What Fraction Of A 15o Sample Decays In 10 Min

What Fraction of a 150 Sample Decays in 10 Minutes? Understanding Radioactive Decay

Radioactive decay is a fascinating and crucial concept in nuclear physics with wide-ranging applications, from medical imaging to geological dating. Understanding the rate at which a radioactive sample decays is essential in various fields. This article delves into the mathematics behind radioactive decay, specifically addressing the question: what fraction of a 150-sample decays in 10 minutes? We will explore the concepts of half-life, decay constant, and exponential decay to arrive at a comprehensive understanding.

Understanding Radioactive Decay: A Fundamental Overview

Radioactive decay is the spontaneous breakdown of an unstable atomic nucleus, resulting in the emission of particles or energy. This process transforms the original atom into a different nuclide, often a more stable one. The rate of decay is not uniform; it follows an exponential pattern, meaning that a larger number of atoms decay in the beginning, and the rate gradually slows down over time.

Several types of radioactive decay exist, each characterized by the type of particle emitted:

• Alpha decay: Emission of an alpha particle (two protons and two neutrons).
• Beta decay: Emission of a beta particle (an electron or positron).
• Gamma decay: Emission of a gamma ray (high-energy photon).

Regardless of the type of decay, the fundamental principle governing the process remains the same: exponential decay.

The Half-Life: A Crucial Parameter

The half-life (t_1/2) of a radioactive isotope is the time it takes for half of the original sample to decay. This is a crucial parameter in characterizing the decay rate. Each radioactive isotope possesses a unique and constant half-life, ranging from fractions of a second to billions of years.

Understanding the Significance: The half-life doesn't mean that after two half-lives, all the sample will have decayed. After one half-life, half remains. After two half-lives, a quarter remains (half of the half). After three half-lives, an eighth remains, and so on. This illustrates the exponential nature of the decay.

The Decay Constant: Quantifying the Decay Rate

The decay constant (λ), often represented by the Greek letter lambda, quantifies the probability of a single nucleus decaying per unit time. It's inversely proportional to the half-life. The relationship is given by:

λ = ln(2) / t_1/2

where:

• λ is the decay constant (in s^-1, min^-1, etc.)
• ln(2) is the natural logarithm of 2 (approximately 0.693)
• t_1/2 is the half-life

The Exponential Decay Equation: Predicting the Remaining Sample

The number of radioactive atoms (N) remaining after a time (t) can be calculated using the following exponential decay equation:

N(t) = N₀ * e^-λt

where:

• N(t) is the number of radioactive atoms remaining after time t.
• N₀ is the initial number of radioactive atoms.
• e is the base of the natural logarithm (approximately 2.718).
• λ is the decay constant.
• t is the elapsed time.

Calculating the Fraction Decayed: Putting it All Together

To determine the fraction of a 150-sample that decays in 10 minutes, we need the half-life of the radioactive isotope. Let's assume, for the sake of this example, that the half-life (t_1/2) is 5 minutes. Now we can proceed step-by-step:

1. Calculate the decay constant (λ):

   λ = ln(2) / t_1/2 = 0.693 / 5 min ≈ 0.1386 min^-1

2. Use the exponential decay equation:

   We want to find N(t), the number of atoms remaining after 10 minutes. N₀ is 150.

   N(10) = 150 * e^{(-0.1386 * 10)} ≈ 150 * e^-1.386 ≈ 150 * 0.25 ≈ 37.5

3. Calculate the fraction decayed:

   The number of atoms that decayed is 150 - 37.5 = 112.5.

   The fraction decayed is 112.5 / 150 = 0.75 or 75%

Therefore, with a half-life of 5 minutes, 75% of the 150-sample would decay in 10 minutes.

Impact of Different Half-Lives: A Comparative Analysis

The half-life significantly influences the fraction of the sample that decays within a given time. Let’s consider different half-lives and observe their effect:

• Half-life of 1 minute: After 10 minutes, significantly more than 99% of the sample will decay. The exponential decay equation would yield a very small remaining amount.

• Half-life of 10 minutes: After 10 minutes, exactly half (50%) of the sample will have decayed. This is because 10 minutes represents exactly one half-life.

• Half-life of 20 minutes: After 10 minutes, less than half the sample will have decayed. The exact fraction can be calculated using the exponential decay equation with the new decay constant.

Applications of Radioactive Decay: Real-World Examples

The principles of radioactive decay have numerous applications across diverse fields:

• Medical Imaging: Techniques like PET (Positron Emission Tomography) and SPECT (Single-Photon Emission Computed Tomography) utilize radioactive tracers to visualize internal organs and detect abnormalities.

• Carbon Dating: Radiocarbon dating, using the decay of carbon-14, is a crucial tool in archaeology and paleontology to determine the age of organic materials.

• Nuclear Medicine: Radioactive isotopes are used in radiation therapy to treat cancer and other diseases.

• Industrial Gauging: Radioactive sources are used in various industrial applications for thickness measurement, level sensing, and density control.

Conclusion: A Deeper Understanding of Decay Processes

This article explored the concept of radioactive decay, its mathematical representation, and its applications. We learned how to calculate the fraction of a sample that decays over a specific time, highlighting the importance of the half-life and decay constant. Understanding these concepts is crucial for anyone working in nuclear science, medicine, archaeology, or related fields. The example calculation provided demonstrates the procedure, emphasizing the impact of the half-life on the decay rate. Remember that the specific fraction decaying will always depend on the half-life of the isotope in question. Further research into specific isotopes and their decay properties will enhance one's comprehension of this fascinating and vital area of science. The exponential nature of radioactive decay ensures that precise predictions can be made, offering valuable insights into a wide range of processes and applications.