Question

The naturally occurring radioactive decay series that begins with ${}_{92}^{235}\mathrm{U}$ stops with formation of the stable ${}_{82}^{207}\mathrm{Pb}$ nucleus. The decays proceed through a series of alpha-particle and beta-particle emissions. How many of each type of emission are involved in this series?

Short answer

There are 7 alpha decays and 4 beta decays involved in the radioactive decay series from ${}^{235}\mathrm{U}$ to ${}^{207}\mathrm{Pb}$.

Step-by-step solution

List down the initial and final isotopes

The initial isotope is ${}^{235}\mathrm{U}$ with 92 protons and 235 nucleons. The final stable isotope is ${}^{207}\mathrm{Pb}$ with 82 protons and 207 nucleons.

Write down the alpha and beta decay process

In an alpha decay, a nucleus loses 2 protons and 2 neutrons, represented by an alpha particle, which is an ${}^4\mathrm{He}$ nucleus: $${}_Z^{A}X{\to }_{Z-2}^{A-4}Y{+}_2^4\mathrm{He}\$Inabetadecay,aneutroninthenucleusisconvertedintoaproton,whileemittingabetaparticle,whichisanelectron:$$_{Z}^{A}X \rightarrow _{Z+1}^{A}Y + _{-1}^0\mathrm{e}$

Set up a system of linear equations

Let x be the number of alpha decays and y be the number of beta decays. After x alpha decays, the number of protons decreases by 2x and the number of nucleons decreases by 4x. After y beta decays, the number of protons increases by y and the number of nucleons remains the same. Now, we can set up a system of equations: 1. Protons: $92-2x+y=82$ 2. Nucleons: $235-4x=207$

Solve the system of linear equations

First, let's solve Equation 2 for x: $4x=235-207$ $x=(235-207)/4=7$ Now, substitute x in Equation 1: $92-2(7)+y=82$ $92-14+y=82$ $y=82-(92-14)=4$ So, there are 7 alpha decays and 4 beta decays involved in this decay series.

Key concepts

Alpha Decay

Alpha decay is a type of radioactive decay where an unstable nucleus releases energy by emitting an alpha particle. An alpha particle consists of two protons and two neutrons, similar to a helium nucleus. This process leads to a decrease in the atomic number of the nucleus by 2, and the mass number by 4.

The equation for alpha decay can be written as:

• Before decay: ${}_Z^{A}X$
• After decay: ${}_{Z-2}^{A-4}Y+{}_2^4\mathrm{He}$

In a radioactive decay series, like the one involving uranium-235, alpha decay is a key process that gradually transforms the parent isotope into a series of other elements until a stable nucleus is reached.

Understanding alpha decay is crucial because it helps to determine the isotopic changes in nuclear reactions and also has important practical applications such as in smoke detectors and the treatment of cancer.

Beta Decay

Beta decay is another process through which unstable nuclei achieve stability. Unlike alpha decay, beta decay involves the transformation of a neutron into a proton or vice versa, accompanied by the emission of a beta particle, which is an electron or a positron.

In beta-minus decay, a neutron in the nucleus is converted into a proton, and an electron (also known as a beta particle) is released. The reaction for beta-minus decay looks like this:

• Before decay: ${}_Z^{A}X$
• After decay: ${}_{Z+1}^{A}Y+{}_{-1}^0\mathrm{e}$

This type of radioactive decay does not change the mass number but increases the atomic number by 1. Beta decay is crucial in the uranium-235 decay series as it contributes to the adjustment of the neutron-to-proton ratio, effectively progressing towards a stable isotope.

Understanding beta decay is important in fields like medical imaging and radiation therapy, where it is utilized for its penetrating ability.

Uranium-235 Decay

Uranium-235 decay is a complex series of transformations that begins with the radioactive isotope uranium-235. This process involves multiple alpha and beta decays that ultimately lead to the formation of a stable isotope, lead-207.

The series starts with the unstable uranium-235 nucleus, which undergoes several emissions:

• 7 alpha decays
• 4 beta decays

Each decay step alters the number of protons and neutrons in the nucleus, bringing it closer to a stable configuration. The balance between these two types of decay is essential for transitioning from the radioactive parent isotope to a stable daughter isotope.

The uranium-235 decay series is significant in both scientific research and practical applications, including radiometric dating which helps in determining the age of geological formations.

Stable Lead-207

Stable lead-207 is the endpoint of the uranium-235 decay series. After passing through a series of alpha and beta decays, uranium-235 finally becomes lead-207, which is stable and does not undergo further radioactive transformations.

Being stable means that lead-207 does not naturally emit radiation, as its nucleus is at an energy level that does not favor decay. This stability is achieved after a carefully balanced sequence of 7 alpha decays and 4 beta decays, which fine-tune the proton and neutron count to a stable configuration.

Understanding the formation of stable lead-207 is essential in fields like archeology and geology, where it serves as a reliable marker in dating rock samples, contributing to our understanding of Earth's history and the timing of events.