Formula Guide: Calculating the Remaining Amount of a Radioactive Isotope After a Certain Time

Understanding Radioactive Decay ☢️

Radioactive decay is the process by which an unstable atomic nucleus loses energy by emitting radiation. A key concept in understanding radioactive decay is the half-life, which is the time it takes for half of the radioactive material to decay. To calculate the remaining amount of a radioactive isotope after a certain time, we use the following formula:

$$N(t) = N_0 * e^{-λt}$$

Where:

• $N(t)$ is the amount of the substance remaining after time $t$.
• $N_0$ is the initial amount of the substance.
• $λ$ (lambda) is the decay constant.
• $t$ is the time elapsed.
• $e$ is the base of the natural logarithm (approximately 2.71828).

Calculating the Decay Constant ⏳

The decay constant ($λ$) is related to the half-life ($t_{1/2}$) by the following equation:

$$λ = \frac{ln(2)}{t_{1/2}} ≈ \frac{0.693}{t_{1/2}}$$

Where:

• $ln(2)$ is the natural logarithm of 2 (approximately 0.693).
• $t_{1/2}$ is the half-life of the radioactive isotope.

Steps to Calculate Remaining Amount 📝

1. Determine the initial amount ($N_0$) of the radioactive isotope.
2. Find the half-life ($t_{1/2}$) of the isotope.
3. Calculate the decay constant ($λ$) using the formula: $λ = \frac{0.693}{t_{1/2}}$.
4. Determine the time elapsed ($t$).
5. Calculate the remaining amount ($N(t)$) using the formula: $N(t) = N_0 * e^{-λt}$.

Example Calculation ➗

Let's say we have 100 grams of a radioactive isotope with a half-life of 5 years. How much will remain after 10 years?

1. $N_0 = 100$ grams
2. $t_{1/2} = 5$ years
3. $λ = \frac{0.693}{5} ≈ 0.1386$ per year
4. $t = 10$ years
5. $N(10) = 100 * e^{-0.1386 * 10}$
6. $N(10) = 100 * e^{-1.386} ≈ 100 * 0.25 = 25$ grams

Therefore, after 10 years, approximately 25 grams of the radioactive isotope will remain.

Code Example (Python) 💻

Here's a Python code snippet to calculate the remaining amount:

import math

def remaining_amount(initial_amount, half_life, time):
    decay_constant = 0.693 / half_life
    remaining = initial_amount * math.exp(-decay_constant * time)
    return remaining

# Example usage:
initial_amount = 100  # grams
half_life = 5         # years
time = 10             # years

remaining = remaining_amount(initial_amount, half_life, time)
print(f"Remaining amount after {time} years: {remaining:.2f} grams")

Important Considerations ⚠️

• Ensure that the units of time for the half-life and the elapsed time are the same.
• This formula assumes first-order kinetics, which is generally valid for radioactive decay.