An investigator receives Co-60 (5.27 year half life) for use in a research study.
Unfortunately the Co-60 is contaminated with Cs-137 (30.0 year half life). The
initial Co-60 activity is 400 times the initial Cs-137 activity. How long after the
initial assay will the Cs-137 activity be 0.2 times the Co-60 activity?
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Oh noes
To answer this question, we need to use the concept of radioactive decay and the formula for calculating the activity of a radioactive substance:
Activity = Initial Activity × (1/2)^(Time / Half-life)
Let's first determine the relationship between the initial activities of Co-60 and Cs-137. Given that the initial Co-60 activity is 400 times the initial Cs-137 activity, we can write the equation:
Co-60 initial activity = 400 × Cs-137 initial activity
Now, the question asks for the time at which the Cs-137 activity is 0.2 times the Co-60 activity. Let's denote the time as "t" and set up the equation:
Cs-137 activity at time t = 0.2 × Co-60 activity at time t
Now, we can substitute the formula for activity into the equation:
Cs-137 initial activity × (1/2)^(t / Cs-137 half-life) = 0.2 × (Co-60 initial activity × (1/2)^(t / Co-60 half-life))
Since we know the half-lives of Co-60 (5.27 years) and Cs-137 (30.0 years), we can rewrite the equation as:
Cs-137 initial activity × (1/2)^(t / 30.0) = 0.2 × (400 × Cs-137 initial activity × (1/2)^(t / 5.27))
Now we can cancel out the Cs-137 initial activity from both sides of the equation:
(1/2)^(t / 30.0) = 0.2 × 400 × (1/2)^(t / 5.27)
Next, let's simplify the equation:
(1/2)^(t / 30.0) = 80 × (1/2)^(t / 5.27)
Now, we can solve for "t" by taking the logarithm (base 2) of both sides:
log2((1/2)^(t / 30.0)) = log2(80 × (1/2)^(t / 5.27))
Using the logarithmic property log2(a^b) = b × log2(a):
(t / 30.0) × log2(1/2) = log2(80) + (t / 5.27) × log2(1/2)
Since log2(1/2) = -1:
(t / 30.0) × (-1) = log2(80) + (t / 5.27) × (-1)
Simplifying further:
-t / 30.0 = log2(80) - t / 5.27
Finally, let's solve for "t" by adding t / 30.0 to both sides:
-t / 30.0 + t / 5.27 = log2(80)
Using a common denominator:
(-5.27t + 30.0t) / (5.27 × 30.0) = log2(80)
Simplifying:
24.73t / 157.71 = log2(80)
Now, we can solve the equation by multiplying both sides by 157.71:
24.73t = 157.71 × log2(80)
Finally, divide both sides by 24.73 to isolate "t":
t = (157.71 × log2(80)) / 24.73
Using a calculator or software, we can evaluate this expression to find the value of "t", which will give us the answer to the question.