A building has become accidentally contaminated with radioactivity. The longest-lived material in the building is strontium-90. (The atomic mass of Sr is 89.9077 u.) If the building initially contained 4.8 kg of this substance and the safe level is less than 10.0 counts/min, how long will the building be unsafe? (in years)
Halflife for Sr-90
T =28.79 yr
Decay constant
λ=ln2/T =0.693/28.79•365•24•3600 =
=7.63•10^-10 s^-1
Initial number of nuclei
Nₒ=m/mₒ=4.8/89.9077•1.66•10^-27 = =3.2•10^25.
Initial activity
Aₒ = λ •Nₒ =7.63•10^-10 • 3.2•10^25= =2.44•10^16 Bq.
A =10 counts/min =
=10/60 counts/s =0.167 Bq.
From the law of radioactive decay N=Nₒ•e^- λ•t
A=Aₒ•e^- λ•t,
A/Aₒ= e^ -λ•t,
A/Aₒ=0.167/2.44•10^16=6.84•10^-18
t = - ln(A/Aₒ)/λ = 39.5/7.63•10^-10 =
=5.18•10^10 s =1643 years.
To calculate the time for the building to become safe, we need to consider the half-life of strontium-90. The half-life of strontium-90 is approximately 28.8 years.
We can use the following formula to calculate the time it takes for a substance to decay to a certain amount:
t = (T₁/2) * log(N₀/N₁)
Where:
- t is the time it takes for the substance to decay.
- T₁/2 is the half-life of the substance.
- N₀ is the initial amount of the substance.
- N₁ is the final amount of the substance that is considered safe.
In this case, N₀ is 4.8 kg and N₁ is the amount of strontium-90 that corresponds to a safe level, which is below 10.0 counts/min.
Since the problem doesn't provide a specific value for N₁, we can assume that it is a negligible amount compared to the initial amount. Let's choose N₁ to be 0.01 kg (or 10 grams) as a conservative estimate.
Plugging the values into the formula, we have:
t = (28.8 years) * log(4.8 kg / 0.01 kg)
Calculating this expression, we get:
t ≈ (28.8 years) * log(480)
Using a calculator, we find:
t ≈ (28.8 years) * 2.681241237
So, the estimated time for the building to become safe is:
t ≈ 77.2923312 years
Therefore, the building will be unsafe for approximately 77.29 years.
To determine how long the building will be unsafe, we need to calculate how long it takes for the radioactivity of strontium-90 to decay to a safe level.
The decay of radioactive substances can be characterized using the half-life. The half-life is the time it takes for half of a sample of radioactive material to decay.
In order to calculate the time it takes for the building to be safe, we need to know the half-life of strontium-90. According to available data, the half-life of strontium-90 is 28.8 years.
To find out how many half-lives it takes for the radioactivity of strontium-90 to decay to a safe level, we can use the following formula:
number of half-lives = (ln(N₀/N)) / (ln(0.5))
Where:
N₀ is the initial amount of strontium-90 in the building
N is the safe level of strontium-90
Given that the initial amount of strontium-90 in the building is 4.8 kg, and the safe level is less than 10.0 counts/min, we need to convert the safe level to an amount of strontium-90 in kg.
To do this, we need to determine the decay constant (λ) using the half-life:
λ = ln(2) / half-life
λ ≈ 0.02403 per year
Using the decay constant (λ), we can find the safe level in kg:
N = N₀ * e^(-λt)
Where t is the time in years.
Since the safe level is less than 10.0 counts/min, we'll take it as 9.9 counts/min for a more conservative estimate.
Now we can substitute the values into the equation and solve for t:
9.9/60 (kg/min) = 4.8 * e^(-0.02403t)
Simplifying the equation, we have:
0.165 (kg/min) = 4.8 * e^(-0.02403t)
Rearranging the equation, we get:
e^(-0.02403t) = 0.165/4.8
Taking the natural logarithm (ln) of both sides of the equation:
-0.02403t = ln(0.165/4.8)
Solving for t, we divide both sides by -0.02403:
t = ln(0.165/4.8) / -0.02403
Using a calculator, we can calculate the value of t.
Please note that the calculated value of t might be an estimate, and additional safety measures may be required to ensure the building's safety.