Question

The half-life of a radioactive isotope is 3 h. If the initial mass of the isotope was 300 g, the mass which remained undecayed after 18 h would be

• A. 4.68 g
• B. 2.34g
• C. 1.17 g
• D. 9.36 g

Answer 1

The Correct Option is A

This problem involves the concept of radioactive decay, specifically the calculation based on the half-life of a substance. Let's solve this step by step.

1. Understanding Half-Life: The half-life of a radioactive isotope is the time required for half of the original quantity of the substance to decay. For this isotope, the half-life is given as 3 hours.
2. Formula for Remaining Mass: The mass of a substance remaining undecayed after a certain time can be calculated using the formula: M = M_0 \left(\frac{1}{2}\right)^{\frac{t}{t_{\frac{1}{2}}}} where M is the remaining mass, M_0 is the initial mass, t is the time elapsed, and t_{\frac{1}{2}} is the half-life.
3. Substitute Known Values:
   • Initial mass, M_0 = 300 \, \text{g}
   • Time elapsed, t = 18 \, \text{h}
   • Half-life, t_{\frac{1}{2}} = 3 \, \text{h}
   Plug these values into the formula: M = 300 \left(\frac{1}{2}\right)^{\frac{18}{3}}
4. Calculate the Exponent: Since \frac{18}{3} = 6, the formula becomes: M = 300 \left(\frac{1}{2}\right)^{6}
5. Compute the Power: Calculate \left(\frac{1}{2}\right)^{6} = \frac{1}{64}
6. Calculate the Remaining Mass: M = 300 \times \frac{1}{64} = \frac{300}{64} = 4.68 \, \text{g}

Conclusion: The mass of the isotope that remains undecayed after 18 hours is 4.68 g. Thus, the correct answer is 4.68 g.