Half-Life of Radium

Radium is a radioactive element that undergoes decay over time. The half-life of radium is the amount of time it takes for half of the original amount of radium to decay. This means that after one half-life, half of the radium will have decayed, and after two half-lives, only one quarter of the original amount will remain.

Calculating the Remaining Amount

In this problem, we are given that the half-life of radium is about 1600 years, and we want to know how much of the original 100 g of radium will remain unchanged after a certain amount of time. We can use the following formula to calculate the remaining amount:

N = N0(1/2)t/T

Where:
N = the remaining amount of radium
N0 = the original amount of radium (100 g in this case)
t = the time elapsed
T = the half-life of radium

Using this formula, we can solve for the remaining amount of radium after a certain amount of time. For example, after one half-life (1600 years), we can calculate:

N = 100(1/2)1600/1600 = 50 g

So after 1600 years, only 50 g of the original 100 g of radium will remain. Similarly, after two half-lives (3200 years), we can calculate:

N = 100(1/2)3200/1600 = 25 g

This means that after 3200 years, only 25 g of the original 100 g of radium will remain.

Solving the Problem

Now we can use this formula to solve the problem given in the question. We are asked to find how much of the original 100 g of radium will remain unchanged after a certain amount of time, and we are given that 25 g will remain unchanged. We can set up the equation as follows:

25 = 100(1/2)t/1600

Solving for t, we get:

t = 1600 log2(4) = 1600(2) = 3200 years

So the answer is option A, 3200 years. After 3200 years, only 25 g of the original 100 g of radium will remain unchanged.