the half life of oxygen-15 is 124 s. if a sample of oxygen-15 has an activity of 4000 Bq, how many minutes will elapse before it reaches an activity of 500Bq
6 minutes and 12 seconds!
since 500 = 4000/8, that would be 3 half-lives.
To solve this problem, we can use the equation for radioactive decay, which relates the activity (A) of a radioactive substance to its initial activity (A₀), decay constant (λ), and time (t):
A = A₀ * e^(-λt)
Given that the half-life of oxygen-15 is 124 s, we can calculate the decay constant (λ) using the formula:
λ = ln(2) / Half-life
Let's calculate the decay constant (λ) for oxygen-15:
λ = ln(2) / 124 s
≈ 0.0056 s^(-1)
Now, we can solve for the time (t) required for the sample to reach an activity of 500 Bq, starting from an initial activity of 4000 Bq:
500 Bq = 4000 Bq * e^(-0.0056 s^(-1) * t)
To isolate t, we need to divide both sides by 4000 Bq and take the natural logarithm of both sides:
e^(-0.0056 s^(-1) * t) = 500 Bq / 4000 Bq
ln(e^(-0.0056 s^(-1) * t)) = ln(1.25 × 10^(-1))
Now we can solve for t:
-0.0056 s^(-1) * t = ln(1.25 × 10^(-1))
t = ln(1.25 × 10^(-1)) / -0.0056 s^(-1)
Using a calculator, we can find the value of t to be approximately 516.85 s.
To convert seconds to minutes, we divide by 60:
t ≈ 516.85 s / 60
t ≈ 8.6142 minutes
Therefore, it would take approximately 8.61 minutes for the activity of the oxygen-15 sample to reach 500 Bq.
To determine the time required for the activity of the sample to decrease from 4000 Bq to 500 Bq, we need to use the decay formula:
N(t) = N₀ * (1/2)^(t/T)
Where:
- N(t) is the current quantity of the substance
- N₀ is the initial quantity of the substance
- t is the time elapsed
- T is the half-life of the substance
In this case, the initial quantity (N₀) is 4000 Bq, and the half-life (T) is 124 seconds. We need to find the time (t) it takes for the quantity to decrease to 500 Bq.
Let's rearrange the formula to solve for t:
t = T * log₂(N(t) / N₀)
Substituting the given values:
t = 124 s * log₂(500 Bq / 4000 Bq)
Now, we can solve this equation to find the value of t. Here is how you can calculate it step by step:
1. Calculate the ratio of N(t) to N₀:
ratio = 500 Bq / 4000 Bq = 0.125
2. Calculate the natural logarithm (ln) of the ratio:
ln_ratio = ln(0.125)
3. Convert the natural logarithm to base 2 logarithm using the logarithmic identity: log₂(a) = ln(a) / ln(2)
log₂_ratio = ln_ratio / ln(2)
4. Calculate the time t:
t = 124 s * log₂_ratio
Now, you can substitute the values into the equation and calculate the time (t) in seconds. To convert it to minutes, divide the result by 60.
Note: Ensure that your calculator is set to use either radians or degrees depending on the logarithmic functions available.