Strontium-90 decays through the emission of beta particles. It has a half-life of 29 years. How long does it take for 80% of a sample of strontium-90 to decay?
Show process of how you got this
k = 0.693/t1/2
ln(No/N) = kt
Use No = 100
N = 20
k from above
solve for t in years.
27
To determine how long it takes for 80% of a sample of strontium-90 to decay, we can use the concept of half-life.
Given that the half-life of strontium-90 is 29 years, this means that every 29 years, the amount of strontium-90 in the sample is reduced by half.
To find the time it takes for 80% of the sample to decay, we can use the formula:
t = t1/2 * ln(1/percent remaining)
Where:
t is the time it takes for the sample to decay (in years),
t1/2 is the half-life of strontium-90 (29 years), and
percent remaining is the decimal form of the remaining percentage (80% = 0.8).
Now, let's substitute the values into the formula:
t = 29 * ln(1/0.8)
Using a calculator, we can calculate the natural logarithm of 1/0.8, which is approximately -0.2231:
t ≈ 29 * (-0.2231)
t ≈ -6.57
The negative value does not make sense in this context since time cannot be negative. Therefore, we can take the absolute value of t to obtain the positive value:
t ≈ | -6.57 |
t ≈ 6.57
Therefore, it takes approximately 6.57 years for 80% of the sample of strontium-90 to decay.
To find out how long it takes for 80% of a sample of strontium-90 to decay, we can use the concept of half-life. The half-life of strontium-90 is given as 29 years.
Let's break down the problem into steps:
Step 1: Determine the number of half-lives required to reach 80% decay.
Since each half-life represents a 50% decay, we need to find out how many half-lives are needed to reach 80% decay. To do this, we can use the formula:
Number of half-lives = (ln(initial amount / final amount)) / ln(1/2)
In this case, the initial amount is 100% and the final amount is 80%. Therefore:
Number of half-lives = (ln(1 / 0.8)) / ln(1/2)
Step 2: Calculate the time required for the number of half-lives obtained in step 1.
The time required for the decay of one half-life is given as 29 years. Hence, the total time required for the number of half-lives obtained in step 1 is:
Total time = Number of half-lives * Half-life duration
So, plugging in the values:
Total time = Number of half-lives * 29 years
Step 3: Calculate the result.
Now, let's execute the calculation to find the time required for 80% decay:
Number of half-lives = (ln(1 / 0.8)) / ln(1/2) = (ln(1.25)) / ln(0.5) ≈ 2
Total time = Number of half-lives * 29 years = 2 * 29 years = 58 years
Therefore, it takes about 58 years for 80% of a sample of strontium-90 to decay.