A building has become accidentally contaminated with radioactivity. THe longest lived material in the building is strontium-90 (atomic mass of 90 Sr 38 is 89.9077). If the building initially contained 5.0kg of this subtstance and the safe level is less than 10.0 counts/min, how long will the building be unsafe?
You need to look up the half life of Sr-90 and also calculate the initial decay rate of 5.0 kg. The number or Sr-90 atoms initially is
(Avogadro's #)*(5000 g/89.9 moles/g) =
3.35*10^25 atoms.
The half life (from a handbook) is
T1/2 = 28 years = 8.84*10^8 s.
The decay constant is
lambda = 0.693/T1/2 = 7.84*10^-10 s^-1
The initial decay rate is therefore
(3.35*10^25) x (7.84*10^-10) =
2.6*10^16 counts/s
The time required for that to decay to 10 counts/s can be computed from the exponential decay law.
10 = 2.6*10^16 * e(-lambda*t)
1623
To find out how long the building will be unsafe, we need to calculate the time it takes for the initial decay rate to reach 10 counts/s.
First, let's calculate the initial decay rate of 5.0 kg of strontium-90. We start by finding the number of Sr-90 atoms initially in the building:
Number of Sr-90 atoms = (Avogadro's number) * (mass in grams / molar mass)
= (6.022 x 10^23 atoms/mol) * (5000 g / 89.9 g/mol)
= 3.35 x 10^25 atoms
Next, we'll find the half-life of Sr-90. According to a handbook, the half-life of Sr-90 is 28 years, which is equivalent to 8.84 x 10^8 seconds.
Then, we can calculate the decay constant (lambda) using the formula:
lambda = 0.693 / (half-life)
= 0.693 / (8.84 x 10^8 seconds)
= 7.84 x 10^-10 s^-1
Now, we can calculate the initial decay rate:
Initial decay rate = (number of Sr-90 atoms) x (decay constant)
= (3.35 x 10^25 atoms) x (7.84 x 10^-10 s^-1)
= 2.6 x 10^16 counts/s
To find the time required for the decay rate to reach 10 counts/s, we'll use the exponential decay law:
10 = (2.6 x 10^16 counts/s) x e^(-lambda x t)
Here, t represents the time in seconds we're looking to find.
To solve for t, we can take the natural logarithm (ln) of both sides and rearrange the equation:
ln(10) = ln(2.6 x 10^16 counts/s) - (lambda x t)
Now, we can solve for t by rearranging the equation:
t = (ln(2.6 x 10^16 counts/s) - ln(10)) / (-lambda)
Plugging in the values:
t = (ln(2.6 x 10^16) - ln(10)) / (-7.84 x 10^-10)
Using a calculator, we can find t to be approximately 2.80 x 10^9 seconds.
Therefore, the building will be unsafe for approximately 2.80 x 10^9 seconds.