Radium-223 has a half-life of 11.4 days.Approximately how long would it take for the activity of a sample of 223Ra to decrease to 2.00 % of its initial value?
Well, let's see. If the half-life of radium-223 is 11.4 days, that means that after 11.4 days, the activity will be halved. So, after the first half-life, we're down to 50% of the initial value.
To find out how long it would take for the activity to decrease to 2.00% of its initial value, we can set up a ratio. We know that after one half-life, the activity is 50% of the initial value. So, after two half-lives, it would be 25% (50% * 50%), and after three half-lives, it would be 12.5% (25% * 50%).
Now, we just need to keep going until we get to 2.00%. Let me fetch my calculator... *beep boop beep*... Okay, after four half-lives, it would be approximately 6.25% (12.5% * 50%), and after five half-lives, it would be about 3.13% (6.25% * 50%).
So, if I had to make an educated guess, I'd say it would take around 5 half-lives for the activity of a sample of radium-223 to decrease to 2.00% of its initial value. Just remember, my calculations may not be 100% accurate, but at least they come with a dash of humor!
To find the time it takes for the activity of a sample of 223Ra to decrease to 2.00% of its initial value, we can use the concept of half-life.
The half-life of a radioactive substance is the time it takes for half of the substance to decay or lose its radioactivity. In this case, the half-life of 223Ra is given as 11.4 days.
To determine the time it takes for the activity to decrease to 2.00% of its initial value, we need to find the number of half-lives that must occur until we reach that point.
Let's define the time we want to find as t.
After one half-life, the activity of the 223Ra sample will be reduced to 50% of its initial value.
After two half-lives, the activity will be reduced to 25% (50% x 50%) of its initial value.
After three half-lives, the activity will be reduced to 12.5% (25% x 50%) of its initial value.
And so on...
We can express the activity of the sample at time t as a fraction of its initial value:
(0.5)^n = 0.02
Where n is the number of half-lives that have occurred.
Solving the equation for n gives:
n = log(0.02) / log(0.5)
Using a calculator, we find that n is approximately 5.64.
Since n represents the number of half-lives, we need to multiply it by the half-life of 223Ra to get the total time it would take for the activity to decrease to 2.00% of its initial value.
t = n x half-life = 5.64 x 11.4 days
Therefore, it would take approximately 64.62 days (or 65 days, rounding to the nearest day) for the activity of a sample of 223Ra to decrease to 2.00% of its initial value.