You have 4,592 grams of a radioactive kind of scandium. How much will be left after 168 days if its half-life is 84 days?
2,296 grams
To calculate the amount of radioactive material remaining after a certain amount of time, we can use the concept of half-life. The half-life is the time it takes for half of the radioactive material to decay.
In this case, the half-life of the radioactive scandium is given as 84 days. This means that after 84 days, half of the initial amount will have decayed, leaving us with half of the original quantity.
To determine how much will be left after 168 days, we need to calculate how many half-lives have passed in that time. Since each half-life is 84 days, dividing 168 by 84 gives us 2. This means that 2 half-lives have passed.
Now, for each half-life, the quantity of radioactive scandium is halved. After the first half-life, we will have half of the initial amount remaining, which is 4,592 / 2 = 2,296 grams.
After the second half-life, we will have half of the remaining amount from the first half-life. Starting with 2,296 grams, halving it gives us 2,296 / 2 = 1,148 grams.
Therefore, after 168 days, there will be approximately 1,148 grams of radioactive scandium remaining.
To calculate the amount of radioactive scandium remaining after 168 days, we can use the formula:
N = N₀ * (1/2)^(t / T₁/₂)
Where:
- N is the final amount remaining,
- N₀ is the initial amount,
- t is the time that has passed,
- T₁/₂ is the half-life of the radioactive substance.
In this case, N₀ = 4,592 grams, t = 168 days, and T₁/₂ = 84 days.
Let's calculate it step-by-step:
Step 1: Calculate the number of half-lives that have passed.
n = t / T₁/₂
n = 168 days / 84 days
n = 2
Step 2: Calculate the amount remaining.
N = N₀ * (1/2)^n
N = 4,592 grams * (1/2)^2
N = 4,592 grams * (1/4)
N = 1,148 grams
Therefore, after 168 days, there will be 1,148 grams of radioactive scandium left.