Question

A radioactive sample decays 7/8 times its original quantity in 15 minutes. The half-life of the sample is

• A. 5 min
• B. 7.5 min
• C. 15 min
• D. 30 min

Answer 1

The Correct Option is A

To determine the half-life of a radioactive sample where it decays to \frac{7}{8} of its original quantity in 15 minutes, follow these step-by-step instructions:

1. Given that \frac{7}{8} of the original sample decays, this implies that \frac{1}{8} of the sample remains after 15 minutes.

2. The formula for radioactive decay is given by:

   N = N_0 \left(\frac{1}{2}\right)^{\frac{t}{T_{\text{half}}}}

   where:

   • N is the remaining quantity of the substance.
   • N_0 is the original quantity of the substance.
   • t is the time elapsed.
   • T_{\text{half}} is the half-life of the substance.

3. Substitute the known values into the equation:

   \frac{1}{8}N_0 = N_0 \left(\frac{1}{2}\right)^{\frac{15}{T_{\text{half}}}}

   After canceling N_0 from both sides, the equation becomes:

   \frac{1}{8} = \left(\frac{1}{2}\right)^{\frac{15}{T_{\text{half}}}}

4. Express \frac{1}{8} as a power of \frac{1}{2}:

   \frac{1}{8} = \left(\frac{1}{2}\right)^3

5. Equate the exponents since the bases are the same:

   \frac{15}{T_{\text{half}}} = 3

6. Solve for T_{\text{half}}:

   T_{\text{half}} = \frac{15}{3} = 5 \text{ min}

Therefore, the correct answer is that the half-life of the sample is 5 minutes.