If compound x has a first-order half-life of 24 seconds, how long would you have to wait for only 33% of the original material to be left?
To answer this question, we need to understand what it means for a compound to have a first-order half-life. In a first-order reaction, the rate of the reaction is directly proportional to the concentration of the reactant. The half-life of a reaction is the time it takes for the concentration of the reactant to decrease by half.
In this case, compound X has a first-order half-life of 24 seconds. This means that every 24 seconds, the concentration of compound X will decrease by half.
To find out how long it would take for only 33% of the original material to be left, we need to calculate the number of half-lives required.
Let's assume we start with 100% concentration of compound X.
After the first half-life, the concentration will decrease to 50%.
After the second half-life, the concentration will decrease to 25% (50% of 50%).
After the third half-life, the concentration will decrease to 12.5% (50% of 25%).
After the fourth half-life, the concentration will decrease to 6.25% (50% of 12.5%).
We can see that after each half-life, the concentration is decreasing exponentially by half. Therefore, after each half-life, the concentration will be halved.
To determine the number of half-lives required for the concentration to reach 33%, we can use the equation:
Final concentration = Initial concentration * (0.5)^n
where n is the number of half-lives.
Plugging in the given values, we have:
0.33 = 1 * (0.5)^n
To solve for n, we can take the logarithm of both sides:
log(0.33) = n * log(0.5)
n = log(0.33) / log(0.5)
Using a calculator, we find that n is approximately 1.54.
As we can't have a fraction of a half-life, we need to round up to the nearest whole number. Therefore, we need at least 2 half-lives for the concentration to reach 33%.
Since each half-life is 24 seconds, the total time required would be:
Total time = Number of half-lives * Half-life time
Total time = 2 * 24 seconds
Total time = 48 seconds
Therefore, you would have to wait for approximately 48 seconds for only 33% of the original material to be left.