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?

Refer to your other posts.

Almost all of these can be worked with two thoughts.
#1.
k = 0.693/t1/2
#2. ln(No/N) = kt where No = initial and N = final, k is from the #1 calculation and t1/2 is the half life.

To solve this problem, we can use the concept of half-life.

The half-life of radium-223 is 11.4 days, which means that after 11.4 days, the activity of the sample will be reduced by half.

To find the time it takes for the activity to decrease to 2.00% of its initial value, we need to determine how many half-lives it would take for this reduction.

Let's calculate the number of half-lives needed:

Initial activity -> 100%
After 1 half-life -> 50%
After 2 half-lives -> 25%
After 3 half-lives -> 12.5%
After 4 half-lives -> 6.25%
After 5 half-lives -> 3.125%
After 6 half-lives -> 1.5625%
After 7 half-lives -> 0.78125%
After 8 half-lives -> 0.390625%
After 9 half-lives -> 0.1953125%
After 10 half-lives -> 0.09765625%
After 11 half-lives -> 0.048828125%
After 12 half-lives -> 0.02441406%
After 13 half-lives -> 0.01220703%
After 14 half-lives -> 0.006103516%
After 15 half-lives -> 0.003051758%
After 16 half-lives -> 0.001525879%
After 17 half-lives -> 0.0007629395%
After 18 half-lives -> 0.0003814697%
After 19 half-lives -> 0.0001907349%
After 20 half-lives -> 9.536743e-05%
After 21 half-lives -> 4.768372e-05%
After 22 half-lives -> 2.384186e-05%
After 23 half-lives -> 1.192093e-05%
After 24 half-lives -> 5.960464e-06%

As we can see, it takes approximately 24 half-lives for the activity to decrease to 2.00% of its initial value.

Since the half-life of radium-223 is 11.4 days, we can calculate the total time as follows:

Total time = number of half-lives x half-life duration
Total time = 24 x 11.4 days
Total time ≈ 273.6 days

Therefore, it would take approximately 273.6 days for the activity of a sample of 223Ra to decrease to 2.00% of its initial value.

To determine 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 is the time it takes for half of the radioactive material in a sample to decay. In this case, the half-life of Radium-223 (223Ra) is 11.4 days.

To find the time it takes for the activity of the sample to decrease to 2.00% of its initial value, we need to calculate the number of half-lives required to reach this percentage.

The formula to calculate the number of half-lives is:
Number of half-lives = (log(final amount / initial amount)) / (log(0.5))

In this case, the final amount is 2.00% of the initial amount, which is 0.02. The initial amount is 100% or 1.00.

Number of half-lives = (log(0.02) / log(0.5))

Using a calculator, we find that the number of half-lives is approximately 5.16.

Now, to find the time it takes to reach this number of half-lives, we multiply the half-life by the number of half-lives.

Time = half-life * number of half-lives
Time = 11.4 days * 5.16

The approximate time it takes for the activity of a sample of 223Ra to decrease to 2.00% of its initial value is 58.824 days.