The rate of decay of a radioactive isotope is directly proportional to the amount remaining.If the half-life of the radioactive isotope,Einsteinium,is 276 days and a sample initially weighs 25 grams,what is its rate of decay on the 150th day? Round your answer to three decimal places.

Question

The rate of decay of a radioactive isotope is directly proportional to the amount remaining.If the half-life of the radioactive isotope,Einsteinium,is 276 days and a sample initially weighs 25 grams,what is its rate of decay on the 150th day?​ Round your answer to three decimal places. ​

Quizplus · Accepted Answer

The rate of decay of a radioactive isotope can be calculated using the formula $N = N_0 \times (1/2)^{\frac{t}{T}}$, where $N$ is the remaining amount after time $t$, $N_0$ is the initial amount, and $T$ is the half-life period. For a half-life of 276 days and an initial amount of 25 grams, the amount remaining after 150 days is $25 \times (1/2)^{\frac{150}{276}}$. The rate of decay at 150 days is the derivative of this amount with respect to time, which gives an average rate of change over a small interval around 150 days. Calculating the amount remaining at 150 days and comparing it to a slightly later time (e.g., 151 days) gives an approximate rate of decay. The correct calculation aligns with choice A, -0.043 grams per day, indicating a gradual decrease in mass.

The Rate of Decay of a Radioactive Isotope Is Directly