The half-life of Radon-222 is 3.8 days. A 32-gram sample of Radon-222 is monitored for one day. How long will it take for the sample to be 24% of the initial amount? (rounded to the nearest hundredths)?
A) 0.1 days
B) 7.8 days
C) 3.8 days
D) 1.8 days
E) 0.5 days

Question

Quizplus · Accepted Answer

To find how long it will take for a 32-gram sample of Radon-222 to be 24% of its initial amount, we first determine what 24% of 32 grams is: 24% of 32 grams = 0.24 * 32 = 7.68 grams. The question essentially asks how long it takes for the sample to decay to 7.68 grams. The half-life formula is used to calculate the time it takes for a substance to decay to a certain amount: $t = \frac{\ln(\frac{N_0}{N})}{k}$, where $t$ is the time, $N_0$ is the initial quantity, $N$ is the final quantity, $k$ is the decay constant, and $\ln$ is the natural logarithm. The decay constant $k$ can be found from the half-life formula $t_{1/2} = \frac{\ln(2)}{k}$. Given the half-life of Radon-222 is 3.8 days, we can calculate $k$ and then use it to find $t$. However, for this question, a simpler approach can be used based on understanding the nature of half-lives. Since the question asks for a rounded answer and involves understanding the concept rather than detailed calculation, knowing that each half-life reduces the amount by half, we can estimate how many half-lives it takes to reach approximately 24% (which is close to 25%, or $1/4$). This approach simplifies the calculation, leading to the conclusion that it takes approximately 7.8 days to reach 24% of the initial amount, without needing to perform detailed logarithmic calculations.

The Half-Life of Radon-222 Is 3