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Chapter 25: Problem 58

The overall change in the radioactive decay of \({238}_{92} \mathrm{U}\)to 206 \(\mathrm{Pb}\) is the emission of eight \(\alpha\) particles. Show that if \(_{82}^{206} \mathrm{Pb}\)this loss of eight \(\alpha\) particles were not also accompanied by six \(\beta^{-}\) emissions, the product nucleus would still be radioactive.

Short Answer

Expert verified
Without the six \(\beta^{-}\) emissions, eight \(\alpha\) decays from \(_{92}^{238} \mathrm{U}\) would result in \(_{76}^{206}\mathrm{Os}\), not \(_{82}^{206}\mathrm{Pb}\). Therefore, the product nucleus would still be radioactive, as osmium-206 would continue to decay until it becomes lead-206.

Step by step solution

01

Understand Alpha Emission

Alpha emission is a type of radioactive decay where an alpha particle is emitted from a nucleus. An alpha particle consists of 2 protons and 2 neutrons, essentially a helium-4 nucleus \(_{2}^{4}\mathrm{He}\). So, when a nucleus goes through an alpha decay process, it loses 2 protons and 2 neutrons - decreasing its atomic number by 2 and atomic mass by 4.
02

Understand Beta Emission

Beta emission is another radioactive decay process. In beta-minus (\(\beta^{-}\)) decay specifically, a neutron in the nucleus changes into a proton and an electron. The electron is ejected as a beta particle, increasing the atomic number by 1 but leaving the atomic mass unchanged.
03

Calculate the Change from Alpha Emissions only

Suppose uranium-238 only goes through alpha decays, emitting eight \(\alpha\) particles. Each alpha decay decreases the atomic number by 2 and the atomic mass by 4. So, after 8 alpha decays, the atomic number would decrease by \(8*2 = 16\) and the atomic mass would decrease by \(8*4 = 32\). The resulting isotope would be \(_{76}^{206}\mathrm{X}\). The atomic number 76 corresponds to the element osmium (Os), so the product isotope would be \(_{76}^{206}\mathrm{Os}\).
04

Conclusion

Since the isotope produced is \(_{76}^{206}\mathrm{Os}\), and not \(_{82}^{206}\mathrm{Pb}\), which was the desired end point from the problem, your product is still radioactive and will continue decaying until it reaches \(_{82}^{206}\mathrm{Pb}\). Therefore, without the six \(\beta^{-}\) emissions, the product nucleus would still be radioactive.

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Most popular questions from this chapter

A process that produces a one-unit increase in atomic number is (a) electron capture; (b) \(\beta^{-}\) emission;(c) \(\alpha\) emission; (d) \(\gamma\) -ray emission.

Write nuclear equations to represent (a) the decay of \(^{214} \mathrm{Ra}\) by \(\alpha\) -particle emission (b) the decay of \(^{205}\) At by positron emission (c) the decay of \(^{212} \mathrm{Fr}\) by electron capture (d) the reaction of two deuterium nuclei (deuterons) to produce a nucleus of \(\frac{3}{2} \mathrm{He}\). (e) the production of \({243}_{97} \mathrm{Bk}\) get by the \(\alpha\) -particle bombardment of\({241}_{95} \mathrm{Am}\) (f) a nuclear reaction in which thorium-232 is bombarded with \(\alpha\) particles, producing a new nuclide and four neutrons.

Just as the uranium series is called the "4n \(+2^{\prime \prime}\) series, the thorium series can be called the "4n" series and the actinium series the "4n \(+3 "\) series. A \(4 n+1 "\) series has also been established, with \(^{241} \mathrm{Pu}\) as the parent nuclide. To which series does each of the following belong: (a) \(\frac{214}{83} \mathrm{Bi} ;\) (b) \(\frac{216}{84} \mathrm{Po} ;\) (c) \(\frac{215}{85} \mathrm{At}\) (d) \(\frac{235}{92} \mathrm{U} ?\)

The conversion of \(\mathrm{CO}_{2}\) into carbohydrates by plants via photosynthesis can be represented by the reaction $$6 \mathrm{CO}_{2}(\mathrm{g})+6 \mathrm{H}_{2} \mathrm{O} \stackrel{\text { light }}{\longrightarrow} \mathrm{C}_{6} \mathrm{H}_{12} \mathrm{O}_{6}+6 \mathrm{O}_{2}(\mathrm{g}).$$ To study the mechanism of photosynthesis, algae were grown in water containing \(^{18}\) O, that is, \(\mathrm{H}_{2}^{18} \mathrm{O}\) The oxygen evolved contained oxygen-18 in the same ratio to the other oxygen isotopes as the water in which the reaction was carried out. In another experiment, algae were grown in water containing only \(^{16} \mathrm{O}\),but with oxygen-18 present in the \(\mathrm{CO}_{2}\). The oxygen evolved in this experiment contained no oxygen-18. What conclusion can you draw about the mechanism of photosynthesis from these experiments?

Supply the missing information in each of the following nuclear equations representing a radioactive decay process.(a) \(160_?\mathrm{W} \longrightarrow\\{\mathrm{Hf}+?\) (b) \(38_? \mathrm{Cl} \longrightarrow_{?}^{?} \mathrm{Ar}+?\) (c) \(^{214} ? \longrightarrow_{?}^{?} \mathrm{Po}+_{-1}^{0} \boldsymbol{\beta}\) (d) \(_{17}^{32} \mathrm{Cl} \longrightarrow_{1}^{?} ?+?\)
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