Imagine a radioactive substance with a half-life of 8 hours. If you start with 160 grams of this substance, how much will remain after 24 hours?
Understand the Problem
The question is asking how much of a radioactive substance will remain after a given period of time, based on its half-life. It requires us to compute the decay of the substance over 24 hours, using the half-life of 8 hours to find the remaining quantity. We will solve it by calculating the number of half-lives that fit into 24 hours and then using that to determine the remaining mass.
Answer
The remaining quantity is $\frac{N_0}{8}$.
Answer for screen readers
The remaining quantity of the radioactive substance after 24 hours will be $\frac{N_0}{8}$, where $N_0$ is the initial amount.
Steps to Solve
- Determine the Number of Half-Lives
First, we find out how many half-lives fit into the total time of 24 hours. The half-life of the substance is 8 hours.
To find the number of half-lives, we calculate: $$ \text{Number of Half-Lives} = \frac{\text{Total Time}}{\text{Half-Life}} = \frac{24 \text{ hours}}{8 \text{ hours}} = 3 $$
- Calculate the Remaining Quantity
Next, we use the formula for exponential decay based on the number of half-lives. If the initial amount of the substance is $N_0$, the remaining quantity $N$ after $n$ half-lives is given by: $$ N = N_0 \left(\frac{1}{2}\right)^n $$
In this case, $n = 3$: $$ N = N_0 \left(\frac{1}{2}\right)^3 = N_0 \left(\frac{1}{8}\right) $$
- Final Expression
Hence, the remaining quantity after 24 hours will be: $$ N = \frac{N_0}{8} $$
The remaining quantity of the radioactive substance after 24 hours will be $\frac{N_0}{8}$, where $N_0$ is the initial amount.
More Information
The concept of half-life is fundamental in nuclear physics and radiochemistry. It is the time taken for a quantity to reduce to half its initial value. This means that every 8 hours (in this case), half of the remaining substance decays.
Tips
- Not calculating the number of half-lives correctly: Ensure you divide the total period by the half-life accurately.
- Confusing remaining quantity formula: Remember that each half-life reduces the remaining quantity by half.
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