Can someone help with this one? Half life of 55Cr is about 2 hours. A sample of the isotopes delivered from the reactor to lab requires 12 hours. What is the minimum amount of material that should be shipped to get 1.0 mg of 55Cr.
I said 32mg but I got it wrong and I don't know why
You can use an equation or you could just do it intuitively; I choose to do this one intuitively.
1.0mg =12 hrs
2.0mg=10 hrs
4.0mg=8 hrs
8.0mg= 6hrs
16.0mg=4hrs
32.0mg=2hr
64.0mg=0hrs
So, you will initially need 64.0mg of 55Cr to have a minimum of 1.0mg after 12-hours.
To find the minimum amount of material that should be shipped to obtain 1.0 mg of 55Cr, we can use the concept of half-life.
The half-life of 55Cr is given as 2 hours. This means that after every 2 hours, the amount of 55Cr in the sample will halve.
Now, let's work through the problem step-by-step:
1. Determine how many half-lives occur within the 12-hour duration:
Number of half-lives = (Total duration) / (Half-life)
= 12 hours / 2 hours
= 6 half-lives
2. Determine the fraction of 55Cr remaining after 6 half-lives:
Fraction remaining = (1/2)^(Number of half-lives)
= (1/2)^6
= 1/64
3. Determine the initial amount of 55Cr required to obtain 1.0 mg after 6 half-lives:
Initial amount = (Final amount) / (Fraction remaining)
= 1.0 mg / (1/64)
= 1.0 mg * 64
= 64 mg
Therefore, the minimum amount of material that should be shipped to obtain 1.0 mg of 55Cr is 64 mg, not 32 mg.
To determine the minimum amount of material that should be shipped to obtain 1.0 mg of 55Cr, we need to consider the half-life of the isotope and the time required for delivery.
The half-life of an isotope is the amount of time it takes for half of the sample to decay. In this case, the half-life of 55Cr is approximately 2 hours.
Given that the delivery time is 12 hours, we need to consider how many half-lives occur during this period. Assuming the decay is exponential, we can calculate this using the formula:
Number of half-lives = (Total time elapsed) / (Half-life)
In this case, the total time elapsed is 12 hours, and the half-life is 2 hours. Plugging in these values, we get:
Number of half-lives = 12 hours / 2 hours = 6 half-lives
Each half-life reduces the amount of the isotope by half. So, after 6 half-lives, the quantity of 55Cr remaining will be:
Fraction remaining = (1/2)^(Number of half-lives)
= (1/2)^6
= 1/64
Therefore, if the delivered sample is 1.0 mg, the initial amount of 55Cr present in the sample must be:
Initial amount = (Delivered sample) / (Fraction remaining)
= 1.0 mg / (1/64)
= 64 mg
Thus, the minimum amount of material that should be shipped to obtain 1.0 mg of 55Cr is 64 mg, not 32 mg.