Question

The half-life of a radioactive nuclide is 100 hours. The fraction of original activity that will remain after 150 hours would be

• A. \(\frac{2}{3\sqrt2}\)
• B. \(\frac{1}{2}\)
• C. \(\frac{1}{2\sqrt2}\)
• D. \(\frac{2}{3}\)

Answer 1

The Correct Option is C

To solve the problem of determining the fraction of original activity of a radioactive nuclide that remains after 150 hours given its half-life is 100 hours, we will use the formula for radioactive decay:

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

1. Here,
   • \(N\) is the remaining quantity of the nuclide.
   • \(N_0\) is the initial quantity of the nuclide.
   • \(t\) is the time elapsed (150 hours).
   • \(T_{1/2}\) is the half-life (100 hours).
2. Substitute the known values into the decay formula:

\(N = N_0 \left(\frac{1}{2}\right)^{\frac{150}{100}}\)

1. Simplify the exponent:

\(N = N_0 \left(\frac{1}{2}\right)^{1.5}\)

1. Calculate the value:

\(\left(\frac{1}{2}\right)^{1.5} = \frac{1}{\sqrt{2^3}} = \frac{1}{\sqrt{8}} = \frac{1}{2\sqrt{2}}\)

Therefore, after 150 hours, the fraction of the original activity that remains is \(\frac{1}{2\sqrt{2}}\).

The correct answer is \(\frac{1}{2\sqrt{2}}\).