Identify The Unknown Isotope X In The Following Decays

Understanding radioactive decay is crucial in nuclear physics, allowing us to identify unknown isotopes through their decay patterns. This article will explore how to identify an unknown isotope X in various decay scenarios. We will cover the fundamental principles of radioactive decay, different types of decay, and step-by-step methods to determine the identity of X.

Radioactive Decay: The Basics

Radioactive decay is the process by which an unstable atomic nucleus loses energy by emitting radiation. This process transforms one nuclide (a specific type of atom characterized by its number of protons and neutrons) into another. The key principles to remember include:

• Conservation Laws: In any nuclear reaction, including radioactive decay, several quantities are conserved:

  • Charge: The total electric charge remains constant.
  • Number of Nucleons: The total number of protons and neutrons (nucleons) is conserved.
  • Energy: The total energy, including mass energy (E = mc²), is conserved.

• Decay Constant (λ): Each radioactive isotope has a specific decay constant, which is the probability of decay per unit time.

• Half-Life (T₁/₂): The half-life is the time required for half of the radioactive nuclei in a sample to decay. It is related to the decay constant by the formula:

  T₁/₂ = ln(2) / λ

Types of Radioactive Decay

Several types of radioactive decay exist, each characterized by the particles emitted and the changes in the nucleus. Understanding these types is essential for identifying unknown isotopes.

Alpha Decay (α)

Alpha decay occurs when a nucleus emits an alpha particle, which consists of two protons and two neutrons (equivalent to a helium nucleus, ⁴₂He). The general form of alpha decay is:

• XA𝑍 → YA-4𝑍₋₂ + ⁴₂He

Where:

• X is the parent nucleus.
• A is the mass number (number of protons + neutrons).
• Z is the atomic number (number of protons).
• Y is the daughter nucleus.

Characteristics of Alpha Decay:

• The mass number decreases by 4.
• The atomic number decreases by 2.
• Alpha particles have relatively low penetration power and can be stopped by a sheet of paper.

Beta Decay (β)

Beta decay involves the emission of a beta particle, which can be either an electron (β⁻) or a positron (β⁺).

Beta-Minus Decay (β⁻)

In beta-minus decay, a neutron in the nucleus is converted into a proton, and an electron (β⁻) and an antineutrino (ν̄ₑ) are emitted. The general form is:

• XA𝑍 → YA𝑍₊₁ + e⁻ + ν̄ₑ

Characteristics of Beta-Minus Decay:

• The mass number remains the same.
• The atomic number increases by 1.
• Beta-minus particles have greater penetration power than alpha particles but can be stopped by a thin sheet of aluminum.

Beta-Plus Decay (β⁺)

In beta-plus decay, a proton in the nucleus is converted into a neutron, and a positron (β⁺) and a neutrino (νₑ) are emitted. The general form is:

• XA𝑍 → YA𝑍₋₁ + e⁺ + νₑ

Characteristics of Beta-Plus Decay:

• The mass number remains the same.
• The atomic number decreases by 1.
• Positrons, after losing their kinetic energy, annihilate with electrons, producing two gamma-ray photons.

Gamma Decay (γ)

Gamma decay involves the emission of a gamma ray, which is a high-energy photon. Gamma decay typically occurs after alpha or beta decay, when the nucleus is in an excited state. The general form is:

• XA𝑍* → XA𝑍 + γ

Characteristics of Gamma Decay:

• The mass number and atomic number remain the same.
• Gamma rays have high penetration power and require thick shielding, such as lead or concrete, to be stopped.

Electron Capture (EC)

Electron capture is a process where a nucleus absorbs an inner-shell electron, typically from the K or L shell. This process converts a proton into a neutron, and a neutrino is emitted. The general form is:

• XA𝑍 + e⁻ → YA𝑍₋₁ + νₑ

Characteristics of Electron Capture:

• The mass number remains the same.
• The atomic number decreases by 1.
• Electron capture results in the emission of characteristic X-rays as other electrons fill the vacancy created by the captured electron.

Identifying Unknown Isotopes: Step-by-Step

To identify an unknown isotope X in a decay series, follow these steps:

Step 1: Analyze the Decay Equation

Start by examining the given decay equation or series of equations. Identify the known particles and isotopes involved. The general form of a decay equation is:

• X → Y + emitted particles

Where:

• X is the parent nucleus (the unknown isotope).
• Y is the daughter nucleus (which may be known or another unknown).
• Emitted particles can be alpha particles, beta particles, gamma rays, etc.

Step 2: Apply Conservation Laws

Use the conservation laws of charge and nucleon number to determine the mass number (A) and atomic number (Z) of the unknown isotope X.

• Conservation of Mass Number (A): The sum of the mass numbers on the right side of the equation must equal the mass number of X.
• Conservation of Atomic Number (Z): The sum of the atomic numbers on the right side of the equation must equal the atomic number of X.

Step 3: Solve for A and Z

Set up equations based on the conservation laws and solve for A and Z of the unknown isotope X. For example, if the decay is:

• XA𝑍 → ²⁰⁶₈₂Pb + ⁴₂He

Then:

• A = 206 + 4 = 210
• Z = 82 + 2 = 84

So, the unknown isotope X has a mass number of 210 and an atomic number of 84.

Step 4: Identify the Element

Use the atomic number (Z) to identify the element on the periodic table. The atomic number uniquely identifies an element. In the example above, Z = 84 corresponds to Polonium (Po).

Step 5: Write the Isotope Notation

Write the complete isotope notation for the unknown isotope X using the element symbol, mass number, and atomic number:

• ²¹⁰₈₄Po

Step 6: Verify the Decay Mode

Check if the identified decay mode is consistent with the properties of the identified isotope. Consult nuclear data tables or databases to confirm the decay mode and half-life of the isotope.

