Determining the Half-Life of a Radioactive Isotope: A Step-by-Step Guide
Introduction
Radioactive isotopes decay over time, and the half-life of an isotope is the time it takes for half of the initial quantity of the isotope to decay. This process is crucial in various fields, including nuclear physics, medicine, and environmental science. In this article, we will walk through a step-by-step guide to determine the half-life of a radioactive isotope, given that after 60 days, 10 grams of a 42-gram sample remains.
Understanding the Problem
Given:
Initial mass (Original amount): 42 grams
Final mass (Remaining amount): 10 grams
Time elapsed (Decay time): 60 days
The task is to determine the half-life of the isotope in days.
Solving the Problem Using Different Methods
There are multiple ways to solve this problem, including the use of natural logarithms and algebraic methods. Let's explore these methods in detail.
Method 1: Using Natural Logarithms
The first decay equation commonly used is:
Af Ao e^(-λt)
Where:
Af Final activity (42 grams)
Ao Initial activity (10 grams)
λ Decay constant
t Decay time (60 days)
Since we are dealing with mass, we can set the equation to:
10 42 e^(-λ * 60)
Let's solve for λ using natural logarithms:
Divide both sides by 42:
10/42 e^(-λ * 60)
Take the natural logarithm of both sides:
ln(10/42) -λ * 60
Solve for λ:
λ -ln(10/42) / 60
Calculate the value:
λ ≈ 0.02457 per day
Find the half-life using the relationship half-life ln(2) / λ:
Half-life ln(2) / 0.02457 ≈ 28.98 days
Round to the nearest whole number:
Half-life ≈ 29 days
Method 2: Direct Calculation
Another approach is to use the following equation:
10 42 * (1/2)^(60/h)
Divide both sides by 42:
10/42 1/2^(60/h)
Take the logarithm of both sides:
log(10/42) -60/h * log(2)
Solve for h:
h -60 * log(2) / log(10/42)
Calculate the value:
h ≈ 28.98 days
Round to the nearest whole number:
h ≈ 29 days
Verification and Simplification
To verify that the half-life is close to 29 days, let's perform a simple check:
After the first half-life (29 days), 21 grams remain (42/2 21).
After the second half-life, 10.5 grams remain (21/2 10.5).
This matches our original problem where 10 grams remain after 60 days, confirming a close approximation to 29 days.
Alternative Method Using Exponential Decay Formula
Another way to approach this problem is to use the exponential decay formula:
N N_0 * (1/2)^(t/h)
Where:
N Remaining mass (10 grams)
N_0 Initial mass (42 grams)
t Time elapsed (60 days)
h Half-life
Solving for h:
Divide both sides by N_0 (42 grams):
10/42 (1/2)^(60/h)
Take the natural logarithm of both sides:
ln(10/42) 60/h * ln(1/2)
Solve for h:
h 60 * ln(2) / ln(10/42)
Calculate the value:
h ≈ 28.98 days
Round to the nearest whole number:
h ≈ 29 days
Conclusion
In conclusion, the half-life of the radioactive isotope, given that 10 grams remain from an original 42-gram sample after 60 days, is approximately 29 days. This can be calculated using various methods, including natural logarithms, direct calculation, and exponential decay formulas. Understanding the half-life of radioactive isotopes is crucial for various applications, from radiation therapy in medicine to carbon dating in archaeology.