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Electricity and Magnetism

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Charge Conservation in Nuclear Decay

EANDM-G16AYD

Most nuclei are not stable. They will decay after some period of time. The half-life of a given nucleus, which is a measure of how long it takes for half of a given sample to decay, can range from $\mu$s to hundreds of thousands of years.

Nuclear decay reactions must satisfy a number of conservation laws, including the law of charge conservation. There are four principle modes by which a nucleus can decay; $\beta$ decay, electron capture, $\alpha$ emission, and fission. The three are distinguished by the type of particles the nucleus breaks into.

A $\beta$ decay is simply the decay of a nucleus through the emission of either an electron ($e^-$), which is called “$\beta$-minus decay”, or a positron ($e^+$), which is called “$\beta$-plus decay”. Electron capture, in which a nucleus captures an electron and thereby changes its nuclear identity, is in many ways the inverse of $\beta$ decay, with the exception that one does not observe the capture of positrons, as they do not stably exist in atomic electron clouds.

In $\alpha$ decay, a nucleus emits an $\alpha$ particle, which is the nucleus of a $\rm{^4He}$ atom. The $\alpha$ particle has an electric charge of $+2e$, where:

$$e = 1.602 \times 10^{-19}\text{ C}$$

…is the electronic charge.

In a fission reaction, a nucleus decays to a pair of relatively heavy nuclei. Oftentimes the daughter nuclei in these reactions are also radioactive, so that a long chain of nuclear decays is initiated by the first decay.

Two other particles that sometimes appear in nuclear decay reactions are the neutrino ($\nu$) and the anti-neutrino ($\bar{\nu}$). These are both electrically neutral and do not affect the balance of charge in a reaction.

In the scientific literature, nuclei are generally labeled by the notation:

$$\rm{^AX}$$

…where the symbol $\rm{X}$ denotes what chemical element the reactant is. It may be $\rm{C}$ (Carbon), or $\rm{Mo}$ (Molybdenum), or whatever element appears in the reaction. This identifies the number of protons the nucleus has. The symbol $\rm{A}$ tells you the total number of nucleons, protons plus neutrons, in the element. This tells you which isotope of a particular element the nucleus is. One can see from the periodic table shown below, for example, that since carbon has six protons, the nucleus $\rm ^{14}C$ is an isotope of carbon that has $8 = 14 - 6$ neutrons.

Sandbh. "Periodic Table." Wikipedia. Wikimedia Foundation, n.d. Web. 19 Mar. 2016.

Consider the following five nuclear decay reactions. Three of these reactions are permitted by charge conservation, and two of them are forbidden:

$\rm{^{60}Co} \rightarrow \rm{^{60}Ni} + e^- + \bar{\nu}$

$\rm{^{175}Pt} \rightarrow \rm{^{171}Os} + \alpha$

$\rm{^{128}Cs} + e^- \rightarrow \rm{^{128}Ba } + \nu$

$\rm{^{230}Pa} \rightarrow \rm{^{230}Th }+ e^+ + \nu$

$\rm{^{252}Es} \rightarrow \rm{^{140}I} + \rm{^{108}Ru} + 4n$

Select the TWO reactions that are forbidden because they violate charge conservation.

A

$\rm{^{60}Co} \rightarrow \rm{^{60}Ni} + e^- + \bar{\nu}$

B

$\rm{^{175}Pt} \rightarrow \rm{^{171}Os} + \alpha$

C

$\rm{^{128}Cs} + e^- \rightarrow \rm{^{128}Ba } + \nu$

D

$\rm{^{230}Pa} \rightarrow \rm{^{230}Th }+ e^+ + \nu$

E

$\rm{^{252}Es} \rightarrow \rm{^{140}I} + \rm{^{108}Ru} + 4n$