GCSE Physics Tutorial - Calculating Net Decline Ratio in Radioactive Emission
In this tutorial, we will learn how to calculate the net decline ratio in a radioactive emission after a given number of half-lives. When a radioactive substance undergoes decay, its population of radioactive nuclei decreases over time. The net decline ratio provides a way to express the reduction in the number of radioactive nuclei as a ratio compared to the initial number. Understanding this concept is essential in radiometric dating and studying the behaviour of radioactive materials. Let's explore how to calculate the net decline ratio.
Steps to Calculate Net Decline Ratio:
Gather Given Information: Collect all the relevant information provided in the problem. This includes the initial number of radioactive nuclei ($N_0$), the remaining number of radioactive nuclei (N), and the number of half-lives that have elapsed (n).
Calculate the Fraction of Remaining Nuclei: The fraction of remaining nuclei ( $ \frac{N}{N_0} $ ) is obtained by dividing the number of remaining nuclei (N) by the initial number of nuclei ($N_0$).
Calculate the Net Decline Ratio: The net decline ratio expresses the reduction in the number of radioactive nuclei as a ratio to the initial number. It can be calculated using the formula: $ [ \text{Net Decline Ratio} = \frac{N}{N_0} = \left( \frac{1}{2} \right)^n ] $
Convert the Ratio to Percentage (Optional): If required, the net decline ratio can be expressed as a percentage by multiplying it by 100. This step helps in better understanding the magnitude of the decline.
Example: Let's work through an example to calculate the net decline ratio in a radioactive emission after a given number of half-lives:
Suppose an initial sample contains 6400 radioactive nuclei. After 4 half-lives, the number of remaining nuclei is 400.
Given information:
$N_0$ (Initial number of nuclei) = 6400
N (Remaining number of nuclei) = 400
n (Number of half-lives elapsed) = 4
Step 1: Calculate the Fraction of Remaining Nuclei: $[ \frac{N}{N_0} = \frac{400}{6400} = 0.0625 ]$
Step 2: Calculate the Net Decline Ratio: $[ \text{Net Decline Ratio} = \left( \frac{1}{2} \right)^4 = 0.0625 ]$
Step 3: Convert the Ratio to Percentage (Optional): The net decline ratio is 0.0625, which can be expressed as 6.25% (0.0625 x 100%).
In this tutorial, we have learned how to calculate the net decline ratio in a radioactive emission after a given number of half-lives. By determining the fraction of remaining nuclei and applying the concept of halving during each half-life, we can express the net decline as a ratio compared to the initial number. The net decline ratio provides valuable information about the reduction in radioactive nuclei over time and is an essential tool in radiometric dating and studying radioactive materials.
To calculate the net decline ratio (NDR) using the half-life of a radioactive substance, you can use the formula:
$ \text{NDR} = \left( \frac{1}{2} \right) ^{ \frac{t}{T_{ \frac{1}{2}}}}$
Where:
$ \text{t}$ = Time elapsed
$ \text{T}_{ \frac{1}{2}}$ = Half-life of the radioactive substance
For example, let's say the half-life of a radioactive substance is 5 years, and 10 years have passed. To find the net decline ratio:
$ \text{NDR} = \left( \frac{1}{2} \right)^{ \frac{10}{5}} = \left( \frac{1}{2} \right)^2 = \frac{1}{4}$
So, the net decline ratio is $ \frac{1}{4}$ or 0.25.
This means that after 10 years, only 25% of the original radioactive substance remains.
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