Examples of Identifying Unknown Isotopes

Let's work through several examples to illustrate the process of identifying unknown isotopes in different decay scenarios.

Example 1: Alpha Decay

Suppose an unknown isotope X undergoes alpha decay to produce Radium-226 (²²⁶₈₈Ra). The decay equation is:

• XA𝑍 → ²²⁶₈₈Ra + ⁴₂He

Applying the conservation laws:

• A = 226 + 4 = 230
• Z = 88 + 2 = 90

The element with atomic number 90 is Thorium (Th). Therefore, the unknown isotope X is ²³⁰₉₀Th.

Example 2: Beta-Minus Decay

An unknown isotope X decays by beta-minus emission to form Protactinium-234 (²³⁴₉₁Pa). The decay equation is:

• XA𝑍 → ²³⁴₉₁Pa + e⁻ + ν̄ₑ

Applying the conservation laws:

• A = 234 + 0 = 234
• Z = 91 - 1 = 90

The element with atomic number 90 is Thorium (Th). Therefore, the unknown isotope X is ²³⁴₉₀Th.

Example 3: Beta-Plus Decay

An unknown isotope X decays by beta-plus emission to form Silicon-26 (²⁶₁₄Si). The decay equation is:

• XA𝑍 → ²⁶₁₄Si + e⁺ + νₑ

Applying the conservation laws:

• A = 26 + 0 = 26
• Z = 14 + 1 = 15

The element with atomic number 15 is Phosphorus (P). Therefore, the unknown isotope X is ²⁶₁₅P.

Example 4: Electron Capture

An unknown isotope X undergoes electron capture to form Vanadium-49 (⁴⁹₂₃V). The decay equation is:

• XA𝑍 + e⁻ → ⁴⁹₂₃V + νₑ

Applying the conservation laws:

• A = 49 + 0 = 49
• Z = 23 + 1 = 24

The element with atomic number 24 is Chromium (Cr). Therefore, the unknown isotope X is ⁴⁹₂₄Cr.

Example 5: Decay Series

Consider a decay series where an unknown isotope X decays to Lead-208 (²⁰⁸₈₂Pb) through the following steps:

1. XA𝑍 → *YA'𝑍' + ⁴₂He
2. *YA'𝑍' → *ZA''*𝑍'' + e⁻ + ν̄ₑ
3. *ZA''*𝑍'' → ²⁰⁸₈₂Pb + ⁴₂He

To identify X, we work backward from the final product.

Step 1: Identify Z and A of ZA''𝑍''

• ²⁰⁸₈₂Pb is formed by alpha decay from *ZA''*𝑍''
• A'' = 208 + 4 = 212
• Z'' = 82 + 2 = 84
• *ZA''*𝑍'' is ²¹²₈₄Po

Step 2: Identify Z and A of YA'𝑍'

• ²¹²₈₄Po is formed by beta-minus decay from *YA'𝑍'
• A' = 212 + 0 = 212
• Z' = 84 - 1 = 83
• *YA'𝑍' is ²¹²₈₃Bi

Step 3: Identify Z and A of XA𝑍

• ²¹²₈₃Bi is formed by alpha decay from XA𝑍*
• A = 212 + 4 = 216
• Z = 83 + 2 = 85
• XA𝑍* is ²¹⁶₈₅At

Therefore, the unknown isotope X is ²¹⁶₈₅At (Astatine-216).

Common Challenges and Solutions

Identifying unknown isotopes can sometimes be challenging. Here are some common issues and their solutions:

• Complex Decay Series: In complex decay series with multiple steps, carefully track each decay and apply the conservation laws at each step. Work backward from the known final product to the unknown parent isotope.
• Isomeric Transitions: Some isotopes can exist in metastable states (isomers) that undergo isomeric transitions (IT), emitting gamma rays without changing A or Z. Account for these transitions in the decay scheme.
• Branching Ratios: Some isotopes can decay through multiple pathways with different branching ratios. Consider all possible decay modes and their probabilities when analyzing the decay.
• Nuclear Data Tables: Always refer to reliable nuclear data tables and databases (e.g., the National Nuclear Data Center at Brookhaven National Laboratory) for accurate information on isotopes, decay modes, half-lives, and branching ratios.

Practical Applications

Identifying unknown isotopes through their decay patterns has numerous practical applications:

• Nuclear Medicine: In nuclear medicine, radioactive isotopes are used for diagnostic imaging and therapeutic treatments. Identifying and characterizing these isotopes is crucial for ensuring patient safety and treatment effectiveness.
• Environmental Monitoring: Radioactive isotopes are used as tracers to study environmental processes and monitor pollution. Identifying unknown radioactive contaminants is essential for assessing environmental risks and implementing remediation strategies.
• Nuclear Forensics: In nuclear forensics, identifying unknown radioactive materials is crucial for tracing the origin of illicit nuclear materials and preventing nuclear terrorism.
• Archaeology and Geology: Radioactive isotopes are used for dating archaeological artifacts and geological samples. Identifying the isotopes and their decay products is essential for accurate dating.
• Nuclear Physics Research: Identifying unknown isotopes is fundamental to nuclear physics research, contributing to our understanding of nuclear structure, reactions, and decay processes.

Conclusion

Identifying unknown isotopes X in radioactive decay requires a solid understanding of the principles of radioactive decay, the different types of decay, and the application of conservation laws. By systematically analyzing the decay equation, applying conservation of mass number and atomic number, and identifying the element using the periodic table, you can determine the identity of the unknown isotope. Remember to verify the decay mode and consult nuclear data tables for accurate information. This knowledge is crucial in various fields, including nuclear medicine, environmental monitoring, nuclear forensics, archaeology, and nuclear physics research.