text{228}}\text{Ra}$ is also radioactive).

${}_{\text{90}}^{\text{232}}{\text{Th}}_{\text{142}}\to {}_{\text{88}}^{\text{228}}{\text{Ra}}_{\text{140}}+{}_{2}^{4}{\text{He}}_{2}$

$\alpha$ decay producing ${}^{\text{208}}\text{Pb}$ . The parent nuclide is in the decay series produced by ${}^{\text{232}}\text{Th}$ , the only naturally occurring isotope of thorium.

When an electron and positron annihilate, both their masses are destroyed, creating two equal energy photons to preserve momentum. (a) Confirm that the annihilation equation ${e}^{+}+{e}^{-}\to \gamma +\gamma$ conserves charge, electron family number, and total number of nucleons. To do this, identify the values of each before and after the annihilation. (b) Find the energy of each $\gamma$ ray, assuming the electron and positron are initially nearly at rest. (c) Explain why the two $\gamma$ rays travel in exactly opposite directions if the center of mass of the electron-positron system is initially at rest.

(a) $\text{charge:}\left(+1\right)+\left(-1\right)=0;\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}\text{electron family number:}\phantom{\rule{0.25em}{0ex}}\left(+1\right)+\left(-1\right)=0;\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}A: 0+0=0$

(b) 0.511 MeV

(c) The two $\gamma$ rays must travel in exactly opposite directions in order to conserve momentum, since initially there is zero momentum if the center of mass is initially at rest.

Confirm that charge, electron family number, and the total number of nucleons are all conserved by the rule for $\alpha$ decay given in the equation ${}_{Z}^{A}{X}_{N}\to {}_{Z-2}^{A-4}{\text{Y}}_{N-2}+{}_{2}^{4}{\text{He}}_{2}$ . To do this, identify the values of each before and after the decay.

Confirm that charge, electron family number, and the total number of nucleons are all conserved by the rule for ${\beta }^{-}$ decay given in the equation ${}_{Z}^{A}{X}_{N}\to {}_{Z+1}^{A}{\text{Y}}_{N-1}+{\beta }^{-}+{\overline{\nu }}_{e}$ . To do this, identify the values of each before and after the decay.

$Z=\left(Z+1\right)-1;\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}A=A\text{;}\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}\text{efn : 0}=\left(+1\right)+\left(-1\right)$

Confirm that charge, electron family number, and the total number of nucleons are all conserved by the rule for ${\beta }^{-}$ decay given in the equation ${}_{Z}^{A}{X}_{N}\to {}_{Z-1}^{A}{\text{Y}}_{N-1}+{\beta }^{-}+{\nu }_{e}$ . To do this, identify the values of each before and after the decay.

Confirm that charge, electron family number, and the total number of nucleons are all conserved by the rule for electron capture given in the equation ${}_{Z}^{A}{X}_{N}+{e}^{-}\to {}_{Z-1}^{A}{\text{Y}}_{N+1}+{\nu }_{e}$ . To do this, identify the values of each before and after the capture.

$Z-1=Z-1;\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}A=A;\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}\text{efn :}\left(+1\right)=\left(+1\right)$

A rare decay mode has been observed in which ${}^{\text{222}}\text{Ra}$ emits a ${}^{\text{14}}C$ nucleus. (a) The decay equation is ${}^{\text{222}}\text{Ra}{\to }^{A}{\text{X+}}^{\text{14}}\text{C}$ . Identify the nuclide ${}^{A}X$ . (b) Find the energy emitted in the decay. The mass of ${}^{\text{222}}\text{Ra}$ is 222.015353 u.

(a) Write the complete $\alpha$ decay equation for ${}^{\text{226}}\text{Ra}$ .

(b) Find the energy released in the decay.

(a) ${}_{\text{88}}^{\text{226}}{\text{Ra}}_{138}\to {}_{\text{86}}^{\text{222}}{\text{Rn}}_{\text{136}}+{}_{2}^{4}{\text{He}}_{2}$

(b) 4.87 MeV

(a) Write the complete $\alpha$ decay equation for ${}^{\text{249}}\text{Cf}$ .

(b) Find the energy released in the decay.

(a) Write the complete ${\beta }^{-}$ decay equation for the neutron. (b) Find the energy released in the decay.

(a) $\text{n}\to \text{p}+{\beta }^{-}+{\overline{\nu }}_{e}$

(b) ) 0.783 MeV

(a) Write the complete ${\beta }^{-}$ decay equation for ${}^{\text{90}}\text{Sr}$ , a major waste product of nuclear reactors. (b) Find the energy released in the decay.

Calculate the energy released in the ${\beta }^{+}$ decay of ${}^{\text{22}}\text{Na}$ , the equation for which is given in the text. The masses of ${}^{\text{22}}\text{Na}$ and ${}^{\text{22}}\text{Ne}$ are 21.994434 and 21.991383 u, respectively.

1.82 MeV

(a) Write the complete ${\beta }^{+}$ decay equation for ${}^{\text{11}}\text{C}$ .

(b) Calculate the energy released in the decay. The masses of ${}^{\text{11}}\text{C}$ and ${}^{\text{11}}\text{B}$ are 11.011433 and 11.009305 u, respectively.

(a) Calculate the energy released in the $\alpha$ decay of ${}^{\text{238}}\text{U}$ .

(b) What fraction of the mass of a single ${}^{\text{238}}\text{U}$ is destroyed in the decay? The mass of ${}^{\text{234}}\text{Th}$ is 234.043593 u.

(c) Although the fractional mass loss is large for a single nucleus, it is difficult to observe for an entire macroscopic sample of uranium. Why is this?

(a) 4.274 MeV

(b) $1\text{.}\text{927}×{\text{10}}^{-5}$

(c) Since U-238 is a slowly decaying substance, only a very small number of nuclei decay on human timescales; therefore, although those nuclei that decay lose a noticeable fraction of their mass, the change in the total mass of the sample is not detectable for a macroscopic sample.

(a) Write the complete reaction equation for electron capture by ${}^{7}\text{Be.}$

(b) Calculate the energy released.

(a) Write the complete reaction equation for electron capture by ${}^{\text{15}}\text{O}$ .

(b) Calculate the energy released.

(a) ${}_{8}^{\text{15}}{O}_{7}+{e}^{-}\to {}_{7}^{\text{15}}{N}_{8}+{\nu }_{e}$

(b) 2.754 MeV

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