Technetium-99m, an important radionuclide in medicine, has a half-life of 6.0 hours. What percentage of a sample of technetium-99m will remain undecayed after the following times?
18.0 hours
1 day
5 half- lives
Well, I must say, technetium-99m must be quite the party animal with that short half-life! But let's see how much it manages to stick around.
After 18.0 hours, we can calculate how many half-lives have passed by dividing 18.0 by 6.0. So, 18.0 / 6.0 equals... 3 half-lives! If we start with 100% of the sample, and each half-life cuts it in half, then after 3 half-lives, we'll have 100% divided by 2, divided by 2, divided by 2, which is... 12.5%!
But let's see how technetium-99m holds up after a day. Since there are 24 hours in a day, we have 24 / 6 equals... 4 half-lives! Applying the same math as before, we get 100% divided by 2, divided by 2, divided by 2, divided by 2, which equals... 6.25%!
Now, if we wait for 5 half-lives, we'll be down to 100% divided by 2, divided by 2, divided by 2, divided by 2, divided by 2, which is... 3.125%. So, after 5 half-lives, only about 3.125% of the sample will be left.
Oh, technetium-99m, always in a hurry to break up and leave the party.
To calculate the percentage of a sample of technetium-99m that remains undecayed after a given time, we need to use the equation:
Percentage remaining = (1/2)^(t / half-life) * 100
Where:
- t is the time that has elapsed
- half-life is the half-life of technetium-99m, which is 6.0 hours
Let's calculate the percentage remaining for each given time:
1. After 18.0 hours:
Percentage remaining = (1/2)^(18.0 / 6.0) * 100
Percentage remaining = (1/2)^3 * 100
Percentage remaining = (1/8) * 100
Percentage remaining = 12.5%
2. After 1 day (24 hours):
Percentage remaining = (1/2)^(24 / 6.0) * 100
Percentage remaining = (1/2)^4 * 100
Percentage remaining = (1/16) * 100
Percentage remaining = 6.25%
3. After 5 half-lives:
Since each half-life of technetium-99m is 6.0 hours, 5 half-lives would be:
t = 5 * 6.0 = 30.0 hours
Percentage remaining = (1/2)^(30.0 / 6.0) * 100
Percentage remaining = (1/2)^5 * 100
Percentage remaining = (1/32) * 100
Percentage remaining = 3.125%
Therefore, after:
- 18.0 hours, approximately 12.5% of the sample will remain undecayed
- 1 day (24 hours), around 6.25% will remain undecayed
- 5 half-lives (30.0 hours), roughly 3.125% will remain undecayed
To determine the percentage of a sample of technetium-99m that remains undecayed after a specific time, we can use the half-life formula:
N(t) = N₀ * (1/2)^(t/T)
Where:
N(t) is the amount of the sample remaining after time t,
N₀ is the initial amount of the sample,
t is the given time, and
T is the half-life of the radionuclide.
Let's solve each scenario step by step:
1. 18.0 hours:
To find the percentage of technetium-99m remaining after 18.0 hours, we need to calculate N(18.0), using the formula above.
Since the half-life is 6.0 hours, we'll assume the initial amount (N₀) is 100% or the total amount.
N(18.0) = 100% * (1/2)^(18.0/6.0)
Simplifying the equation:
N(18.0) = 100% * (1/2)^3 = 100% * (1/8) = 12.5%
Therefore, after 18.0 hours, 12.5% of the technetium-99m sample will remain undecayed.
2. 1 day (24.0 hours):
Now let's calculate the percentage of technetium-99m remaining after 1 day or 24.0 hours.
Using the same formula:
N(24.0) = 100% * (1/2)^(24.0/6.0)
Simplifying the equation:
N(24.0) = 100% * (1/2)^4 = 100% * (1/16) = 6.25%
Therefore, after 1 day or 24.0 hours, 6.25% of the technetium-99m sample will remain undecayed.
3. 5 half-lives:
To calculate the percentage of technetium-99m remaining after 5 half-lives, we need to use the formula repeatedly.
N(5T) = 100% * (1/2)^(5 * 6.0/6.0)
Simplifying the equation:
N(5T) = 100% * (1/2)^5 = 100% * (1/32) = 3.125%
Therefore, after 5 half-lives, which is equivalent to 30.0 hours for technetium-99m, 3.125% of the sample will remain undecayed.
k = 0.693/t1/2
ln(No/N) = kt
No = 100 (but you can start with any number you wish but with any other starting value you must calculate percentage differently.
N = ?
k from above
time = from the problem
Solve for N and since you started with 100 that is the percent remaining undecayed.