A typical dose of a radioactive sample is 27.0 mCi. How long does it take for the activity to reduce to 0.100 mCi? The half life of the sample is 211,000 y.
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To calculate the time it takes for the activity of a radioactive sample to reduce to a certain level, we can use the formula for radioactive decay:
A(t) = A₀ * (1/2)^(t / t½)
Where:
A(t) is the activity at time t
A₀ is the initial activity
t is the time elapsed
t½ is the half-life of the sample
In this case, we want to find the time it takes for the activity to reduce to 0.100 mCi, starting with an initial activity of 27.0 mCi, and a half-life of 211,000 years.
First, let's convert the given values to the same unit. Since the half-life is given in years, we'll convert the dose to mCi/year:
Initial activity (A₀) = 27.0 mCi
Final activity (A(t)) = 0.100 mCi
Half-life (t½) = 211,000 years
Now, let's rearrange the formula to solve for time (t):
t = t½ * log₂(A(t) / A₀)
Substituting the values we know:
t = 211,000 * log₂(0.100 / 27.0)
Using a calculator, we can evaluate the logarithm to find the time:
t ≈ 211,000 * log₂(0.0037)
t ≈ 211,000 * (-8.52)
t ≈ -1,798,920 years
Since time cannot be negative, we can conclude that it will take approximately 1,798,920 years for the activity to reduce to 0.100 mCi.
To determine how long it takes for the activity of a radioactive sample to reduce to a specific amount, we can use the concept of half-life. The half-life is the time it takes for half of the radioactive substance to decay.
Given:
Initial activity (A₀) = 27.0 mCi
Final activity (A) = 0.100 mCi
Half-life (t₁/₂) = 211,000 years
Let's proceed with the calculations:
1. Determine the decay factor:
The decay factor (D) is calculated using the equation D = (1/2)^(t / t₁/₂), where "t" is the time we are trying to find.
D = (A / A₀)
D = (0.100 mCi / 27.0 mCi)
D ≈ 0.00370
2. Calculate the elapsed time (t):
We can rearrange the decay factor equation to solve for "t":
t = t₁/₂ * (log(D) / log(1/2))
t = (211,000 years) * (log(0.00370) / log(1/2))
t ≈ 1,308,385 years
Therefore, it will take approximately 1,308,385 years for the activity of the sample to reduce to 0.100 mCi.