A scientist is analyzing a sample that decays at a rate of 12% per year. If the initial mass is 200 grams, what will be the mass after 5 years? - Coaching Toolbox

Understanding the Context

Introduction

Radioactive decay is a fundamental concept in nuclear physics and environmental science, describing how unstable atomic nuclei lose energy by emitting radiation over time. For scientists tracking decay, precise calculations are essential—especially when measuring how much a sample remains after years of slow decay. One common decay scenario involves a substance decaying at a steady rate, such as 12% per year. But what does this mean mathematically, and how do researchers compute the mass after a specific period?

In this article, we examine a real-world scientific scenario: a scientist analyzing a 200-gram sample decays at 12% per year and determine its mass after 5 years using exponential decay models.

The Science Behind Radioactive Decay

Radioactive decay follows an exponential decay model rather than linear. The decay rate, expressed as a percentage per time unit (annually here), directly affects the formula:

Key Insights

Remaining mass = Initial mass × (1 – decay rate)^t

Where:

• Initial mass = 200 grams
  
• Decay rate = 12% = 0.12 per year
  
• Time t = 5 years

This formula captures how each year, only a fraction—denoted by (1 – 0.12) = 0.88—of the remaining mass survives over time.

Calculating the Remaining Mass After 5 Years

Applying the formula step-by-step:

Final Thoughts

1. Decay factor per year: 1 – 0.12 = 0.88
   
2. Number of years: 5
   
3. Compute the decay exponent: (0.88)^5 ≈ 0.5277 (using calculator or logarithmic tables)
   
4. Remaining mass = 200 grams × 0.5277 ≈ 105.54 grams

Thus, after 5 years, the mass of the sample decays to approximately 105.54 grams.

Why This Matters in Science

Understanding decay rates is critical in fields like nuclear medicine, environmental monitoring, and archaeology. For instance:

• Medical isotopes decay at known rates to safely diagnose or treat conditions.
  
• Environmental scientists track pollutants or radioactive waste to assess long-term risks.
  
• Archaeologists use decay models (like carbon-14 dating, though significantly slower) to date ancient organic materials.

Knowing the exact mass after years allows precise experimental design and data interpretation, ensuring safety and accuracy.