Two 5-grain aspirin tablets contain 650 mg of the drug. With aspirins half-life of 29 minutes, how much is left in the bloodstream after 2 hours? How long does it take for the level to be equivalent to 10 mg of aspirin? If an individual takes two tablets every 4 hours, what is the maintenance level of the aspirin?
the fraction left after t minutes is
(1/2)^(t/29)
so, after two hours (120 minutes), the fraction left is
1/2^(120/29) = 0.0568
To find how log it takes to get to 10mg,
650(1/2)^(t/29) = 10
(1/2)^(t/29) = 1/65
t/29 = log(1/65)/log(1/2) = 6.022
t = 174.65
or, almost 3 hours
After 4 hours, the fraction left is
(1/2)^(240/29) = 0.00323
So, over the long term, the amount left is
650(.00323 + .00323^2 + ...)
= 2.1 mg
so the maintenance level is 2.1 mg?
To answer these questions, we need to understand the concept of half-life and how it affects the concentration of a substance in the bloodstream over time.
Firstly, let's calculate how much is left in the bloodstream after 2 hours.
1. Start by converting the dose of 650 mg to the number of tablets. Since two 5-grain aspirin tablets contain 650 mg, each tablet contains 650 mg / 2 = 325 mg of the drug.
2. The half-life of aspirin is given as 29 minutes, which means that after every 29 minutes, the concentration of the drug in the bloodstream reduces by half. To determine the amount left after 2 hours, we need to calculate how many half-lives have elapsed during this time.
Since each half-life is 29 minutes, there are 120 minutes in 2 hours. Therefore, the number of half-lives elapsed is 120 minutes / 29 minutes ≈ 4.14 half-lives.
3. To calculate the amount of drug remaining after 2 hours, we can use the formula:
Amount remaining = Initial amount × (1/2)^(number of half-lives)
In this case, the initial amount is 2 tablets × 325 mg per tablet = 650 mg.
Amount remaining = 650 mg × (1/2)^4.14 ≈ 650 mg × 0.0874 ≈ 56.71 mg.
Therefore, after 2 hours, approximately 56.71 mg of the drug will be left in the bloodstream.
Next, let's determine how long it takes for the level to be equivalent to 10 mg of aspirin.
Using the same formula as above, we can rearrange it to solve for the number of half-lives:
Number of half-lives = log(amount remaining / initial amount) / log(1/2)
Number of half-lives = log(10 mg / 650 mg) / log(1/2) ≈ log(0.0154) / -0.301 ≈ 4.042.
Therefore, it would take approximately 4.042 half-lives for the level of aspirin to be equivalent to 10 mg.
Finally, let's determine the maintenance level of aspirin for an individual who takes two tablets every 4 hours.
Since the half-life of aspirin is 29 minutes, we need to determine the amount of drug remaining after 4 hours or 240 minutes (as we've done in the first question).
Amount remaining = Initial amount × (1/2)^(number of half-lives)
Considering 2 tablets × 325 mg per tablet = 650 mg as the initial amount, and using the same formula as before:
Amount remaining = 650 mg × (1/2)^(240 minutes / 29 minutes) ≈ 650 mg × (1/2)^8.276 ≈ 650 mg × 0.0237 ≈ 15.4 mg.
Therefore, the maintenance level of aspirin for an individual who takes two tablets every 4 hours would be approximately 15.4 mg.