Two radioactive elements A and B have half-lives of 100 and 50 years respectively. Samples of A and B initially contain equal amounts/number of atoms. What ...

Two radioactive elements A and B have half-lives of 100 and 50 years respectively. Samples of A and B initially contain equal amounts/number of atoms. What is the ratio of the number of the remaining atoms of A to that of B after 200 years?

Answer Details

The half-life of a radioactive element is the amount of time it takes for half of the initial number of atoms to decay. For element A, the half-life is 100 years, which means that after 100 years, half of the initial number of atoms will have decayed, leaving half remaining. Similarly, for element B, the half-life is 50 years, which means that after 50 years, half of the initial number of atoms will have decayed, leaving half remaining. After 200 years, we can calculate the remaining amount of each element using the following formula: Remaining amount = Initial amount * (1/2)^(time/half-life) For element A, we have: Remaining amount of A = Initial amount of A * (1/2)^(200/100) = (1/2)^2 * Initial amount of A = (1/4) * Initial amount of A For element B, we have: Remaining amount of B = Initial amount of B * (1/2)^(200/50) = (1/2)^4 * Initial amount of B = (1/16) * Initial amount of B Since the initial amounts of A and B are equal, we can compare their remaining amounts by simply dividing one by the other: Remaining amount of A / Remaining amount of B = [(1/4) * Initial amount of A] / [(1/16) * Initial amount of B] = (1/4) / (1/16) = 4 Therefore, the ratio of the number of remaining atoms of A to that of B after 200 years is 4:1. Option A is the correct answer.