The radioactive isotope sodium-24 is used as a tracer to measure the rate of flow in an artery or vein. The half life of sodium-24 is 14.9 hours.Suppose that a hospital buys a sample of sodium-24
a. how much of the sample will remain AFTER 48 Hours?
b. How long will it be until one gram remains
amount left = a (.5)^(t/14.9) , where a is the initial amount, and t is the time in hours
a) sub in t = 48
amount left = a(.5)(48/14.9) = .1072a
= appr 10.72% of what you started with
b) can't be answered since you did not say what they started with.
set 1 = a(.5)^(t/14.9) , (once you know a, you will have to use logs)
To answer these questions, we need to use the concept of half-life. The half-life of an isotope is the time it takes for half of the initial amount to decay.
a. How much of the sample will remain AFTER 48 Hours?
The half-life of sodium-24 is 14.9 hours. So, after each half-life, the amount of sodium-24 is halved. Let's calculate the number of half-lives in 48 hours.
Number of half-lives = 48 hours / 14.9 hours per half-life = 3.22 half-lives
Therefore, after 3.22 half-lives, the amount of sodium-24 remaining will be:
Remaining amount = initial amount / (2^3.22)
We don't have the initial amount mentioned in the question, so we cannot provide the exact remaining amount. However, you can substitute the initial amount of the sodium-24 sample and calculate the remaining amount using the above formula.
b. How long will it be until one gram remains?
We want to find the time it takes for the amount of sodium-24 to reduce to 1 gram.
Number of half-lives = log(base 2)(final amount / initial amount)
We can rearrange the formula to find the time:
Time = Number of half-lives × half-life duration
Let's assume the initial amount is 100 grams:
Number of half-lives = log(base 2)(1 g / 100 g) = log(base 2)(0.01) ≈ -6.64
Since the number of half-lives cannot be negative, we can take the absolute value:
Number of half-lives = |-6.64| = 6.64
Time = 6.64 half-lives × 14.9 hours per half-life ≈ 98.94 hours
Therefore, it will take approximately 98.94 hours for one gram of the sodium-24 sample to remain.
To answer these questions, we need to understand the concept of half-life and how it relates to the decay of radioactive isotopes.
The half-life of an isotope is the amount of time it takes for half of the sample to decay. In this case, the half-life of sodium-24 is given as 14.9 hours. This means that after 14.9 hours, half of the initial sample will have decayed.
a. To determine how much of the sodium-24 sample will remain after 48 hours, we need to calculate the number of half-lives within that time period.
Since each half-life is 14.9 hours, dividing 48 hours by 14.9 hours will give us the number of half-lives:
48 hours / 14.9 hours ≈ 3.22 half-lives
Now, we can calculate the amount of the sample that will remain after 3.22 half-lives. For each half-life, the remaining amount is halved. So, we will multiply the initial sample by (1/2) raised to the power of the number of half-lives:
Remaining amount = Initial amount * (1/2)^(number of half-lives)
Let's assume the initial sample size is 1 gram:
Remaining amount = 1 gram * (1/2)^(3.22)
Using a calculator, we find that (1/2)^(3.22) is approximately 0.195.
Therefore, the amount of sodium-24 remaining after 48 hours would be:
Remaining amount = 1 gram * 0.195 ≈ 0.195 grams
b. To find out how long it will take until one gram of sodium-24 remains, we need to determine the number of half-lives required for the sample to reach that amount.
Using the same formula as before, we can rearrange it to solve for the number of half-lives:
(1/2)^(number of half-lives) = Remaining amount / Initial amount
Plugging in the numbers:
(1/2)^(number of half-lives) = 1 gram / Initial amount
Since Initial amount is also 1 gram, we have:
(1/2)^(number of half-lives) = 1
To solve for the number of half-lives, we can take the logarithm (base 1/2) of both sides of the equation:
log(base 1/2)(1/2)^(number of half-lives) = log(base 1/2)(1)
This simplifies to:
number of half-lives = log(base 1/2)(1)
The logarithm (base 1/2) of 1 is 0, so we have:
number of half-lives = 0
This means that it would take 0 half-lives for the sample to reach one gram. However, this situation is not physically possible since the initial sample size is 1 gram. In this case, one gram will never be reached because the decay process continues indefinitely.