# Half-Life of Isotope Calculation Explained Joyfully

## How can we determine the half-life of an isotope that decays to 3.125% of its original activity in 50.4 hours?

Let's explore the fun process of calculating the half-life of an isotope in an exciting way!

## Calculating the Half-Life of the Isotope

Calculating the half-life of an isotope can be an engaging task, especially when we have specific data to work with. In this scenario, where the isotope decays to 3.125% of its original activity in 50.4 hours, we can follow a systematic approach to find the half-life.

Here's a cheerful breakdown of the steps to determine the half-life:

## 1. Calculate the Ratio of Final Activity to Initial Activity

To start our calculation journey, we first need to find the ratio of the final activity to the initial activity of the isotope. In this case, the final activity ratio is 3.125% or 0.03125.

## 2. Apply Exponential Decay Equation

Next, we can utilize the equation for exponential decay to establish a relationship between the final activity ratio and the half-life of the isotope. The equation states that the final activity ratio is equal to 0.5 raised to the power of (time elapsed / half-life).

## 3. Solve for the Half-Life

After setting up the exponential decay equation, we can proceed to solve for the half-life by equating the final activity ratio to the expression involving 0.5 raised to a certain power. By solving this equation, we will uncover the magical half-life of the isotope.

## Dive Deeper into the Calculation Adventure

Let's embark on an even more thrilling part of the calculation journey by exploring the detailed explanation and mathematical manipulation involved in finding the half-life of the isotope.

## The Fascinating Exploration of Half-Life Calculation

When it comes to determining the half-life of an isotope, the process can be both educational and enjoyable. By following the steps outlined above, we can uncover the mystery behind how long it takes for the isotope to decay to half of its original activity.

## Embracing the Mathematical Joy

To delve deeper into the calculation adventure, we venture into the realm of logarithmic functions to isolate the exponent and derive the half-life of the isotope based on the given data. By applying logarithmic properties and rearranging the equation, we arrive at the exhilarating solution that reveals the time duration for the isotope's decay process.

## Unveiling the Half-Life Wonder

In summary, the process of calculating the half-life of an isotope that decays to 3.125% of its original activity in 50.4 hours is a captivating journey filled with mathematical discoveries. By using exponential decay principles and applying logarithmic functions, we can unravel the enigmatic concept of half-life and appreciate the beauty of scientific calculations.

**Join the Half-Life Adventure Now!**