Problem 13 - University Physics with Modern Physics

Question

What nuclide is produced in the following radioactive decays? (a) \(\alpha\) decay of \(^{239} \mathrm{Pu} ;\) (b) \(\beta^{-}\) decay of \(_{11}^{24} \mathrm{Na} ;(\mathrm{c}) \beta^{+}\) decay of \(_{8}^{15} \mathrm{O}\)

Step-by-Step Solution

Answer

(a) \(^{235}_{92}\mathrm{U}\), (b) \(^{24}_{12}\mathrm{Mg}\), (c) \(^{15}_{7}\mathrm{N}\).

Step 1: Understanding Alpha Decay

In an alpha (\(\alpha\)) decay, the nucleus emits an alpha particle, which consists of 2 protons and 2 neutrons (a helium-4 nucleus). This means the new atom will have an atomic number reduced by 2 and a mass number reduced by 4.

Step 2: Calculating Alpha Decay of Pu-239

For the alpha decay of Pu-239, the atomic number decreases from 94 to 92 (since 2 protons are lost), and the mass number decreases from 239 to 235 (since 4 nucleons in total are lost). Thus, the resulting nuclide is \(^{235}_{92}\mathrm{U}\).

Step 3: Understanding Beta-minus Decay

In a beta-minus (\(\beta^{-}\)) decay, a neutron is converted into a proton while emitting an electron (\(e^{-}\)) and an antineutrino. This results in an increase of the atomic number by 1, while the mass number remains unchanged.

Step 4: Calculating Beta-minus Decay of Na-24

During the \(\beta^{-}\) decay of Na-24, its atomic number increases from 11 to 12 (turning it into magnesium) while the mass number stays the same. The resulting nuclide is \(^{24}_{12}\mathrm{Mg}\).

Step 5: Understanding Beta-plus Decay

In a beta-plus (\(\beta^{+}\)) decay, a proton is converted into a neutron while emitting a positron (\(e^{+}\)) and a neutrino. This results in a decrease of the atomic number by 1, while the mass number remains unchanged.

Step 6: Calculating Beta-plus Decay of O-15

For the \(\beta^{+}\) decay of O-15, the atomic number decreases from 8 to 7 (turning it into nitrogen) while the mass number remains 15. The resulting nuclide is \(^{15}_{7}\mathrm{N}\).

Key Concepts

Alpha Decay

Alpha decay is a type of nuclear decay where the atom's nucleus releases an alpha particle. An alpha particle contains 2 protons and 2 neutrons, similar to a helium-4 nucleus. When an atom undergoes alpha decay, its atomic number decreases by 2 since it loses 2 protons. Additionally, the mass number decreases by 4 because the atom loses 2 neutrons and 2 protons, totaling 4 nucleons.

• Alpha particles are relatively large and can be stopped by a sheet of paper.
• This form of decay is common among heavy elements such as uranium and radium.

For example, in the alpha decay of plutonium-239 ( Pu-239):

• The atomic number reduces from 94 (Pu) to 92, resulting in uranium (U).
• The mass number decreases from 239 to 235.

Thus, the resulting nuclide is uranium-235 ( U-235). This process happens because the nucleus seeks a more stable configuration by releasing excess protons and neutrons.

Beta-Minus Decay

Beta-minus decay is a process that involves the conversion of a neutron into a proton, with the emission of an electron (known as beta particle) and an antineutrino. The exciting part is that the atomic number of the atom increases by one because a new proton is formed in place of the neutron, while the mass number remains unchanged.

• This decay can penetrate further than alpha decay and requires a thin layer of aluminum to stop it.
• Beta-minus decay is commonly observed in nuclides with an excess of neutrons.

During the beta-minus decay of sodium-24 ( Na-24):

• The atomic number increases from 11 to 12, thus converting sodium (Na) into magnesium (Mg).
• The mass number remains the same at 24.

Consequently, the resulting nuclide is magnesium-24 ( Mg-24). This transformation helps balance the ratio of protons to neutrons, thereby stabilizing the atom.

Beta-Plus Decay

Beta-plus decay occurs when a proton is converted into a neutron, emitting a positron (the electron's antimatter counterpart) and a neutrino in the process. This transformation results in the decrease of the atomic number by one, while, similar to beta-minus decay, the mass number doesn't change.

• Beta-plus decay is typical in isotopes with an excess of protons.
• This type of decay is less penetrating than beta-minus decay and is effectively blocked by a plastic or glass container.

Take the example of oxygen-15 ( O-15) undergoing beta-plus decay:

• The atomic number decreases from 8 to 7, transforming oxygen (O) into nitrogen (N).
• The mass number stays constant at 15.

Thus, the end product is nitrogen-15 ( N-15). This decay pathway assists the nucleus in attaining a more stable energy state by adjusting its proton-to-neutron ratio.