Technetium-99m is an ideal radioisotope for scanning organs because it has a half-life of 6.0 {\rm hr} and is a pure gamma emitter. Suppose that 280{\rm mg} were prepared in the technetium generator this morning. How many milligrams would remain after the following intervals?
Technetium-99m is an ideal radioisotope for scanning organs because it has a half-life of 6.0 {\rm hr} and is a pure gamma emitter. Suppose that 280{\rm mg} were prepared in the technetium generator this morning. How many milligrams would remain after the following intervals?
To calculate the remaining amount of technetium-99m after a given interval, we need to use the concept of half-life. The half-life of technetium-99m is provided as 6.0 hours.
To determine the remaining amount after a given interval, we can use the following formula:
Remaining amount = Initial amount * (1/2)^(t / half-life),
where t is the time interval.
Let's calculate the remaining amount of technetium-99m after the provided time intervals:
1. After 6 hours:
Remaining amount = 280 mg * (1/2)^(6/6) = 280 mg * (1/2)^1 = 280 mg * 1/2 = 140 mg.
2. After 12 hours:
Remaining amount = 280 mg * (1/2)^(12/6) = 280 mg * (1/2)^2 = 280 mg * 1/4 = 70 mg.
3. After 18 hours:
Remaining amount = 280 mg * (1/2)^(18/6) = 280 mg * (1/2)^3 = 280 mg * 1/8 = 35 mg.
Therefore, the remaining amounts of technetium-99m after the respective time intervals are as follows:
- After 6 hours: 140 mg
- After 12 hours: 70 mg
- After 18 hours: 35 mg.
To find the remaining mass of Technetium-99m after a certain interval, we need to use the radioactive decay formula. The formula for radioactive decay is given by:
N(t) = N0 * e^(-λt)
Where:
- N(t) is the remaining mass at time t
- N0 is the initial mass
- λ is the decay constant
- t is the time interval
In this case, we are given that the half-life of Technetium-99m is 6.0 hours. The half-life is the time it takes for half of the radioactive substance to decay. We can use this information to determine the decay constant (λ) using the formula:
λ = ln(2) / T1/2
Where:
- ln(2) is the natural logarithm of 2 (approximately 0.693)
- T1/2 is the half-life
Given that T1/2 = 6.0 hours, we can calculate the decay constant:
λ = ln(2) / 6.0
Now we can use the radioactive decay formula to determine the remaining mass at different time intervals. Let's calculate it for the intervals provided:
1. After 2 hours:
- t = 2 hours
- N0 = 280 mg
Plug these values into the formula:
N(2) = 280 * e^(-λ * 2)
2. After 6 hours:
- t = 6 hours
- N0 = 280 mg
Plug these values into the formula:
N(6) = 280 * e^(-λ * 6)
3. After 24 hours:
- t=24 hours
- N0 = 280 mg
Plug these values into the formula:
N(24) = 280 * e^(-λ * 24)
By calculating these three equations, we can determine the remaining mass of Technetium-99m after each time interval.
no intervals but here is how you do it.
k = 0.693/t1/2
ln(No/N) = kt
No = 280 mg
N = unknown
k from above
t = intervals in the problem (in hours).