A certian radioactive isotope has a half-life of 1 hour. If you start with 1 gram of the material at noon, how much of the original isotope will the sample contain at 3 pm? At 6 pm? At 10 pm?
A half-life is how long it takes for a radioactive isotope to decay by half.
At 2 PM, you will have .5 grams of the material.
Each hour thereafter, divide the material by half to see how much remains.
A better way is with the exponential function on your calculator.
Amountremaining= Originalamount* e^(-.692t/halflife)
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To determine how much of the radioactive isotope remains at different times, we can use the concept of half-life. The half-life of an isotope is the time it takes for half of the initial quantity to decay.
In this case, the given isotope has a half-life of 1 hour. Let's break down the problem step by step.
1. Number of half-lives:
To find out how many half-lives have passed from noon to a specific time, divide the time elapsed by the half-life. Since the half-life is 1 hour, we divide the elapsed time (in hours) by 1.
a) From noon to 3 pm, 3 hours have passed, so there are 3 half-lives.
b) From noon to 6 pm, 6 hours have passed, so there are 6 half-lives.
c) From noon to 10 pm, 10 hours have passed, so there are 10 half-lives.
2. Remaining amount after each half-life:
Since each half-life leaves behind half of the previous amount, we multiply the original amount by (1/2) raised to the power of the number of half-lives.
Let's calculate the remaining amount:
a) From noon to 3 pm (3 half-lives):
Remaining amount = Initial amount * (1/2)^(number of half-lives)
= 1 gram * (1/2)^3
= 1 gram * (1/8)
= 1/8 gram
= 0.125 grams
b) From noon to 6 pm (6 half-lives):
Remaining amount = Initial amount * (1/2)^(number of half-lives)
= 1 gram * (1/2)^6
= 1 gram * (1/64)
= 1/64 gram
= 0.015625 grams
c) From noon to 10 pm (10 half-lives):
Remaining amount = Initial amount * (1/2)^(number of half-lives)
= 1 gram * (1/2)^10
= 1 gram * (1/1024)
= 1/1024 gram
= 0.0009765625 grams
So, at 3 pm, the sample will contain 0.125 grams of the original isotope.
At 6 pm, the sample will contain 0.015625 grams of the original isotope.
At 10 pm, the sample will contain 0.0009765625 grams of the original isotope.