12. Aluminum in blood plasma was shown to undergo a first order elimination reaction with a half-life of 160 days. If the initial concentration was 8.2 µg/L, how long will it take for the concentration to drop to 2.5 µg/L?
To solve this problem, we need to use the formula for a first-order reaction, which is:
\[ \text{concentration} = \text{initial concentration} \times e^{-kt} \]
Where:
- concentration is the current concentration of aluminum in blood plasma
- initial concentration is the initial concentration of aluminum in blood plasma
- e is the natural logarithm base (approximately 2.71828)
- k is the rate constant for the reaction
- t is the time elapsed
We can rearrange the formula to solve for time (t):
\[ t = \frac{\ln(\frac{\text{concentration}}{\text{initial concentration}})}{-k} \]
Given that the initial concentration is 8.2 µg/L, the concentration we want to find is 2.5 µg/L, and the half-life is 160 days, we can find the rate constant (k) using the half-life formula:
\[ t_{1/2} = \frac{\ln(2)}{k} \]
Substituting the given half-life of 160 days, we can solve for k:
\[ 160 = \frac{\ln(2)}{k} \]
\[ k = \frac{\ln(2)}{160} \]
Now, we can substitute the initial and desired concentrations along with the calculated value of k into the formula to find the time (t):
\[ t = \frac{\ln(\frac{2.5}{8.2})}{-\frac{\ln(2)}{160}} \]
Calculating this expression will give us the time it takes for the concentration to drop to 2.5 µg/L.