The half-life of a radioactive isotope s th ee hours. If the initial mass of the isotope were 256 g, the mass of it remaining undecayed after 18 hours would be

To determine the remaining mass of a radioactive isotope after a certain period, we can use the concept of half-life. The half-life is the time it takes for half of the radioactive substance to decay. In your case, the half-life is three hours, and you want to find out how much of the initial 256 grams remains after 18 hours.

Calculating the Number of Half-Lives

First, we need to figure out how many half-lives fit into the 18-hour period. Since each half-life is three hours, we can calculate the number of half-lives by dividing the total time by the half-life duration:

• Number of half-lives = Total time / Half-life duration
• Number of half-lives = 18 hours / 3 hours = 6

Understanding the Decay Process

Each half-life reduces the remaining mass of the isotope by half. Starting with 256 grams, we can see how the mass decreases over each half-life:

• After 1 half-life (3 hours): 256 g / 2 = 128 g
• After 2 half-lives (6 hours): 128 g / 2 = 64 g
• After 3 half-lives (9 hours): 64 g / 2 = 32 g
• After 4 half-lives (12 hours): 32 g / 2 = 16 g
• After 5 half-lives (15 hours): 16 g / 2 = 8 g
• After 6 half-lives (18 hours): 8 g / 2 = 4 g

Final Calculation

After completing all six half-lives, we find that the remaining mass of the radioactive isotope after 18 hours is 4 grams. This systematic reduction illustrates how radioactive decay works over time, emphasizing the exponential nature of the process.

Real-World Application

This principle is not just theoretical; it has practical applications in fields like archaeology (carbon dating), medicine (radiotherapy), and nuclear energy. Understanding half-lives helps scientists and professionals make informed decisions based on the stability and decay of isotopes.

In summary, after 18 hours, the mass of the radioactive isotope remaining is 4 grams. This example highlights the predictable nature of radioactive decay and the importance of half-life in understanding how substances change over time.