Problem 46 - Physics for Scientists and Engineers with Modern Physics

Question

(II) In a series of decays, the nuclide \({ }_{92}^{235} \mathrm{U}\) becomes \({ }_{82}^{207} \mathrm{~Pb}\). How many \(\alpha\) and \(\beta^{-}\) particles are emitted in this series?

Step-by-Step Solution

Answer

7 alpha and 4 beta particles are emitted.

Step 1: Understand the Decay Rules

An alpha (\(\alpha\)) particle emission decreases the atomic number by 2 and the mass number by 4 (\(_{2}^{4}He\)). A beta-minus (\(\beta^{-}\)) particle emission increases the atomic number by 1 without changing the mass number (as a neutron is converted into a proton).

Step 2: Calculate Change in Mass Number

Determine the change in the mass number. The original nuclide has a mass number of 235 and the final nuclide has a mass number of 207. Thus, the change (\(\Delta A\)) is: \[235 - 207 = 28\]. As each \(\alpha\) particle emission decreases the mass number by 4, 28 units decrease corresponds to \(\frac{28}{4} = 7\) \(\alpha\) particles.

Step 3: Calculate Change in Atomic Number

Determine the change in the atomic number. The original nuclide has an atomic number of 92 and the final nuclide has an atomic number of 82. Thus, the change (\(\Delta Z\)) is: \[92 - 82 = 10\]. Each \(\alpha\) particle decreases the atomic number by 2, so 7 \(\alpha\) particles will change the atomic number by \(7 \times 2 = 14\).

Step 4: Account for Excess Change in Atomic Number

The required change in atomic number is only 10, but the 7 \(\alpha\) particles change it by 14. The remaining difference must be adjusted using \(\beta^{-}\) emissions, which increase the atomic number. Therefore, the number of \(\beta^{-}\) particles needed is: \[14 - 10 = 4\]

Key Concepts

Alpha Decay

Alpha decay is a type of radioactive decay where an unstable nucleus releases an alpha (). particle. An alpha particle is similar to a helium-4 nucleus, consisting of 2 protons and 2 neutrons, symbolized as \(_{2}^{4}He\).This decay process decreases both the atomic number and the mass number of the original nucleus. Specifically:

• The atomic number decreases by 2 because 2 protons are lost.
• The mass number decreases by 4 due to the loss of 2 protons and 2 neutrons.

Let's use an example: Consider an atom with an atomic number of 92 and a mass number of 235 undergoing alpha decay. The emitted alpha particle lowers the atomic number to 90 and the mass number to 231.Alpha decay is common among heavy elements with more protons than neutrons. It helps make the nucleus more stable by losing excess nuclear mass.

Beta Decay

In radioactive decay, beta decay plays a significant role. It occurs when a neutron in the nucleus transforms into a proton while releasing a beta-minus ( ). particle and an antineutrino. Here's how beta decay influences atomic structure:

• Increases the atomic number by 1, since a neutron becomes a proton.
• Keeps the mass number unchanged, as the total count of protons and neutrons remains the same.

For instance, if a nuclide has an atomic number of 82 and undergoes beta decay, the atomic number will become 83, but the mass number won't change. This decay process aids in stabilizing the nucleus by altering the neutron-to-proton ratio, which can be crucial in both naturally occurring and artificially created radioactive materials.

Atomic Number

The atomic number is a fundamental property of an element, representing the number of protons in the nucleus. It determines the element's identity on the periodic table and influences chemical properties. In radioactive decay processes, changes to the atomic number are critical:

• Alpha decay decreases the atomic number by 2, moving the element back one or more places on the periodic table.
• Beta decay increases the atomic number by 1, moving the element forward one place on the periodic table.

For example, uranium-235 starts with an atomic number of 92. Through a series of decays, its atomic number can eventually reduce to 82, turning it into lead. This transformation is part of radioactive decay chains crucial in nuclear physics and energy production. Understanding atomic numbers helps us predict chemical behavior and allows scientists to identify and categorize elements effectively.