To determine the amount of a radioactive substance that has decayed in a given time, we need to understand the concept of "half-life." The half-life of a radioactive substance is the time required for half of the sample to decay.
In this case, the half-life of the substance is \(10\,\text{min}\). We are asked to find the amount of substance decayed in \(40\,\text{min}\).
First, calculate how many half-lives have passed in \(40\,\text{min}\):
\(\text{Number of half-lives} = \frac{40\,\text{min}}{10\,\text{min}} = 4\)
After each half-life, the amount of remaining substance is halved. Initially, we start with the full amount \(N_0\). After one half-life, the remaining amount is \(\frac{N_0}{2}\), after two half-lives it's \(\frac{N_0}{4}\), and so on.
Following this sequence for four half-lives:
So, after 4 half-lives, the remaining substance is \(\frac{N_0}{16}\).
The amount that has decayed is:
\(\text{Decayed amount} = N_0 - \frac{N_0}{16} = N_0 \left(1 - \frac{1}{16}\right) = N_0 \cdot \frac{15}{16}\)
The percentage of the decayed substance can be calculated as:
\(\text{Decayed percentage} = \left(\frac{15}{16}\right) \times 100\% = 93.75\%\)
Thus, the percentage of substance decayed in \(40\,\text{min}\) is \(93.75\%\).
Therefore, the correct answer is "None of these" as \(93.75\%\) does not match any of the given options.
The decay constant for a radioactive nuclide is \(1.5 × 10^{−5}s^{−1}\). Atomic weight of the substance is 60 g mole−1. (\(N_A = 6×10^{23}\)). The activity of 1.0 µg of the substance is _____\(×10^{10}\) Bq.